commit 50f7b1159e2b1e49c0bc8b37c470d0f9eb349d68 Author: geoffrey Date: Sun Jan 11 09:19:22 2026 +0100 First commit diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..d271b56 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +__pycache__/ +build/** +Cryptotools.egg-info/** +dist/** +**.swp diff --git a/Cryptotools/Cryptotools.egg-info/PKG-INFO b/Cryptotools/Cryptotools.egg-info/PKG-INFO new file mode 100644 index 0000000..1f32233 --- /dev/null +++ b/Cryptotools/Cryptotools.egg-info/PKG-INFO @@ -0,0 +1,6 @@ +Metadata-Version: 2.1 +Name: Cryptotools +Version: 1.0 +Summary: Cryptography library +Author: Geoffrey Bucchino +Author-email: contact@bucchino.org diff --git a/Cryptotools/Cryptotools.egg-info/SOURCES.txt b/Cryptotools/Cryptotools.egg-info/SOURCES.txt new file mode 100644 index 0000000..9c3a71f --- /dev/null +++ b/Cryptotools/Cryptotools.egg-info/SOURCES.txt @@ -0,0 +1,13 @@ +README.md +setup.py +Cryptotools/Cryptotools.egg-info/PKG-INFO +Cryptotools/Cryptotools.egg-info/SOURCES.txt +Cryptotools/Cryptotools.egg-info/dependency_links.txt +Cryptotools/Cryptotools.egg-info/top_level.txt +Cryptotools/Numbers/__init__.py +Cryptotools/Numbers/carmi.py +Cryptotools/Numbers/numbers.py +Cryptotools/Numbers/primeNumber.py +tests/test_carmi.py +tests/test_numbers.py +tests/test_phi.py \ No newline at end of file diff --git a/Cryptotools/Cryptotools.egg-info/dependency_links.txt b/Cryptotools/Cryptotools.egg-info/dependency_links.txt new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/Cryptotools/Cryptotools.egg-info/dependency_links.txt @@ -0,0 +1 @@ + diff --git a/Cryptotools/Cryptotools.egg-info/top_level.txt b/Cryptotools/Cryptotools.egg-info/top_level.txt new file mode 100644 index 0000000..c4a38ac --- /dev/null +++ b/Cryptotools/Cryptotools.egg-info/top_level.txt @@ -0,0 +1 @@ +Numbers diff --git a/Cryptotools/Encryptions/RSA.py b/Cryptotools/Encryptions/RSA.py new file mode 100644 index 0000000..5e1fdd3 --- /dev/null +++ b/Cryptotools/Encryptions/RSA.py @@ -0,0 +1,174 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.coprime import phi +from Cryptotools.Numbers.carmi import carmichael_lambda +from Cryptotools.Utils.utils import gcd +from Cryptotools.Numbers.primeNumber import getPrimeNumber +from sys import getsizeof + +class RSAKey: + """ + This class store the RSA key with the modulus associated + The key is a tuple of the key and the modulus n + + Attributes: + key: It's the exponent key, can be public or private + modulus: It's the public modulus + length: It's the key length + """ + def __init__(self, key, modulus, length): + """ + Contain the RSA Key. An object of RSAKey can be a public key or a private key + + Args: + key (Integer): it's the exponent of the key + modulus (Integer): it's the public modulus of the key + length (Integer): length of the key + """ + + self._key = key + self._modulus = modulus + self._length = length + + @property + def key(self): + return self._key + + @property + def modulus(self): + return self._modulus + + @property + def length(self): + return self.length + +class RSA: + """ + This class generate public key based on RSA algorithm + + Attributes: + p: it's the prime number for generating the modulus + q: it's the second prime number for the modulus + public: Object of the RSAKey which is the public key + private: Object of the RSAKey for the private key + """ + + def __init__(self): + """ + Build a RSA Key + """ + self._p = None + self._q = None + self._public = None + self._private = None + + def generateKeys(self, size=512): + """ + This function generate both public and private keys + Args: + size: It's the size of the key and must be multiple of 64 + """ + # p and q must be coprime + self._p = getPrimeNumber(size) + self._q = getPrimeNumber(size) + + # compute n = pq + n = self._p * self._q + + phin = (self._p - 1) * (self._q - 1) + + # e must be coprime with phi(n) + # According to the FIPS 186-5, the public key exponent must be odd + # and the minimum size is 65536 (Cf. Section 5.4 PKCS #1) + # https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-5.pdf + e = 65535 # Works + for i in range(2, phin - 1): + if gcd(phin, e) == 1: + break + e += 1 + # print(gcd(phin, e)) + self._public = RSAKey(e, n, getsizeof(e)) + + # d is the reverse modulo of phi(n) + # d = self._inverseModular(e, phin) + d = pow(e, -1, phin) # Works in python 3.8 + self._private = RSAKey(d, n, getsizeof(d)) + + @property + def e(self): + return self._public.key + + @property + def d(self): + return self._private.key + + @property + def n(self): + return self._public.modulus + + @property + def p(self): + return self._p + + @property + def q(self): + return self._q + + def encrypt(self, data) -> list: + """ + This function return a list of data encrypted with the public key + + Args: + data (str): it's the plaintext which need to be encrypted + + Returns: + return a list of the data encrypted, each entries contains the value encoded + """ + dataEncoded = self._str2bin(data) + return list( + pow(int(x), self._public.key, self._public.modulus) + for x in dataEncoded + ) + + def decrypt(self, data) -> list: + """ + This function return a list decrypted with the private key + + Args: + data (str): It's the encrypted data which need to be decrypted + + Returns: + Return the list of data uncrypted into plaintext + """ + decrypted = list() + for x in data: + d = pow(x, self._private.key, self._private.modulus) + decrypted.append(chr(d)) + return ''.join(decrypted) + + def _str2bin(self, data) -> list: + """ + This function convert a string into the unicode value + + Args: + data: the string which need to be converted + + Returns: + Return a list of unicode values of data + """ + return list(ord(x) for x in data) + + def _inverseModular(self, a, n): + """ + This function compute the modular inverse for finding d, the decryption key + + Args: + a (Integer): the base of the exponent + n (Integer): the modulus + """ + for b in range(1, n): + #if pow(a, b, n) == 1: + if (a * b) % n == 1: + inv = b + break + return inv diff --git a/Cryptotools/Encryptions/__init__.py b/Cryptotools/Encryptions/__init__.py new file mode 100644 index 0000000..e5a0d9b --- /dev/null +++ b/Cryptotools/Encryptions/__init__.py @@ -0,0 +1 @@ +#!/usr/bin/env python3 diff --git a/Cryptotools/Groups/__init__.py b/Cryptotools/Groups/__init__.py new file mode 100644 index 0000000..e5a0d9b --- /dev/null +++ b/Cryptotools/Groups/__init__.py @@ -0,0 +1 @@ +#!/usr/bin/env python3 diff --git a/Cryptotools/Groups/cyclic.py b/Cryptotools/Groups/cyclic.py new file mode 100644 index 0000000..fe63ce4 --- /dev/null +++ b/Cryptotools/Groups/cyclic.py @@ -0,0 +1,91 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.group import Group + + +class Cyclic(Group): + """ + This object contain a list of the Group for a cyclic group. This class find all generator of the group. + This class is inherited from Group object + + Attributes: + G (list): list of all elements in the group + n (Integer): it's the value where the group has been generated + operation (Function): it's the operation generating the group + generators (list): contain all generators of the group + generatorChecked (boolean): Check if generators has been found + """ + def __init__(self, G:list, n, ope): + super().__init__(n, G, ope) # Call the Group's constructor + self._G = G + self._n = n + self._operation = ope + self._generators = list() + self._generatorChecked = False + + def generator(self): + """ + This function find all generators in the group G + """ + + index = 1 + G = sorted(self._g) + for g in range(2, self._n): + z = list() + for entry in range(1, self._n): + res = self._operation(g, index, self._n) + if res not in z: + z.append(res) + index = index + 1 + + # We check if that match with G + # If yes, we find a generator + if sorted(z) == G: + self._generators.append(g) + self._generatorChecked = True + + def getPrimitiveRoot(self): + """ + This function return the primitive root modulo of n + + Returns: + Return the primitive root of the group. None if no primitive has been found + """ + + index = 1 + G = sorted(self._g) + for g in range(2, self._n): + z = list() + for entry in range(1, self._n): + res = self._operation(g, index, self._n) + if res not in z: + z.append(res) + index += 1 + + # If the group is the same has G, we found a generator + if sorted(z) == G: + return g + return None + + def getGenerators(self) -> list: + """ + This function return all generators of that group + + Returns: + Return the list of generators found. The function generators() must be called before to call this function + """ + if not self._generatorChecked: + self.generator() + self._generatorChecked = True + return self._generators + + def isCyclic(self) -> bool: + """ + Check if the group is a cyclic group, means we have at least one generator + + Returns: + REturn a boolean, False if the group is not Cyclic otherwise return True + """ + if len(self.getGenerators()) == 0: + return False + return True diff --git a/Cryptotools/Groups/galois.py b/Cryptotools/Groups/galois.py new file mode 100644 index 0000000..741ec2a --- /dev/null +++ b/Cryptotools/Groups/galois.py @@ -0,0 +1,227 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.group import Group + + +class Galois: + """ + This class contain the Galois Field (Finite Field) + + + + Attributes: + q (Integer): it's the number of the GF + operation (Function): Function for generating the group + identityAdd (Integer): it's the identity element in the GF(n) for the addition operation + identityMul (Integer): it's the identity element in the GF(n) for the multiplicative operation + """ + def __init__(self, q, operation): + self._q = q + self._operation = operation + self._identityAdd = 0 + self._identityMul = 1 + self._F = [x for x in range(q)] + self._add = [[0 for x in range(q)] for y in range(q)] + + # TODO: May do we do a deep copy between all groups ? + self._div = [[0 for x in range(q)] for y in range(q)] + self._mul = [[0 for x in range(q)] for y in range(q)] + self._sub = [[0 for x in range(q)] for y in range(q)] + self._primitiveRoot = list() + + def primitiveRoot(self): + """ + In this function, we going to find the primitive root modulo n of the galois field + + Returns: + Return the list of primitive root of the GF(q) + """ + for x in range(2, self._q): + z = list() + for entry in range(1, self._q): + res = self._operation(x, entry, self._q) + if res not in z: + z.append(res) + + if self.isPrimitiveRoot(z, self._q - 1): + if x not in self._primitiveRoot: # It's dirty, need to find why we have duplicate entry + self._primitiveRoot.append(x) + return z + + def getPrimitiveRoot(self): + """ + Return the list of primitives root + + Returns: + Return the primitive root + """ + return self._primitiveRoot + + def isPrimitiveRoot(self, z, length): + """ + Check if z is a primitive root + + Args: + z (list): check if z is a primitive root + length (Integer): Length of the GF(q) + """ + if len(z) == length: + return True + return False + + def add(self): + """ + This function do the operation + on the Galois Field + + Returns: + Return a list of the group with the addition operation + """ + for x in range(0, self._q): + for y in range(0, self._q): + self._add[x][y] = (x + y) % self._q + return self._add + + def _inverseModular(self, a, n): + """ + This function find the reverse modular of a by n + + Returns: + Return the reverse modular + """ + for b in range(1, n): + if (a * b) % n == 1: + inv = b + break + return inv + + def div(self): + """ + This function do the operation / on the Galois Field + + Returns: + Return a list of the group with the division operation + """ + for x in range(1, self._q): + for y in range(1, self._q): + inv = self._inverseModular(y, self._q) + + self._div[x][y] = (x * inv) % self._q + return self._div + + def mul(self): + """ + This function do the operation * on the Galois Field + + Returns: + Return a list of the group with the multiplication operation + """ + for x in range(0, self._q): + for y in range(0, self._q): + self._mul[x][y] = (x * y) % self._q + return self._mul + + def sub(self): + """ + This function do the operation - on the Galois Field + + Returns: + Return a list of the group with the subtraction operation + """ + for x in range(0, self._q): + for y in range(0, self._q): + self._sub[x][y] = (x - y) % self._q + return self._sub + + def check_closure_law(self, arithmetic): + """ + This function check the closure law. + By definition, every element in the GF is an abelian group, which respect the closure law: for a and b belongs to G, a + b belongs to G, the operation is a binary operation + + Args: + Arithmetics (str): contain the operation to be made, must be '+', '*', '/' or /-' + """ + if arithmetic not in ['+', '*', '/', '-']: + raise Exception("The arithmetic need to be '+', '*', '/' or '-'") + + if arithmetic == '+': + G = self._add + elif arithmetic == '*': + G = self._mul + elif arithmetic == '/': + G = self._div + else: + G = self._sub + + start = 0 + """ + In case of multiplicative, we bypass the first line, because all elements are zero, otherwise the test fail + """ + if arithmetic == '*' or arithmetic == '/': + start = 1 + + isClosure = True + for x in range(start, self._q): + gr = Group(self._q, G[x], self._operation) + if not gr.closure(): + isClosure = False + del gr + + if isClosure: + print(f"The group {arithmetic} respect closure law") + else: + print(f"The group {arithmetic} does not respect closure law") + + def check_identity_add(self): + """ + This function check the identity element and must satisfy this condition: $a + 0 = a$ for each element in the GF(n) + In Group Theory, an identity element is an element in the group which do not change the value every element in the group + + Returns: + + """ + for x in self._F: + if not self._identityAdd + x == x: + raise Exception( + f"The identity element {self._identityAdd} "\ + "do not satisfy $a + element = a$" + ) + + def check_identity_mul(self): + """ + This function check the identity element and must satisfy this condition: $a * 1 = a$ for each element in the GF(n) + In Group Theory, an identity element is an element in the group which do not change the value every element in the group + + Returns: + + """ + for x in self._F: + if not self._identityMul * x == x: + raise Exception( + f"The identity element {self._identityAdd} "\ + "do not satisfy $a * element = a$" + ) + + def printMatrice(self, m): + """ + This function print the GF(m) + + Args: + m (list): Matrix of the GF + """ + header = str() + header = " " + for x in range(0, self._q): + header += f"{x} " + + header += "\n--|" + "-" * (len(header)- 3) +"\n" + + s = str() + + for x in range(0, self._q): + s += f"{x} | " + for y in range(0, self._q): + s += f"{m[x][y]} " + s += "\n" + + s = header + s + print(s) diff --git a/Cryptotools/Groups/group.py b/Cryptotools/Groups/group.py new file mode 100644 index 0000000..b96484d --- /dev/null +++ b/Cryptotools/Groups/group.py @@ -0,0 +1,124 @@ +#!/usr/bin/env python3 + +class Group: + """ + This class generate a group self._g based on the operation (here denoted +) + with the function ope: (a ** b) % n + + In group theory, any group has an identity element (e), which with the binary operation, do not change the value and must satisfy the condition: $a + e = 0$ + + Attributes: + n (Integer): It's the G_n elements in the Group + g (List): The set of the Group + operation (Function): Function for generating the Group + identity (Integer): The identity of the group. + reverse (Dict): For each elements of the Group of n, we have the reverse value + """ + def __init__(self, n, g, ope): + self._n = n + self._g = g + self._operation = ope + self._identity = 0 + self._reverse = dict() + + def getG(self) -> list(): + """ + This function return the Group + + Returns: + List of all elements in the Group + """ + return self._g + + def closure(self) -> bool: + """ + Check the closure law + In a group, each element a, b belongs to G, such as a + b belongs to G + + Returns: + Return a Boolean if the closure law is respected. True if yes otherwise it's False + """ + for e1 in self._g: + for e2 in self._g: + res = self._operation(e1, e2, self._n) + if not res in self._g: + # raise Exception(f"{res} not in g. g is not a group") + return False + return True + + def associative(self) -> bool: + """ + Check the associative law. + In a group, for any a, b and c belongs to G, + they must respect this condition: (a + b) + c = a + (a + b) + + Returns: + Return a boolean if the Associative law is respected. True if yes otherwise it's False + """ + a = self._g[0] + b = self._g[1] + c = self._g[2] + + res_ope = self._operation(a, b, self._n) + res1 = self._operation(res_ope, c, self._n) + + res_ope = self._operation(b, c, self._n) + res2 = self._operation(a, res_ope, self._n) + if res1 != res2: + # raise Exception(f"{res1} is different from {res2}. g is not a group") + return False + return True + + def identity(self) -> bool: + """ + Check the identity law. + In a group, an identity element exist and must be uniq + + Returns: + Return a Boolean if the identity elements has been found. True if found otherwise it's False + """ + for a in self._g: + for b in self._g: + if not self._operation(a, b, self._n) == b: + break + + self._identity = a + return True + return False + + def getIdentity(self) -> int: + """ + Return the identity element. The function identitu() must be called before. + + Returns: + Return the identity element if it has been found + """ + return self._identity + + def reverse(self) -> bool: + """ + Check the inverse law + In a group, for each element belongs to G + they must have an inverse a ^ (-1) = e (identity) + + Returns: + Return a Boolean if the all elements ha a reverse element. True if yes otherwise it's False + """ + reverse = False + for a in self._g: + for b in self._g: + if self._operation(a, b, self._n) == self._identity: + self._reverse[a] = b + reverse = True + break + return reverse + + def getReverses(self) -> dict: + """ + This function return the dictionary of all reverses elements. The key is the element in G and the value is the reverse element + + Returns: + Return the reverse dictionary + + """ + return self._reverse diff --git a/Cryptotools/Numbers/__init__.py b/Cryptotools/Numbers/__init__.py new file mode 100644 index 0000000..e5a0d9b --- /dev/null +++ b/Cryptotools/Numbers/__init__.py @@ -0,0 +1 @@ +#!/usr/bin/env python3 diff --git a/Cryptotools/Numbers/__pycache__/__init__.cpython-310.pyc b/Cryptotools/Numbers/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000..4dd26cb Binary files /dev/null and b/Cryptotools/Numbers/__pycache__/__init__.cpython-310.pyc differ diff --git a/Cryptotools/Numbers/__pycache__/__init__.cpython-39.pyc b/Cryptotools/Numbers/__pycache__/__init__.cpython-39.pyc new file mode 100644 index 0000000..3a1d189 Binary files /dev/null and b/Cryptotools/Numbers/__pycache__/__init__.cpython-39.pyc differ diff --git a/Cryptotools/Numbers/__pycache__/carmi.cpython-310.pyc b/Cryptotools/Numbers/__pycache__/carmi.cpython-310.pyc new file mode 100644 index 0000000..b046345 Binary files /dev/null and b/Cryptotools/Numbers/__pycache__/carmi.cpython-310.pyc differ diff --git a/Cryptotools/Numbers/__pycache__/carmi.cpython-39.pyc b/Cryptotools/Numbers/__pycache__/carmi.cpython-39.pyc new file mode 100644 index 0000000..c5b283b Binary files /dev/null and b/Cryptotools/Numbers/__pycache__/carmi.cpython-39.pyc differ diff --git a/Cryptotools/Numbers/carmi.py b/Cryptotools/Numbers/carmi.py new file mode 100644 index 0000000..71cfde0 --- /dev/null +++ b/Cryptotools/Numbers/carmi.py @@ -0,0 +1,177 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import gcd, isPrimeNumber +from Cryptotools.lcm import lcm +from Cryptotools.Numbers.coprime import phi +from Cryptotools.Utils.utils import gcd +from sympy import factorint +from functools import reduce + + +def carmichael_lambda(n: int) -> int: + """ + This function is a carmichael lambda for identifying the smallest integer m + and saisfy this condition: a ** m congrue 1 modulo n + This function is relatively closed with the euler's totient (phi of n) + + Args: + n (Integer): found the smallest integer of n + + Returns: + REturn the smallest integer of n + """ + + # First, factorizing n number + # factorint return a list of prime factors of n + # Base on that theorem: + # https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic + factorsN = factorint(n) + + """ + The result of the factorint, we have the following: + n = p ** e1, p ** e2, ..., p ** ek + """ + + factors = list() + + for p, e in factorsN.items(): + # Base on the theorem of arithmetic, p is always prime + if isPrimeNumber(p): + if p == 2 and e >= 3: + factors.append(phi(pow(p, e)) // 2) + elif p >= 2 and e > 0: + factors.append(phi(pow(p, e))) + elif p > 2: + factors.append(phi(pow(p, e))) + + # Now, we can compute lcm + result = reduce(lcm, factors) + + return result + +def carmichael_lambda2(n): + """ + Deprecated function + """ + # Get all coprimes with n + coprimes = list() + for a in range(1, n): + if gcd(a, n) == 1: + coprimes.append(a) + + # Need to find the smallest value + """ + Need to find the smallest value m. Must to satisfy this condition: + a ** m congrus 1 mod n + """ + for m in range(1, n): + print(f"{m} {a ** m % n}") + + print(coprimes) + +def carmi_numbers(n): + """ + This function get all GCD of the number if these number are prime or not + + Args: + n (Integer): Iterate to n elements + + Returns: + Return a list of prime numbers + """ + # https://kconrad.math.uconn.edu/blurbs/ugradnumthy/carmichaelkorselt.pdf + primes = list() + for i in range(2, n - 1): + if gcd(n, i) == i: + if isPrimeNumber(i): + primes.append(i) + + return primes + +def is_carmichael(n, primes): + """ + This function check if the n number is a carmichael number. The arguments primes at least 3 prime numbers + As it's said in the Carmichael theorem, a carmichael number has three factors and they are all primes: https://en.wikipedia.org/wiki/Carmichael_number + With the Korselt's criterion, to check if the number is a carmichael number p - 1 / n - 1. + + Args: + n (Integer): the carmichael number + primes (list): list of prime numbers + + Returns: + Boolean of the carmichael's number result + """ + r = True + n = n - 1 + + if len(primes) < 3: + return False + + # Korselt's criterion + for p in primes: + p = p - 1 + if n % p != 0: + r = False + break + return r + +def is_carmi_number(n) -> bool: + """ + This function check if the number n is a carmichael number (pseudoprime) + + Args: + n (Integer): Iterate to n elements + + Returns: + Return a Boolean if n is a carmichael number or not. True if yes + """ + j = 0 + for i in range(n): + if i ** n % n == i: + j += 1 + if j == n: + return True + return False + +def generate_carmi_numbers(nbr) -> list: + """ + This function generate carmichael numbers, they are called pseudoprimes + For instance: a = 545, n = 561 + gcd(a, n) = 1 + pow(a, n - 1) % n = 1 + + Args: + nbr (Integer): Find all pseudoprimes until nbr is achieves + + Returns: + Return the list of pseudoprimes + """ + carmi = list() + + count = 0 + n = 2 + while count < nbr: + # First, we need to find a, coprime with n + for a in range(1, n): + # We check if n is coprime with a + # All integers must be coprime with n, a belong to n + if gcd(a, n) == 1: + # Check if is a carmichael number + # Fermat's little theorem, a ** (n - 1) % n, must be congruent to 1 + if pow(a, (n - 1)) % n == 1: + #if (a ** (n - 1)) % n == 1: + #print(f"{gcd(a, n)} {a} {n}") + if n not in carmi: + count += 1 + carmi.append(n) + else: + print(f"{gcd(a, n)} {a} {n}") + + break + n += 1 + print(carmi) + #for cop in range(2, n): + # if gcd(cop, n) == 1: + # coprimes.append(cop) + + return carmi diff --git a/Cryptotools/Numbers/coprime.py b/Cryptotools/Numbers/coprime.py new file mode 100644 index 0000000..6bf355e --- /dev/null +++ b/Cryptotools/Numbers/coprime.py @@ -0,0 +1,22 @@ +#!/usr/bin/env python3 + +from Cryptotools.Utils.utils import gcd + + +# https://oeis.org/A000010 +def phi(n): + """ + This function count all phi(n) + + Args: + n (integer): it's the phi(n) value + + Returns: + Return the phi(n) + """ + y = 1 + for i in range(2, n): + if gcd(n, i) == 1: + y += 1 + return y + diff --git a/Cryptotools/Numbers/factorization.py b/Cryptotools/Numbers/factorization.py new file mode 100644 index 0000000..9211df0 --- /dev/null +++ b/Cryptotools/Numbers/factorization.py @@ -0,0 +1,56 @@ +#!/usr/bin/env python3 + +from random import randint +from Cryptotools.Numbers.primeNumber import _millerRabinTest, isPrimeNumber +from Cryptotools.Utils.utils import gcd +from Cryptotools.lcm import lcm +from math import floor, log + + +def pollard_p_minus_1(n, B=None) -> int: + """ + This function factorize the number n into factor based on the Pollard's p - 1 algorithm + + The first is to choose B, which is a positive integer and B < n + In the second step, we need to define $M=\prod _{primes~q\leq B}q^{\lfloor \log_{q}{B}\rfloor }$ + + Args: + n (Integer): The number n to factorize + + Returns: + Return the factorize of the number n or None + """ + + if B is None: + B = randint(1, n - 1) + # B = randint(1, n - 1) + + #### We define M + # Select all prime numbers 1 < B + q = list() + for x in range(2, B): + # with the _millerRabinTest, the program crash, because the randint() is empty + # because, both min value and max value are 2 + # if _millerRabinTest(x): + if isPrimeNumber(x): + q.append(x) + + M = 1 + for prime in q: + e = floor(log(B, prime)) + # print(prime, e) + M *= pow(prime, e) + + # We choose a and coprime with n + a = 2 + while True: + if gcd(a, n) == 1: + break + a += 1 + + # We can compute g = gcd((a ** M) - 1, n) + g = gcd(pow(a, M, n) - 1, n) + if g > 1: + return g + + return None diff --git a/Cryptotools/Numbers/numbers.py b/Cryptotools/Numbers/numbers.py new file mode 100644 index 0000000..3295de3 --- /dev/null +++ b/Cryptotools/Numbers/numbers.py @@ -0,0 +1,19 @@ +#!/usr/bin/env python3 + + +def fibonacci(n) -> list: + """ + This function generate the list of Fibonacci + + Args: + n (integer): it's maximum value of Fibonacci sequence + + Returns: + Return a list of n element in the Fibonacci sequence + """ + fibo = [0, 1] + fibo.append(fibo[0] + fibo[1]) + for i in range(2, n): + fibo.append(fibo[i - 1] + fibo[i]) + return fibo + diff --git a/Cryptotools/Numbers/primeNumber.py b/Cryptotools/Numbers/primeNumber.py new file mode 100644 index 0000000..4622872 --- /dev/null +++ b/Cryptotools/Numbers/primeNumber.py @@ -0,0 +1,288 @@ +#!/usr/bin/env python3 + +from random import randint, seed, getrandbits +from math import log2 +from Cryptotools.Utils.utils import gcd + + +def isPrimeNumber(p): + """ + Check if the number p is a prime number or not. The function iterate until p is achieve. + + This function is not the efficient way to determine if p is prime or a composite. + + To identify if p is prime or not, the function check the result of the Euclidean Division has a remainder. If yes, it's means it's not a prime number + + Args: + p (Integer): number if possible prime or not + + Returns: + Return a boolean if the number p is a prime number or not. True if yes + """ + for i in range(2, p): + if p % i == 0: + return False + return True + +# https://people.csail.mit.edu/rivest/Rsapaper.pdf +# https://arxiv.org/pdf/1912.11546 +# https://link.springer.com/content/pdf/10.1007/3-540-44499-8_27.pdf +# https://link.springer.com/article/10.1007/BF00202269 +# https://www.geeksforgeeks.org/dsa/how-to-generate-large-prime-numbers-for-rsa-algorithm/ +# https://crypto.stackexchange.com/questions/20548/generation-of-strong-primes +# https://crypto.stackexchange.com/questions/71/how-can-i-generate-large-prime-numbers-for-rsa + +def getPrimeNumber(n, safe=True): + """ + This function generate a large prime number + based on "A method for Obtaining Digital Signatures + and Public-Key Cryptosystems" + Section B. How to Find Large Prime Numbers + https://dl.acm.org/doi/pdf/10.1145/359340.359342 + + Args: + n (Integer): The size of the prime number. Must be a multiple of 64 + safe (Boolean): When generating the prime number, the function must find a safe prime number, based on the Germain primes (2p + 1 is also a prime number) + + Returns: + Return the prime number + """ + if n % 64 != 0: + print("Must be multiple of 64") + return + # from sys import getsizeof + upper = getrandbits(n) + lower = upper >> 7 + r = randint(lower, upper) + + while r % 2 == 0: + r += 1 + + # Now, we are going to compute r as prime number + i = 100 + while 1: + # Check if it's a prime number + if _millerRabinTest(r): + break + + # TODO: it's dirty, need to change that + # i = int(log2(r)) + # i *= randint(2, 50) + + r += i + # print(f"{i} {r}") + i += 1 + + + # print(_fermatLittleTheorem(r)) + # print(getsizeof(r)) + return r + +def getSmallPrimeNumber(n) -> int: + """ + This function is deprecated + + Args: + n (Integer): Get the small prime number until n + + Returns: + Return the first prime number found + """ + is_prime = True + + while True: + for i in range(2, n): + # If n is divisible by i, it's not a prime, we break the loop + if n % i == 0: + is_prime = False + break + + if is_prime: + break + is_prime = True + n = n + 1 + + p = n + return p + +def get_prime_numbers(n) -> list: + """ + This function get all prime number of the n + + Args: + n (Integer): find all prime number until n is achieves + + Returns: + Return a list of prime numbers + """ + l = list() + for i in range(2, n): + if n % i == 0: + l.append(i) + return l + +def get_n_prime_numbers(n) -> list: + """ + This function return a list of n prime numbers + + Args: + n (Integer): the range, must be an integer + + Returns: + Return a list of n prime numbers + """ + l = list() + + count = 2 + index = 0 + while index < n: + is_prime = True + for x in range(2, count): + if count % x == 0: + is_prime = False + break + + if is_prime: + l.append(count) + index += 1 + + count += 1 + + return l + +def are_coprime(p1, p2): + """ + This function check if p1 and p2 are coprime + + Args: + p1 (list): list of prime numbers of the first number + p2 (list): list of prime number of the second number + + Returns: + Return a boolean result + """ + r = True + for entry in p1: + if entry in p2: + r = False + break + return r + +def sieveOfEratosthenes(n) -> list: + """ + This function build a list of prime number based on the Sieve of Erastosthenes + + Args: + n (Integer): Interate until n is achives + + Returns: + Return a list of all prime numbers to n + """ + if n < 1: + return list() + + eratost = dict() + for i in range(2, n): + eratost[i] = True + + for i in range(2, n): + if eratost[i]: + for j in range(i*i, n, i): + eratost[j] = False + + sieve = list() + for i in range(2, n): + if eratost[i]: + sieve.append(i) + + return sieve + +def _fermatLittleTheorem(n) -> bool: + """ + The Fermat's little theorem. if n is a prime number, any number from 0 to n- 1 is a multiple of n + + Args: + n (Integer): Check if n is prime or not + + Returns: + Return True if the number a is a multiple of n, otherwise it's False + """ + a = randint(1, n - 1) + # We compute a ** (n - 1) % n + if pow(a, n - 1, n) == 1: + return True + return False + +def _millerRabinTest(n) -> bool: + """ + This function execute a Miller Rabin test + For the algorithm, it's based on the pseudo-algo provided by this document: + * https://www.cs.cornell.edu/courses/cs4820/2010sp/handouts/MillerRabin.pdf + The Mille-Rabin test is an efficient way to determine if n is prime or not + + Args: + n (Integer): Check if n is a prime number + Returns: + Return a boolean. True for a prime otherwise, it's a composite number + + """ + if n < 2 or (n > 2 and n % 2 == 0): # If n is even, it's a composite + return False + + k = 0 + q = n - 1 + #while (q & 1) == 0: + # k += 1 + # q >>= 1 + while q % 2 == 0: + k += 1 + q //= 2 + + # We choose a: a < n - 1 + for _ in range(40): + a = randint(2, n - 1) + # We compute a ** q % n + x = pow(a, q, n) + + # If it's a composite number + if x == 1 or x == n - 1: + continue + + for _ in range(k): + z = pow(x, 2, n) + if z == 1 or z == n - 1: + return False + else: + return False + return True + +def sophieGermainPrime(p) -> bool: + """ + Check if the number p is a safe prime number: 2p + 1 is also a prime + + Args: + p (Integer): Possible prime number + + Returns: + Return True if p is a safe prime number, otherwise it's False + """ + pp = 2 * p + 1 + return _millerRabinTest(pp) + +def isSafePrime(n) -> bool: + """ + This function has not been implemented yet, but check if the number n is a safe prime number. This function will test different properties of the possible prime number n + + Args: + n (Integer): the prime number to check + + Returns: + Return a Boolean if the prime number n is safe or not. True if yes, otherwise it's False + """ + if n.bit_length() >= 256: + return True + + # Do Sophie Germain's test + + return False + diff --git a/Cryptotools/Utils/__init__.py b/Cryptotools/Utils/__init__.py new file mode 100644 index 0000000..e5a0d9b --- /dev/null +++ b/Cryptotools/Utils/__init__.py @@ -0,0 +1 @@ +#!/usr/bin/env python3 diff --git a/Cryptotools/Utils/number.py b/Cryptotools/Utils/number.py new file mode 100644 index 0000000..538e4bd --- /dev/null +++ b/Cryptotools/Utils/number.py @@ -0,0 +1,9 @@ +#!/usr/bin/env python3 + +COMPOSITE = 0 +PRIME = 1 + +class Number: + def __init__(self): + self._int = int() + self._kind = COMPOSITE diff --git a/Cryptotools/Utils/utils.py b/Cryptotools/Utils/utils.py new file mode 100644 index 0000000..9090efe --- /dev/null +++ b/Cryptotools/Utils/utils.py @@ -0,0 +1,34 @@ +#!/usr/bin/env python3 + +def gcd(a, b): + """ + This function calculate the GCD (Greatest Common Divisor of the number a of b + Args: + a (Integer): the number a + b (integer): the number b + + Return: + the GCD + """ + + if b == 0: + return a + return gcd(b, a%b) + +def egcd(a, b): + """ + This function compute the Extended Euclidean algorithm + https://user.eng.umd.edu/~danadach/Cryptography_20/ExtEuclAlg.pdf + https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm + + Args: + a (Integer): the number a + b (Integer): the number b + """ + if a == 0: + return b, 0, 1 + gcd, x1, y1 = egcd(b % a, a) + x = y1 - (b // a) * x1 + y = x1 + return gcd, x, y + diff --git a/Cryptotools/__init__.py b/Cryptotools/__init__.py new file mode 100644 index 0000000..9c3c3bd --- /dev/null +++ b/Cryptotools/__init__.py @@ -0,0 +1,3 @@ +#!/usr/bin/env python3 + +__all__ = ["Numbers", "Groups"] diff --git a/Cryptotools/lcm.py b/Cryptotools/lcm.py new file mode 100644 index 0000000..ebdcd7c --- /dev/null +++ b/Cryptotools/lcm.py @@ -0,0 +1,33 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import gcd + +def compute_multiple(a): + """ + This function compute the 20 first multiples of the number a + + Args: + a (Integer): the number + + Return: + list of multiple of a + """ + + r = list() + for i in range(1, 20): + r.append(a * i) + return r + +def lcm(a, b): + """ + This function get the LCM (Least Common Multiple) a of b + + Args: + a (Integer): the number + b (Integer): the number + + Return: + the LCM + """ + return (b // gcd(a, b)) * a + diff --git a/README.md b/README.md new file mode 100644 index 0000000..36c40cd --- /dev/null +++ b/README.md @@ -0,0 +1,31 @@ +# Introduction +This project contains the module *cryptotools* which can used for cryptography + +# Install the program +``` +$ virtualenv ~/venv/myenv +$ source ~/venv/myenv/bin/activate +$ python3 setup.py install +``` + +## Examples +In the directory *examples/*, you have different python files for using the module + +## Unitests +### Test phi +``` +$ python3 tests/test_phi.py -v +``` + +### Test Fibonacci +For testing Fibonacci sequence, I generate a list from OEIS wich contains 40 first numbers of Fibonacci. The `test_numbers.py` test the Fibonacci sequence generated from the functions `fibonacci` in `utils/numbers.py`: + +``` +$ python3 tests/test_numbers.py -v +``` + +# Build the doc + +``` +$ mkdocs build +``` diff --git a/docs/examples-rsa-keys.md b/docs/examples-rsa-keys.md new file mode 100644 index 0000000..0b66fb7 --- /dev/null +++ b/docs/examples-rsa-keys.md @@ -0,0 +1,27 @@ +# Generating RSA Keys +In the following section, we are going to see how to generate RSA Keys and how to encrypt data with these keys. + +``` +#!/usr/bin/env python3 + +from Cryptotools.Encryptions.RSA import RSA + + +rsa = RSA() +rsa.generateKeys(size=512) + +e = rsa.e +d = rsa.d +n = rsa.n + +s = "I am encrypted with RSA" +print(f"plaintext: {s}") +encrypted = rsa.encrypt(s) + +# Encrypt data +print(f"ciphertext: {encrypted}") + +# We decrypt +plaintext = rsa.decrypt(encrypted) +print(f"Plaintext: {plaintext}") +``` diff --git a/docs/group-theory.md b/docs/group-theory.md new file mode 100644 index 0000000..dc18237 --- /dev/null +++ b/docs/group-theory.md @@ -0,0 +1,9 @@ +# Group Theory +## Group +::: Cryptotools.Groups.group + +## Cyclic group +::: Cryptotools.Groups.cyclic + +## Galois Field (Finite Field) +::: Cryptotools.Groups.galois diff --git a/docs/index.md b/docs/index.md new file mode 100644 index 0000000..b575c8e --- /dev/null +++ b/docs/index.md @@ -0,0 +1,13 @@ +# Welcome to CryptoTools + +* [Introduction](/introduction) +* [Installation](/installation) +* Low-Level Cryptographic + * [Number Theory](/number-theory) + * [Group Theory](/group-theory) +* Public Keys: + * [RSA](/rsa) +* Examples: + * [Generating RSA keys](/examples-rsa-keys) + + diff --git a/docs/installation.md b/docs/installation.md new file mode 100644 index 0000000..adc2c0a --- /dev/null +++ b/docs/installation.md @@ -0,0 +1,18 @@ +# Installation +To install the project, you should install from my own Python repository but, you first need to deploy your virtual enviroment. + +## Installation from source +You can install from the source. Download the [project](https://gitea.bucchino.org/gbucchino/cryptotools.git) and install it in your virtual environment: + +``` +$ virtualenv ~/venv/cryptotools +$ source ~/venv/cryptotools/bin/activate +``` + +``` +$ git clone https://gitea.bucchino.org/gbucchino/cryptotools.git +$ cd cryptotools +$ python3 setup.py install +``` + +The installation is completed. You can know use Cryptotools package into your project. In the directory `examples` you may find some examples scripts diff --git a/docs/introduction.md b/docs/introduction.md new file mode 100644 index 0000000..29f95f8 --- /dev/null +++ b/docs/introduction.md @@ -0,0 +1,5 @@ +# CryptoTools +CryptoTools is a Python package that provides low-level cryptographic primitives for generating strong numbers. With this project, it's possible to generate public key cryptosystems such as RSA. +This project has a academic purpose and can not be used in production enviroment yet. + +So far, my cryptographic modules are not compliant with [FIPS 140-3](https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.140-3.pdf) but in the future, that will be. diff --git a/docs/number-theory.md b/docs/number-theory.md new file mode 100644 index 0000000..9efa817 --- /dev/null +++ b/docs/number-theory.md @@ -0,0 +1,9 @@ +# Number Theory +For generating keys, we need strong prime number, they are the basis. CryptoTools provides functions for generating these numbers. + +## prime numbers +::: Cryptotools.Numbers.primeNumber + +## Fibonacci sequence +::: Cryptotools.Numbers.numbers + diff --git a/docs/rsa.md b/docs/rsa.md new file mode 100644 index 0000000..19d8ad6 --- /dev/null +++ b/docs/rsa.md @@ -0,0 +1,3 @@ +CryptoTools provides several functions for generating a RSA Keys + +::: Cryptotools.Encryptions.RSA diff --git a/examples/carmi.py b/examples/carmi.py new file mode 100644 index 0000000..bf131af --- /dev/null +++ b/examples/carmi.py @@ -0,0 +1,32 @@ +#!/usr/bin/env python3 + +from Cryptotools.Utils.utils import gcd + + +def carmi_first_method(n) -> int: + coprimes = list() + for i in range(1, n): + if gcd(i, n) == 1: + coprimes.append(i) + print(coprimes) + smallest = list() + for m in range(1, n): + l = 0 + for a in coprimes: + if a ** m % n == 1: + l += 1 + if l == len(coprimes): + smallest.append(m) + + #print(smallest) + return min(smallest) + +print(f"lambda(12) = {carmi_first_method(12)}") +print(f"lambda(19) = {carmi_first_method(19)}") +#print(f"lambda(28) = {carmi_first_method(28)}") +#print(f"lambda(32) = {carmi_first_method(32)}") +#print(f"lambda(33) = {carmi_first_method(33)}") +#print(f"lambda(35) = {carmi_first_method(35)}") +#print(f"lambda(36) = {carmi_first_method(36)}") +#print(f"lambda(135) = {carmi_first_method(135)}") +#print(f"lambda(1200) = {carmi_first_method(1200)}") diff --git a/examples/coprime.py b/examples/coprime.py new file mode 100644 index 0000000..09368ab --- /dev/null +++ b/examples/coprime.py @@ -0,0 +1,23 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.coprime import phi +from Cryptotools.lcm import lcm +from Cryptotools.Numbers.carmi import carmi_numbers, is_carmichael, carmichael_lambda + +print(f"phi(19) = {phi(19)}") + +print("Carmichael's number") +l = carmi_numbers(561) +print(is_carmichael(561, l)) + +# Test Carmichael lambda +print("Carmichael") +print(f"12 {carmichael_lambda(12)}") +print(f"19 {carmichael_lambda(19)}") +print(f"28 {carmichael_lambda(28)}") +print(f"32 {carmichael_lambda(32)}") +print(f"33 {carmichael_lambda(33)}") +print(f"35 {carmichael_lambda(35)}") +print(f"36 {carmichael_lambda(36)}") +print(f"135 {carmichael_lambda(135)}") +print(f"1200 {carmichael_lambda(1200)}") diff --git a/examples/cyclic.py b/examples/cyclic.py new file mode 100644 index 0000000..cb21afb --- /dev/null +++ b/examples/cyclic.py @@ -0,0 +1,41 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Numbers.primeNumber import isPrimeNumber +from random import choice + + +def operation(a, b, n): + return (a ** b) % n + +n = 19 +g = list() +for i in range(1, n): + g.append(i) + +print(f"n = {n}") +print(f"G = {g}") + +cyclic = Cyclic(g, n, operation) +order = len(g) # Length of the group +print(f"len: {order}") +print(f"prime: {isPrimeNumber(order)}") + +cyclic.generator() + +print(f"All generators: {cyclic.getGenerators()}") +print(f"Is cyclic: {cyclic.isCyclic()}") + +# Check if the abelian group is respected +print(f"It is an abelian group: {cyclic.closure()}\n") + + +e = choice(cyclic.getGenerators()) +z = list() +for a in range(1, n): + res = operation(e, a, n) + z.append(res) + +print(z) +if z == g: + print(f"The group generated with the generator {e} works") diff --git a/examples/cyclic_dlp.py b/examples/cyclic_dlp.py new file mode 100644 index 0000000..eecfdcf --- /dev/null +++ b/examples/cyclic_dlp.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python3 +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Numbers.primeNumber import getPrimeNumber +from random import randint, choice +from math import log, log10, log2 + +""" + Here, we will try to understand why we need to have a generator when we encrypt data for Diffie-Hellman + https://crypto.stackexchange.com/questions/25489/why-does-diffie-hellman-need-be-a-cyclic-group +""" +def operation(a, b, n): + return (a ** b) % n + +def getGenerator(gr, p): + index = 1 + for g in range(2, p): + z = list() + for entry in range(1, p): + res = operation(g, index, p) + #if res not in z: + z.append(res) + index = index + 1 + print(f"{g}: {sorted(z)}") + +def generateGroupByG(gr, p, g): + index = 1 + z = list() + for entry in range(1, p): + res = operation(g, index, p) + #if res not in z: + z.append(res) + index = index + 1 + return z + +def computePublicKey(key, p, g): + return (g ** key) % p + + +gr = list() +# Public value +p = 5 +g = 0 +for i in range(1, p): + gr.append(i) + +print(f"p = {p}") +print(f"G = {gr}") +# We try with a generator which is not in list +cyclic = Cyclic(gr, p, operation) +generators = cyclic.getGenerators() +print(f"All generators: {generators}") + +# Generate group with g = 2 +grWithG = generateGroupByG(gr, p, 2) +print(f"Group generated with g = 2: {grWithG}") + +for a in grWithG: + # print(f"log2({a}) = {log(a, 2)}") # Same as below + print(f"log2({a}) = {log2(a)}") + +print() + +for a in range(1, len(grWithG) + 1): + print(f"log2({a}) = {log2(a)}") diff --git a/examples/cyclic_generator.py b/examples/cyclic_generator.py new file mode 100644 index 0000000..76775fd --- /dev/null +++ b/examples/cyclic_generator.py @@ -0,0 +1,32 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.cyclic import Cyclic +from random import choice + + +def operation(a, b, n): + return (a ** b) % n + +n = 19 +g = list() +for i in range(1, n): + g.append(i) + +print(f"n = {n}") +print(f"G = {g}") + +cyclic = Cyclic(g, n, operation) + +cyclic.generator() +generators = cyclic.getGenerators() +print(f"All generators: {generators}") + +e = choice(generators) +z = list() +for a in range(1, n): + res = operation(e, a, n) + z.append(res) + +z = sorted(z) +if z == g: + print(f"Working with the generator {e}") diff --git a/examples/cyclic_subgroup.py b/examples/cyclic_subgroup.py new file mode 100644 index 0000000..a101581 --- /dev/null +++ b/examples/cyclic_subgroup.py @@ -0,0 +1,49 @@ +#!/usr/bin/env python3 +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Numbers.primeNumber import getPrimeNumber +from random import randint, choice +from math import log, log10 + +""" + Here, we will try to understand why we need to have a generator when we encrypt data for Diffie-Hellman + https://crypto.stackexchange.com/questions/25489/why-does-diffie-hellman-need-be-a-cyclic-group +""" +def operation(a, b, n): + return (a ** b) % n + +def getGenerator(gr, p): + cyclic = Cyclic(gr, p, operation) + generators = cyclic.getGenerators() + print(f"All generators: {generators}") + cyclic.identity() + print(f"Identity: {cyclic.getIdentity()}") + + return generators + + +gr = list() +# Public value +p = 13 +g = 0 +for i in range(1, p): + gr.append(i) + +print(f"p = {p}") +print(f"G = {gr}") +# We try with a generator which is not in list +generators = getGenerator(gr, p) + +g = 3 # In the group +#g = # Not in the group +#print(f"g = {g}") + + + +# Try to compute the cyclic subgroup +for G in range(p + 1): + res = 0 + for a in range(G): + r = operation(G, a, p) + res = res + r + if res == p: + print(f"G = {G}; res = {res}") diff --git a/examples/cyclic_test.py b/examples/cyclic_test.py new file mode 100644 index 0000000..8351322 --- /dev/null +++ b/examples/cyclic_test.py @@ -0,0 +1,73 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Utils.utils import gcd +from Cryptotools.Numbers.primeNumber import isPrimeNumber +from random import choice + + +def operation(a, b, n): + return (a ** b) % n + +def test1(n): + """ + We test with n is not prime but the order of the group is a prime number + """ + g = list() + g2 = list() + for i in range(1, n): + #if gcd(i, n) == 1: + g.append(i) + + print(f"n = {n}") + print(f"G = {g}") + Gsorted = sorted(g) + + cyclic = Cyclic(g, n, operation) + order = len(g) # https://en.wikipedia.org/wiki/Order_(group_theory) + print(f"len: {order}") + print(f"prime: {isPrimeNumber(order)}") + g2 = list() + for i in range(1, order): + if gcd(i, order) == 1: + g2.append(i) + pass + + print(g2) + # Pick a number in the previous list and check if we can generate all number with it + item = choice(g2) + print() + + # if the order is prime, the group is cyclic ? + + # Check if we have all items + g3 = list() + for i in range(1, n): + res = operation(i, item, n) + if gcd(res, item) == 1: + g3.append(res) + #print(res) + pass + + G2sorted = sorted(g2) + G3sorted = sorted(g3) + print() + # print(f"{g} = {cyclic.generator(g)}") + gen = cyclic.generator() + + print(f"All generators: {cyclic.getGenerators()}") + print(f"Is cyclic: {cyclic.isCyclic()}") + + print() + print(G2sorted) + print(G3sorted) + if G3sorted == Gsorted: + print(f"Matching with item {item}") # Always match with item 1 + + # Check if the abelian group is respected + print(f"It is an abelian group: {cyclic.closure()}\n") + + + +test1(19) +test1(12) diff --git a/examples/dh_cyclic.py b/examples/dh_cyclic.py new file mode 100644 index 0000000..c587129 --- /dev/null +++ b/examples/dh_cyclic.py @@ -0,0 +1,74 @@ +#!/usr/bin/env python3 +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Numbers.primeNumber import getPrimeNumber +from random import randint, choice +from math import log, log10 + +""" + Here, we will try to understand why we need to have a generator when we encrypt data for Diffie-Hellman + https://crypto.stackexchange.com/questions/25489/why-does-diffie-hellman-need-be-a-cyclic-group +""" +def operation(a, b, n): + return (a ** b) % n + +def getGenerator(gr, p): + cyclic = Cyclic(gr, p, operation) + generators = cyclic.getGenerators() + print(f"All generators: {generators}") + + # Test with a no generator + item = 2 + while item in generators: # we loop until we found an item which is not a generators of the group + item = randint(1, p) + return item + +def computePublicKey(key, p, g): + return (g ** key) % p + +def computeEphemeralKey(public, secret, p): + return (public ** secret) % p + +gr = list() +# Public value +#p = getPrimeNumber(n = 8) +#p = 257 +p = 19 +g = 0 +for i in range(1, p): + gr.append(i) + +print(f"p = {p}") +print(f"G = {gr}") +# We try with a generator which is not in list +g = getGenerator(gr, p) +#g = [3, 5, 6, 7, 10, 12, 14, 19, 20, 24, 27, 28, 33, 37, 38, 39, 40, 41, 43, 45, 47, 48, 51, 53, 54, 55, 56, 63, 65, 66, 69, 71, 74, 75, 76, 77, 78, 80, 82, 83, 85, 86, 87, 90, 91, 93, 94, 96, 97, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 115, 119, 125, 126, 127, 130, 131, 132, 138, 142, 145, 147, 148, 149, 150, 151, 152, 154, 155, 156, 160, 161, 163, 164, 166, 167, 170, 171, 172, 174, 175, 177, 179, 180, 181, 182, 183, 186, 188, 191, 192, 194, 201, 202, 203, 204, 206, 209, 210, 212, 214, 216, 217, 218, 219, 220, 224, 229, 230, 233, 237, 238, 243, 245, 247, 250, 251, 252, 254] +g = 29 # Not in the group +print(f"g = {g}") + +# We can compute with the secret key +secretKeyA = 5 +secretKeyB = 10 +publicKeyA = computePublicKey(secretKeyA, p, g) +publicKeyB = computePublicKey(secretKeyB, p, g) +print(f"Public key A: {publicKeyA}") +print(f"Public key B: {publicKeyB}") + +# Eve sniff the traffic and knows p, g and publicKeyA and B + + +# Generator need to be use, because that avoid to Eve to try to find a secret key of Alice or Bob ??? +# Utiliser un generateur qui genere la tout le groupe, va permettre d'éviter à Eve de trouver la secret key de Alice ou Bob ???? +# https://eitca.org/cybersecurity/eitc-is-acc-advanced-classical-cryptography/diffie-hellman-cryptosystem/diffie-hellman-key-exchange-and-the-discrete-log-problem/examination-review-diffie-hellman-key-exchange-and-the-discrete-log-problem/what-are-the-roles-of-the-prime-number-p-and-the-generator-alpha-in-the-diffie-hellman-key-exchange-process/ +# https://www.perplexity.ai/search/why-generator-in-cyclic-group-QRYR6.rxSI218hs_x5CvnQ#0 + +# They exchange their public keys +ephemeralKeyA = computeEphemeralKey(publicKeyB, secretKeyA, p) +ephemeralKeyB = computeEphemeralKey(publicKeyA, secretKeyB, p) +print(f"Ephemeral key A: {ephemeralKeyA}") +print(f"Ephemeral key B: {ephemeralKeyB}") + +# Test log10 +#for i in range(1, 1000): +# r = log10(i) +# if isinstance(r, int): +# print(f"{i} = {r}") diff --git a/examples/dh_no_generator.py b/examples/dh_no_generator.py new file mode 100644 index 0000000..8649f89 --- /dev/null +++ b/examples/dh_no_generator.py @@ -0,0 +1,116 @@ +#!/usr/bin/env python3 +from Cryptotools.Groups.cyclic import Cyclic +from Cryptotools.Numbers.primeNumber import getPrimeNumber +from random import randint, choice +from math import log, log10 + +""" + Here, we will try to understand why we need to have a generator when we encrypt data for Diffie-Hellman + https://crypto.stackexchange.com/questions/25489/why-does-diffie-hellman-need-be-a-cyclic-group +""" +def operation(a, b, n): + return (a ** b) % n + +def getGenerator(gr, p): + index = 1 + for g in range(2, p): + z = list() + for entry in range(1, p): + res = operation(g, index, p) + #if res not in z: + z.append(res) + index = index + 1 + print(f"{g}: {sorted(z)}") + +def getGenerator2(gr, p, g): + index = 1 + z = list() + for entry in range(1, p): + res = operation(g, index, p) + if res not in z: + z.append(res) + index = index + 1 + return z + + +def computePublicKey(key, p, g): + return (g ** key) % p + +def computeEphemeralKey(public, secret, p): + return (public ** secret) % p + +gr = list() +# Public value +p = 43 +g = 0 +for i in range(1, p): + gr.append(i) + +print(f"p = {p}") +print(f"G = {gr}") +# We try with a generator which is not in list +cyclic = Cyclic(gr, p, operation) +generators = cyclic.getGenerators() +print(f"All generators: {generators}") + +g = 3 # In the group +print(f"g = {g}") + +# We can compute with the secret key +secretKeyA = 5 +secretKeyB = 10 +publicKeyA = computePublicKey(secretKeyA, p, g) +publicKeyB = computePublicKey(secretKeyB, p, g) +print(f"Public key A: {publicKeyA}") +print(f"Public key B: {publicKeyB}\n") + +# Here, g is in the generator, we need to compute all values until that match with publicKeyA +for a in range(1, p): + res = operation(g, a, p) + if res == publicKeyA: + print(f"Brute forced secret key of A: {a}") + +for a in range(1, p): + res = operation(g, a, p) + if res == publicKeyB: + print(f"Brute forced secret key of B: {a}") + +print() + +#for a in generators: +# print(a) +# +#print() + +# Here, we generate all key withthe same generator +keys = list() +for a in range(1, p): + keys.append(operation(g, a, p)) + +print((keys)) +print(sorted(keys)) + +print() + +print(f"Keys with g = 4: {sorted(getGenerator2(gr, p, 4))}") + +# Do the same, but g is not a generator +g = 4 # Not in the generator group +publicKeyA = computePublicKey(secretKeyA, p, g) +publicKeyB = computePublicKey(secretKeyB, p, g) +print(f"Public key A: {publicKeyA}") +print(f"Public key B: {publicKeyB}") + +keys = list() +for a in range(1, p): + keys.append(operation(g, a, p)) +print(keys) + +# Eve sniff the traffic and knows p, g and publicKeyA and B +# Eve, knows p and g, because it's public +# Eve need to guess the secretKey of A. +# For doing that, we iterate all posibility until that match with the publicKeyA + +#for a in range(1, 4): +# res = operation(g, a, p) +# print(res) diff --git a/examples/fermat_factorization.py b/examples/fermat_factorization.py new file mode 100644 index 0000000..02dd2d1 --- /dev/null +++ b/examples/fermat_factorization.py @@ -0,0 +1,80 @@ +#!/usr/bin/env python3 + +import unittest +from Cryptotools.Numbers.primeNumber import isPrimeNumber, _millerRabinTest, getPrimeNumber +from math import sqrt, isqrt, ceil +from sympy import sqrt as sq + +def get_closest_prime(p): + q = p + 1 + l = list() + while True: + if _millerRabinTest(q): + l.append(q) + if len(l) == 20: + break + q += 1 + return l + +def test(): + while True: + p = getPrimeNumber(64) + b = sqrt(p) + if b % 2 == 0.0: + break + print(p) + +#test() +#exit(1) +p = 7901 +q = 7817 +#p = getPrimeNumber(64) +#q = get_closest_prime(p)[1] +p = 7943202761666983 # Works +q = 7943202761667119 +#p = 314159200000000028138418196395985880850000485810513 +#q = 314159200000000028138415196395985880850000485810479 +print(f"p = {p}") +print(f"q = {q}") +n = p * q +print(f"n = {n}") + +a = ceil(sqrt(n)) +#print(a) +#print(sqrt(n)) +#print(sqrt(n) < n) # True +#print(a * a) +#print(a * a < n) +#print() +#a = isqrt(n) + 1 +iteration = 0 +while True: + # b2 = (a ** 2) - n + iteration += 1 + b2 = pow(a, 2) - n + #b2 = ceil(a*a) - n + # print(b2) + sqb2 = ceil(sqrt(b2)) + # print(b2, isqrt(pow(b2, 2))) + if b2 % 2 == 0.0: + #if isqrt(pow(b2, 2)) == b2: + b = isqrt(b2) + break + a = a + 1 + +print(f"Iteration: {iteration}") +print(f"a = {a}") +print(f"b2 = {b2}") +print(f"b = {b}") +p = int((a + b)) +q = int((a - b)) +print(p) +print(q) +#N = (a + b) * (a - b) +N = p * q +print(_millerRabinTest(p)) +print(_millerRabinTest(q)) +print(f"N = {N}") +print(n == N) +print() + diff --git a/examples/fibonacci.py b/examples/fibonacci.py new file mode 100644 index 0000000..1730e5e --- /dev/null +++ b/examples/fibonacci.py @@ -0,0 +1,5 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.numbers import fibonacci + +print(fibonacci(40)) diff --git a/examples/galois.py b/examples/galois.py new file mode 100644 index 0000000..2b1a018 --- /dev/null +++ b/examples/galois.py @@ -0,0 +1,36 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.galois import Galois +import matplotlib.pyplot as plt # pip install numpy==1.26.4 + + +def operation(a, b, n): + return (a ** b) % n + +q = 13 +g = Galois(q, operation) +add = g.add() +div = g.div() +mul = g.mul() +sub = g.sub() +g.check_closure_law('+') +g.check_closure_law('*') +g.check_closure_law('-') +g.check_closure_law('/') + +print("Addition") +g.printMatrice(add) +print("Division") +g.printMatrice(div) +print("Multiplication") +g.printMatrice(mul) +print("Substraction") +g.printMatrice(sub) + + +print("Primitives root") +print(g.primitiveRoot(), end="\n\n") + +#print("Identity elements") +g.check_identity_add() +g.check_identity_mul() diff --git a/examples/group.py b/examples/group.py new file mode 100644 index 0000000..1931d3c --- /dev/null +++ b/examples/group.py @@ -0,0 +1,55 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.group import Group +from Cryptotools.Utils.utils import gcd +import matplotlib.pyplot as plt + + +n = 18 +def operation(e1, e2, n) -> int: + return e1 * e2 % n + +# Generate the group G +g = list() +for i in range(1, n): + #if gcd(i, n) == 1: + g.append(i) + +G = Group(n = n, g = g, ope=operation) +print(f"n = {n}") +print(f"G = {G.getG()}") + +if G.closure() is False: + print("The closure law is not respected. It's not an abelian (commutative) group") +else: + print("It is a closure group") + +if G.associative() is False: + print("The associative law is not respected. It's not a group") +else: + print("It is an associative group") + +if G.identity() is False: + print("The group hasn't an identity element") +else: + print(f"Identity: {G.getIdentity()}") + +if G.reverse() is False: + print(f"The group hasn't a reverse") +else: + print(f"Reverses: {G.getReverses()}") + +#reverses = G.getReverses() +#plt.rcParams["figure.figsize"] = [7.00, 3.50] +#plt.rcParams["figure.autolayout"] = True +#for key, value in reverses.items(): +# x = key +# y = value +# plt.plot(x, y, marker="o", markersize=2, markeredgecolor="blue") +# #plt.Circle((0, n), 0.2, color='r') +# +#plt.xlim(0, n) +#plt.ylim(0, n) +#plt.grid() +#plt.show() + diff --git a/examples/main.py b/examples/main.py new file mode 100644 index 0000000..2bdc08d --- /dev/null +++ b/examples/main.py @@ -0,0 +1,6 @@ +#!/usr/bin/env python3 + +from utils.primeNumber import primeNumber + +print(primeNumber(19)) +print(primeNumber(20)) diff --git a/examples/plotlib.py b/examples/plotlib.py new file mode 100644 index 0000000..da545c5 --- /dev/null +++ b/examples/plotlib.py @@ -0,0 +1,25 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import get_n_prime_numbers +import matplotlib.pyplot as plt +#import numpy as np + +# Get list of n prime numbers +n = 100 +primes = get_n_prime_numbers(n) +print(primes) + +# With matplotlib, show into a graphics and try to explain that +plt.rcParams["figure.figsize"] = [7.00, 3.50] +plt.rcParams["figure.autolayout"] = True +for i in range(0, n): + x = primes[i] + y = primes[i] + plt.plot(x, y, marker="o", markersize=2, markeredgecolor="blue") + + +plt.xlim(0, n) +plt.ylim(0, n) +plt.grid() +plt.show() + diff --git a/examples/pollard.py b/examples/pollard.py new file mode 100644 index 0000000..39c0e5f --- /dev/null +++ b/examples/pollard.py @@ -0,0 +1,53 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.factorization import pollard_p_minus_1 +from Cryptotools.Numbers.primeNumber import _millerRabinTest, getPrimeNumber +from sympy.ntheory.factor_ import pollard_pm1 +from sympy import factorint + + + +def pollard(n): + p = pollard_p_minus_1(n, B=10) + if p is not None: + print(p, n / p) + print(pollard_pm1(n, B=10)) + print() + return True + return False + +def safePrime(p): + nP = 2 * p + 1 + return _millerRabinTest(nP) + +#pollard(299) +#pollard(257 * 1009) +#pollard(1684) # Doesn't works + +# Test for RSA +#p = 281 +while True: + while True: + p = getPrimeNumber(64) + if ((p - 1) / 2520) % 2 == 0.0: + #print(p) + break + #p = 322506219347091343 + #q = 13953581789873249851 + # n = p * q + while True: + q = getPrimeNumber(64) + if int(((q - 1) / 2520) % 2) == 1: + #print(q) + break + + #q = 223 + n = p * q + ##print(n) + #print(pollard(n)) + #break + if pollard(n): # We can factorize n + break + +print(safePrime(p)) +#print(safePrime(q)) diff --git a/examples/prime.py b/examples/prime.py new file mode 100644 index 0000000..a36e613 --- /dev/null +++ b/examples/prime.py @@ -0,0 +1,8 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import getPrimeNumber, sophieGermainPrime + +p = getPrimeNumber(512) +print(p) +if sophieGermainPrime(p): + print("It's a safe prime") diff --git a/examples/pseudoprimes.py b/examples/pseudoprimes.py new file mode 100644 index 0000000..e4b44a8 --- /dev/null +++ b/examples/pseudoprimes.py @@ -0,0 +1,17 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import _millerRabinTest, _fermatLittleTheorem +from Cryptotools.Numbers.carmi import carmi_numbers, is_carmi_number, generate_carmi_numbers + +print(_fermatLittleTheorem(1729)) +print(_fermatLittleTheorem(1105)) +print(_fermatLittleTheorem(6601)) +print(_millerRabinTest(1729)) + +print(f"1729: {is_carmi_number(1729)}") +print(f"1105: {is_carmi_number(1105)}") +print(f"110: {is_carmi_number(110)}") +print(f"2465: {is_carmi_number(2465)}") +print(f"6601: {is_carmi_number(6601)}") +#print(carmi_numbers(1729)) +print(generate_carmi_numbers(10)) diff --git a/examples/rsa.py b/examples/rsa.py new file mode 100644 index 0000000..2f9e5e2 --- /dev/null +++ b/examples/rsa.py @@ -0,0 +1,22 @@ +#!/usr/bin/env python3 + +from Cryptotools.Encryptions.RSA import RSA + + +rsa = RSA() +rsa.generateKeys(size=512) + +e = rsa.e +d = rsa.d +n = rsa.n + +s = "I am encrypted with RSA" +print(f"plaintext: {s}") +encrypted = rsa.encrypt(s) + +# Encrypt data +# print(f"ciphertext: {encrypted}") + +# We decrypt +plaintext = rsa.decrypt(encrypted) +print(f"Plaintext: {plaintext}") diff --git a/examples/rsa_problem.py b/examples/rsa_problem.py new file mode 100644 index 0000000..f4dbb08 --- /dev/null +++ b/examples/rsa_problem.py @@ -0,0 +1,52 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.coprime import phi +from Cryptotools.Utils.utils import gcd + +def generate_keys(): + p = 7853 + q = 7919 + n = p * q + e = 65536 # It's public value, must be coprime with phi n + + print(n) + #phin = phi(n) + phin = (p - 1) * (q - 1) + print(phin) + + for _ in range(2, phin): + if gcd(phin, e) == 1: + break + e += 1 + + print(e) + plaintext = 'A' + ciphertext = pow(ord(plaintext), e, n) + print(f"Ciphertext: {ciphertext}") + +# Now, we can test + +# To resolve that formula: C = x ** e mod n +# Where C is the ciphertext, and C, e and n are known (public values) +# First, we need to find the reverse modular of phi(n) or carmi(n) +# z = e -1 mod phi(n) +# After that, we have our decryption key, we can resolve x +# x = C ** z mod n +# The RSA Problem is to decrypt with the public-key +# We just need to find the decryption key with the public-key and the modulus + +# First, we need to find phi(n) +# phin = phi(n) # I computed here, result = 62172136 + +n = 62187907 +phin = 62172136 +e = 65537 +ciphertext = 38605768 +# print(phin) + +# Find the reverse modular +d = pow(e, -1, phin) +# print(d) +plaintext = pow(ciphertext, d, n) +print(chr(plaintext)) + diff --git a/examples/sieveOfEratosthenes.py b/examples/sieveOfEratosthenes.py new file mode 100644 index 0000000..e278ab7 --- /dev/null +++ b/examples/sieveOfEratosthenes.py @@ -0,0 +1,6 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import sieveOfEratosthenes + +print(sieveOfEratosthenes(100)) +print() diff --git a/examples/test_gcd.py b/examples/test_gcd.py new file mode 100644 index 0000000..7ccd4d5 --- /dev/null +++ b/examples/test_gcd.py @@ -0,0 +1,15 @@ + +from Cryptotools.Utils.utils import gcd + + +for i in range(1, 30): + if 30 % i == 0: + print(i) + +print() + +for i in range(1, 40): + if 40 % i == 0: + print(i) + +print(gcd(30, 40)) diff --git a/mkdocs.yml b/mkdocs.yml new file mode 100644 index 0000000..04cebae --- /dev/null +++ b/mkdocs.yml @@ -0,0 +1,17 @@ +site_name: CryptoTools documentation +theme: + name: "readthedocs" + +plugins: + - mkdocstrings + +nav: + - Introduction: introduction.md + - Installation: installation.md + - Low-level cryptographic: + - Number theory: number-theory.md + - Group theory: group-theory.md + - Public Keys: + - RSA: rsa.md + - Examples: + - Generating RSA Keys: examples-rsa-keys.md diff --git a/pyproject.toml b/pyproject.toml new file mode 100644 index 0000000..747f68a --- /dev/null +++ b/pyproject.toml @@ -0,0 +1,16 @@ +[project] +name = "Cryptotools" +version = "0.1.0" +description = "" +authors = [ + {name = "gbucchino",email = "gbucchino@fortinet-us.com"} +] +readme = "README.md" +requires-python = ">=3.10" +dependencies = [ +] + + +[build-system] +requires = ["poetry-core>=2.0.0,<3.0.0"] +build-backend = "poetry.core.masonry.api" diff --git a/requirements.txt b/requirements.txt new file mode 100644 index 0000000..79cdabf --- /dev/null +++ b/requirements.txt @@ -0,0 +1,5 @@ +sympy +mkdocs +mkdocstrings[python] +mkdocs-readthedocs +mkdocs-material diff --git a/setup.py b/setup.py new file mode 100644 index 0000000..37daa98 --- /dev/null +++ b/setup.py @@ -0,0 +1,22 @@ +from setuptools import setup, find_packages + + +_version = "1.0" +_project_name = "Cryptotools" + +_packages = [ + "Numbers", + "Groups", + "Utils", +] + +setup( + name=_project_name, + version=_version, + description="Cryptography library", + author="Geoffrey Bucchino", + author_email="contact@bucchino.org", + packages=find_packages(), + #packages=_packages, + #package_dir={"": "Cryptotools"} +) diff --git a/sieves_base.py b/sieves_base.py new file mode 100644 index 0000000..db5cdf0 --- /dev/null +++ b/sieves_base.py @@ -0,0 +1,29 @@ +#!/usr/bin/env python3 + +from Cryptotools.Numbers.primeNumber import isPrimeNumber + +def get_bases(n): + # Generate the n first prime numbers + bases = list() + i = 2 + while (len(bases) < n): + if isPrimeNumber(i): + bases.append(i) + i = i + 1 + + return bases + +bases = get_bases(1000) + + + +print("(") +index = 0 +for i in range(0, len(bases)): + if index == 10: + index = 0 + print(f"\n", end="") + else: + print(f"{bases[i]}", end=",") + index = index + 1 +print(")") diff --git a/site/404.html b/site/404.html new file mode 100644 index 0000000..7a70374 --- /dev/null +++ b/site/404.html @@ -0,0 +1,122 @@ + + + + + + + + CryptoTools documentation + + + + + + + + + + + + +
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Using flexbox, we + * achieve it in such a way that it will look like the following: + * + * [No repo_name] + * Next >> // On the first page + * << Previous Next >> // On all subsequent pages + * + * [With repo_name] + * Next >> // On the first page + * << Previous Next >> // On all subsequent pages + * + * https://github.com/mkdocs/mkdocs/issues/2012 + */ +.rst-versions .rst-current-version { + padding: 0 12px; + display: flex; + font-size: initial; + justify-content: space-between; + align-items: center; + line-height: 40px; +} + +/* + * Please note that this amendment also involves removing certain inline-styles + * from the file ./mkdocs/themes/readthedocs/versions.html. + * + * https://github.com/mkdocs/mkdocs/issues/2012 + */ +.rst-current-version span { + flex: 1; + text-align: center; +} diff --git a/site/examples-rsa-keys/index.html b/site/examples-rsa-keys/index.html new file mode 100644 index 0000000..af1636d --- /dev/null +++ b/site/examples-rsa-keys/index.html @@ -0,0 +1,158 @@ + + + + + + + + Generating RSA Keys - CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Examples
    • +
  • Generating RSA Keys
    • +
  • +
    • +
+
+
+
+
+ +

Generating RSA Keys

+

In the following section, we are going to see how to generate RSA Keys and how to encrypt data with these keys.

+
#!/usr/bin/env python3
+
+from Cryptotools.Encryptions.RSA import RSA
+
+
+rsa = RSA()
+rsa.generateKeys(size=512)
+
+e = rsa.e
+d = rsa.d
+n = rsa.n
+
+s = "I am encrypted with RSA"
+print(f"plaintext: {s}")
+encrypted = rsa.encrypt(s)
+
+# Encrypt data
+print(f"ciphertext: {encrypted}")
+
+# We decrypt
+plaintext = rsa.decrypt(encrypted)
+print(f"Plaintext: {plaintext}")
+
+ +
+
+ +
+
+ +
+ +
+ +
+ + + + « Previous + + + +
+ + + + + + + + diff --git a/site/group-theory/index.html b/site/group-theory/index.html new file mode 100644 index 0000000..98bd03c --- /dev/null +++ b/site/group-theory/index.html @@ -0,0 +1,3059 @@ + + + + + + + + Group theory - CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Low-level cryptographic
    • +
  • Group theory
    • +
  • +
    • +
+
+
+
+
+ +

Group Theory

+

Group

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + +
+ + + +

+ Group + + +

+ + +
+ + +

This class generate a group self._g based on the operation (here denoted +) +with the function ope: (a ** b) % n

+

In group theory, any group has an identity element (e), which with the binary operation, do not change the value and must satisfy the condition: $a + e = 0$

+ + + + + + + + + + + + + +
Attributes: +
    +
  • + n + (Integer) + – +
    +

    It's the G_n elements in the Group

    +
    +
    • +
  • + g + (List) + – +
    +

    The set of the Group

    +
    +
    • +
  • + operation + (Function) + – +
    +

    Function for generating the Group

    +
    +
    • +
  • + identity + (Integer) + – +
    +

    The identity of the group.

    +
    +
    • +
  • + reverse + (Dict) + – +
    +

    For each elements of the Group of n, we have the reverse value

    +
    +
    • +
+
+ + + + + + + +
+ Source code in Cryptotools/Groups/group.py + 
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class Group:
+    """
+        This class generate a group self._g based on the operation (here denoted +)
+        with the function ope: (a ** b) % n
+
+        In group theory, any group has an identity element (e), which with the binary operation, do not change the value and must satisfy the condition: $a + e = 0$
+
+        Attributes:
+            n (Integer): It's the G_n elements in the Group
+            g (List): The set of the Group
+            operation (Function): Function for generating the Group
+            identity (Integer): The identity of the group.
+            reverse (Dict): For each elements of the Group of n, we have the reverse value
+    """
+    def __init__(self, n, g, ope):
+        self._n = n
+        self._g = g
+        self._operation = ope
+        self._identity = 0
+        self._reverse = dict()
+
+    def getG(self) -> list():
+        """
+            This function return the Group
+
+            Returns:
+                List of all elements in the Group
+        """
+        return self._g
+
+    def closure(self) -> bool:
+        """
+            Check the closure law
+            In a group, each element a, b belongs to G, such as a + b belongs to G
+
+            Returns:
+                Return a Boolean if the closure law is respected. True if yes otherwise it's False
+        """
+        for e1 in self._g:
+            for e2 in self._g:
+                res = self._operation(e1, e2, self._n)
+                if  not res in self._g:
+                    # raise Exception(f"{res} not in g. g is not a group")
+                    return False
+        return True 
+
+    def associative(self) -> bool:
+        """
+            Check the associative law. 
+            In a group, for any a, b and c belongs to G,
+            they must respect this condition: (a + b) + c = a + (a + b)
+
+            Returns:
+                Return a boolean if the Associative law is respected. True if yes otherwise it's False
+        """
+        a = self._g[0]
+        b = self._g[1]
+        c = self._g[2]
+
+        res_ope = self._operation(a, b, self._n)
+        res1 = self._operation(res_ope, c, self._n)
+
+        res_ope = self._operation(b, c, self._n)
+        res2 = self._operation(a, res_ope, self._n)
+        if res1 != res2:
+            # raise Exception(f"{res1} is different from {res2}. g is not a group")
+            return False
+        return True
+
+    def identity(self) -> bool:
+        """
+            Check the identity law.
+            In a group, an identity element exist and must be uniq
+
+            Returns:
+                Return a Boolean if the identity elements has been found. True if found otherwise it's False
+        """
+        for a in self._g:
+            for b in self._g:
+                if not self._operation(a, b, self._n) == b:
+                    break
+
+                self._identity = a
+                return True
+        return False
+
+    def getIdentity(self) -> int:
+        """
+            Return the identity element. The function identitu() must be called before.
+
+            Returns:
+                Return the identity element if it has been found
+        """
+        return self._identity
+
+    def reverse(self) -> bool:
+        """
+            Check the inverse law
+            In a group, for each element belongs to G
+            they must have an inverse a ^ (-1) = e (identity)
+
+            Returns:
+                Return a Boolean if the all elements ha a reverse element. True if yes otherwise it's False
+        """
+        reverse = False
+        for a in self._g:
+            for b in self._g:
+                if self._operation(a, b, self._n) == self._identity:
+                    self._reverse[a] = b
+                    reverse = True
+                    break
+        return reverse        
+
+    def getReverses(self) -> dict:
+        """
+            This function return the dictionary of all reverses elements. The key is the element in G and the value is the reverse element
+
+            Returns:
+                Return the reverse dictionary
+
+        """
+        return self._reverse
+
+
+ + + +
+ + + + + + + + + + +
+ + +

+ associative() + +

+ + +
+ +

Check the associative law. +In a group, for any a, b and c belongs to G, +they must respect this condition: (a + b) + c = a + (a + b)

+ + + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return a boolean if the Associative law is respected. True if yes otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def associative(self) -> bool:
+    """
+        Check the associative law. 
+        In a group, for any a, b and c belongs to G,
+        they must respect this condition: (a + b) + c = a + (a + b)
+
+        Returns:
+            Return a boolean if the Associative law is respected. True if yes otherwise it's False
+    """
+    a = self._g[0]
+    b = self._g[1]
+    c = self._g[2]
+
+    res_ope = self._operation(a, b, self._n)
+    res1 = self._operation(res_ope, c, self._n)
+
+    res_ope = self._operation(b, c, self._n)
+    res2 = self._operation(a, res_ope, self._n)
+    if res1 != res2:
+        # raise Exception(f"{res1} is different from {res2}. g is not a group")
+        return False
+    return True
+
+
+
+ +
+ +
+ + +

+ closure() + +

+ + +
+ +

Check the closure law +In a group, each element a, b belongs to G, such as a + b belongs to G

+ + + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return a Boolean if the closure law is respected. True if yes otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def closure(self) -> bool:
+    """
+        Check the closure law
+        In a group, each element a, b belongs to G, such as a + b belongs to G
+
+        Returns:
+            Return a Boolean if the closure law is respected. True if yes otherwise it's False
+    """
+    for e1 in self._g:
+        for e2 in self._g:
+            res = self._operation(e1, e2, self._n)
+            if  not res in self._g:
+                # raise Exception(f"{res} not in g. g is not a group")
+                return False
+    return True 
+
+
+
+ +
+ +
+ + +

+ getG() + +

+ + +
+ +

This function return the Group

+ + + + + + + + + + + + + +
Returns: +
    +
  • + list() + – +
    +

    List of all elements in the Group

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def getG(self) -> list():
+    """
+        This function return the Group
+
+        Returns:
+            List of all elements in the Group
+    """
+    return self._g
+
+
+
+ +
+ +
+ + +

+ getIdentity() + +

+ + +
+ +

Return the identity element. The function identitu() must be called before.

+ + + + + + + + + + + + + +
Returns: +
    +
  • + int + – +
    +

    Return the identity element if it has been found

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def getIdentity(self) -> int:
+    """
+        Return the identity element. The function identitu() must be called before.
+
+        Returns:
+            Return the identity element if it has been found
+    """
+    return self._identity
+
+
+
+ +
+ +
+ + +

+ getReverses() + +

+ + +
+ +

This function return the dictionary of all reverses elements. The key is the element in G and the value is the reverse element

+ + + + + + + + + + + + + +
Returns: +
    +
  • + dict + – +
    +

    Return the reverse dictionary

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def getReverses(self) -> dict:
+    """
+        This function return the dictionary of all reverses elements. The key is the element in G and the value is the reverse element
+
+        Returns:
+            Return the reverse dictionary
+
+    """
+    return self._reverse
+
+
+
+ +
+ +
+ + +

+ identity() + +

+ + +
+ +

Check the identity law. +In a group, an identity element exist and must be uniq

+ + + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return a Boolean if the identity elements has been found. True if found otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def identity(self) -> bool:
+    """
+        Check the identity law.
+        In a group, an identity element exist and must be uniq
+
+        Returns:
+            Return a Boolean if the identity elements has been found. True if found otherwise it's False
+    """
+    for a in self._g:
+        for b in self._g:
+            if not self._operation(a, b, self._n) == b:
+                break
+
+            self._identity = a
+            return True
+    return False
+
+
+
+ +
+ +
+ + +

+ reverse() + +

+ + +
+ +

Check the inverse law +In a group, for each element belongs to G +they must have an inverse a ^ (-1) = e (identity)

+ + + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return a Boolean if the all elements ha a reverse element. True if yes otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/group.py + 
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def reverse(self) -> bool:
+    """
+        Check the inverse law
+        In a group, for each element belongs to G
+        they must have an inverse a ^ (-1) = e (identity)
+
+        Returns:
+            Return a Boolean if the all elements ha a reverse element. True if yes otherwise it's False
+    """
+    reverse = False
+    for a in self._g:
+        for b in self._g:
+            if self._operation(a, b, self._n) == self._identity:
+                self._reverse[a] = b
+                reverse = True
+                break
+    return reverse        
+
+
+
+ +
+ + + +
+ +
+ +
+ + + + +
+ +
+ +

Cyclic group

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + +
+ + + +

+ Cyclic + + +

+ + +
+

+ Bases: Group

+ + +

This object contain a list of the Group for a cyclic group. This class find all generator of the group. +This class is inherited from Group object

+ + + + + + + + + + + + + +
Attributes: +
    +
  • + G + (list) + – +
    +

    list of all elements in the group

    +
    +
    • +
  • + n + (Integer) + – +
    +

    it's the value where the group has been generated

    +
    +
    • +
  • + operation + (Function) + – +
    +

    it's the operation generating the group

    +
    +
    • +
  • + generators + (list) + – +
    +

    contain all generators of the group

    +
    +
    • +
  • + generatorChecked + (boolean) + – +
    +

    Check if generators has been found

    +
    +
    • +
+
+ + + + + + + +
+ Source code in Cryptotools/Groups/cyclic.py + 
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class Cyclic(Group):
+    """
+        This object contain a list of the Group for a cyclic group. This class find all generator of the group.
+        This class is inherited from Group object
+
+        Attributes:
+            G (list): list of all elements in the group
+            n (Integer): it's the value where the group has been generated
+            operation (Function): it's the operation generating the group
+            generators (list): contain all generators of the group 
+            generatorChecked (boolean): Check if generators has been found
+    """
+    def __init__(self, G:list, n, ope):
+        super().__init__(n, G, ope) # Call the Group's constructor
+        self._G = G
+        self._n = n
+        self._operation = ope
+        self._generators = list()
+        self._generatorChecked = False
+
+    def generator(self):
+        """
+            This function find all generators in the group G
+        """
+
+        index = 1
+        G = sorted(self._g)
+        for g in range(2, self._n):
+            z = list()
+            for entry in range(1, self._n):
+                res = self._operation(g, index, self._n)
+                if res not in z:
+                    z.append(res)
+                    index = index + 1
+
+            # We check if that match with G
+            # If yes, we find a generator
+            if sorted(z) == G:
+                self._generators.append(g)
+        self._generatorChecked = True
+
+    def getPrimitiveRoot(self):
+        """
+            This function return the primitive root modulo of n
+
+            Returns:
+                Return the primitive root of the group. None if no primitive has been found
+        """
+
+        index = 1
+        G = sorted(self._g)
+        for g in range(2, self._n):
+            z = list()
+            for entry in range(1, self._n):
+                res = self._operation(g, index, self._n)
+                if res not in z:
+                    z.append(res)
+                    index += 1
+
+            # If the group is the same has G, we found a generator
+            if sorted(z) == G:
+                return g
+        return None
+
+    def getGenerators(self) -> list:
+        """
+            This function return all generators of that group
+
+            Returns:
+                Return the list of generators found. The function generators() must be called before to call this function
+        """
+        if not self._generatorChecked:
+            self.generator()
+            self._generatorChecked = True
+        return self._generators
+
+    def isCyclic(self) -> bool:
+        """
+            Check if the group is a cyclic group, means we have at least one generator
+
+            Returns:
+                REturn a boolean, False if the group is not Cyclic otherwise return True
+        """
+        if len(self.getGenerators()) == 0:
+            return False
+        return True
+
+
+ + + +
+ + + + + + + + + + +
+ + +

+ generator() + +

+ + +
+ +

This function find all generators in the group G

+ + +
+ Source code in Cryptotools/Groups/cyclic.py + 
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def generator(self):
+    """
+        This function find all generators in the group G
+    """
+
+    index = 1
+    G = sorted(self._g)
+    for g in range(2, self._n):
+        z = list()
+        for entry in range(1, self._n):
+            res = self._operation(g, index, self._n)
+            if res not in z:
+                z.append(res)
+                index = index + 1
+
+        # We check if that match with G
+        # If yes, we find a generator
+        if sorted(z) == G:
+            self._generators.append(g)
+    self._generatorChecked = True
+
+
+
+ +
+ +
+ + +

+ getGenerators() + +

+ + +
+ +

This function return all generators of that group

+ + + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return the list of generators found. The function generators() must be called before to call this function

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/cyclic.py + 
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def getGenerators(self) -> list:
+    """
+        This function return all generators of that group
+
+        Returns:
+            Return the list of generators found. The function generators() must be called before to call this function
+    """
+    if not self._generatorChecked:
+        self.generator()
+        self._generatorChecked = True
+    return self._generators
+
+
+
+ +
+ +
+ + +

+ getPrimitiveRoot() + +

+ + +
+ +

This function return the primitive root modulo of n

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return the primitive root of the group. None if no primitive has been found

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/cyclic.py + 
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def getPrimitiveRoot(self):
+    """
+        This function return the primitive root modulo of n
+
+        Returns:
+            Return the primitive root of the group. None if no primitive has been found
+    """
+
+    index = 1
+    G = sorted(self._g)
+    for g in range(2, self._n):
+        z = list()
+        for entry in range(1, self._n):
+            res = self._operation(g, index, self._n)
+            if res not in z:
+                z.append(res)
+                index += 1
+
+        # If the group is the same has G, we found a generator
+        if sorted(z) == G:
+            return g
+    return None
+
+
+
+ +
+ +
+ + +

+ isCyclic() + +

+ + +
+ +

Check if the group is a cyclic group, means we have at least one generator

+ + + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    REturn a boolean, False if the group is not Cyclic otherwise return True

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/cyclic.py + 
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def isCyclic(self) -> bool:
+    """
+        Check if the group is a cyclic group, means we have at least one generator
+
+        Returns:
+            REturn a boolean, False if the group is not Cyclic otherwise return True
+    """
+    if len(self.getGenerators()) == 0:
+        return False
+    return True
+
+
+
+ +
+ + + +
+ +
+ +
+ + + + +
+ +
+ +

Galois Field (Finite Field)

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + +
+ + + +

+ Galois + + +

+ + +
+ + +

This class contain the Galois Field (Finite Field)

+ + + + + + + + + + + + + +
Attributes: +
    +
  • + q + (Integer) + – +
    +

    it's the number of the GF

    +
    +
    • +
  • + operation + (Function) + – +
    +

    Function for generating the group

    +
    +
    • +
  • + identityAdd + (Integer) + – +
    +

    it's the identity element in the GF(n) for the addition operation

    +
    +
    • +
  • + identityMul + (Integer) + – +
    +

    it's the identity element in the GF(n) for the multiplicative operation

    +
    +
    • +
+
+ + + + + + + +
+ Source code in Cryptotools/Groups/galois.py + 
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class Galois:
+    """
+        This class contain the Galois Field (Finite Field)
+
+
+
+        Attributes:
+            q (Integer): it's the number of the GF
+            operation (Function): Function for generating the group
+            identityAdd (Integer): it's the identity element in the GF(n) for the addition operation
+            identityMul (Integer): it's the identity element in the GF(n) for the multiplicative operation
+    """
+    def __init__(self, q, operation):
+        self._q = q
+        self._operation = operation
+        self._identityAdd = 0
+        self._identityMul = 1
+        self._F = [x for x in range(q)] 
+        self._add = [[0 for x in range(q)] for y in range(q)] 
+
+        # TODO: May do we do a deep copy between all groups ?
+        self._div = [[0 for x in range(q)] for y in range(q)] 
+        self._mul = [[0 for x in range(q)] for y in range(q)] 
+        self._sub = [[0 for x in range(q)] for y in range(q)] 
+        self._primitiveRoot = list()
+
+    def primitiveRoot(self):
+        """
+            In this function, we going to find the primitive root modulo n of the galois field
+
+            Returns:
+               Return the list of primitive root of the GF(q) 
+        """
+        for x in range(2, self._q):
+            z = list()
+            for entry in range(1, self._q):
+                res = self._operation(x, entry, self._q)
+                if res not in z:
+                    z.append(res)
+
+            if self.isPrimitiveRoot(z, self._q - 1):
+                if x not in self._primitiveRoot: # It's dirty, need to find why we have duplicate entry
+                    self._primitiveRoot.append(x)
+        return z
+
+    def getPrimitiveRoot(self):
+        """
+            Return the list of primitives root
+
+            Returns:
+                Return the primitive root
+        """
+        return self._primitiveRoot
+
+    def isPrimitiveRoot(self, z, length):
+        """
+            Check if z is a primitive root 
+
+            Args:
+                z (list): check if z is a primitive root
+                length (Integer): Length of the GF(q)
+        """
+        if len(z) == length:
+            return True
+        return False
+
+    def add(self):
+        """
+            This function do the operation + on the Galois Field
+
+            Returns:
+                Return a list of the group with the addition operation
+        """
+        for x in range(0, self._q):
+            for y in range(0, self._q):
+                self._add[x][y] = (x + y) % self._q
+        return self._add
+
+    def _inverseModular(self, a, n):
+        """
+            This function find the reverse modular of a by n
+
+            Returns:
+                Return the reverse modular
+        """
+        for b in range(1, n):
+            if (a * b) % n == 1:
+                inv = b
+                break
+        return inv
+
+    def div(self):
+        """
+            This function do the operation / on the Galois Field
+
+            Returns:
+                Return a list of the group with the division operation
+        """
+        for x in range(1, self._q):
+            for y in range(1, self._q):
+                inv = self._inverseModular(y, self._q)
+
+                self._div[x][y] = (x * inv) % self._q
+        return self._div
+
+    def mul(self):
+        """
+            This function do the operation * on the Galois Field
+
+            Returns:
+                Return a list of the group with the multiplication operation
+        """
+        for x in range(0, self._q):
+            for y in range(0, self._q):
+                self._mul[x][y] = (x * y) % self._q
+        return self._mul
+
+    def sub(self):
+        """
+            This function do the operation - on the Galois Field
+
+            Returns:
+                Return a list of the group with the subtraction operation
+        """
+        for x in range(0, self._q):
+            for y in range(0, self._q):
+                self._sub[x][y] = (x - y) % self._q
+        return self._sub
+
+    def check_closure_law(self, arithmetic):
+        """
+            This function check the closure law.
+            By definition, every element in the GF is an abelian group, which respect the closure law: for a and b belongs to G, a + b belongs to G, the operation is a binary operation
+
+            Args:
+                Arithmetics (str): contain the operation to be made, must be '+', '*', '/' or /-'
+        """
+        if arithmetic not in ['+', '*', '/', '-']:
+            raise Exception("The arithmetic need to be '+', '*', '/' or '-'")
+
+        if arithmetic == '+':
+            G = self._add
+        elif arithmetic == '*':
+            G = self._mul
+        elif arithmetic == '/':
+            G = self._div
+        else:
+            G = self._sub
+
+        start = 0
+        """
+            In case of multiplicative, we bypass the first line, because all elements are zero, otherwise the test fail
+        """
+        if arithmetic == '*' or arithmetic == '/':
+            start = 1
+
+        isClosure = True
+        for x in range(start, self._q):
+            gr = Group(self._q, G[x], self._operation)
+            if not gr.closure():
+                isClosure = False
+            del gr
+
+        if isClosure:
+            print(f"The group {arithmetic} respect closure law")
+        else:
+            print(f"The group {arithmetic} does not respect closure law")
+
+    def check_identity_add(self):
+        """
+            This function check the identity element and must satisfy this condition: $a + 0 = a$ for each element in the GF(n)
+            In Group Theory, an identity element is an element in the group which do not change the value every element in the group
+
+            Returns:
+
+        """
+        for x in self._F:
+            if not self._identityAdd + x == x:
+                raise Exception(
+                        f"The identity element {self._identityAdd} "\
+                         "do not satisfy $a + element = a$"
+                )
+
+    def check_identity_mul(self):
+        """
+            This function check the identity element and must satisfy this condition: $a * 1 = a$ for each element in the GF(n)
+            In Group Theory, an identity element is an element in the group which do not change the value every element in the group
+
+            Returns:
+
+        """
+        for x in self._F:
+            if not self._identityMul * x == x:
+                raise Exception(
+                        f"The identity element {self._identityAdd} "\
+                         "do not satisfy $a * element = a$"
+                )
+
+    def printMatrice(self, m):
+        """
+            This function print the GF(m)
+
+            Args:
+                m (list): Matrix of the GF
+        """
+        header = str()
+        header = "    "
+        for x in range(0, self._q):
+            header += f"{x} "
+
+        header += "\n--|" + "-" * (len(header)- 3) +"\n"
+
+        s = str()
+
+        for x in range(0, self._q):
+            s += f"{x} | "
+            for y in range(0, self._q):
+                s += f"{m[x][y]} "
+            s += "\n"
+
+        s = header + s
+        print(s)
+
+
+ + + +
+ + + + + + + + + + +
+ + +

+ add() + +

+ + +
+ +

This function do the operation + on the Galois Field

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a list of the group with the addition operation

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
def add(self):
+    """
+        This function do the operation + on the Galois Field
+
+        Returns:
+            Return a list of the group with the addition operation
+    """
+    for x in range(0, self._q):
+        for y in range(0, self._q):
+            self._add[x][y] = (x + y) % self._q
+    return self._add
+
+
+
+ +
+ +
+ + +

+ check_closure_law(arithmetic) + +

+ + +
+ +

This function check the closure law. +By definition, every element in the GF is an abelian group, which respect the closure law: for a and b belongs to G, a + b belongs to G, the operation is a binary operation

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + Arithmetics + (str) + – +
    +

    contain the operation to be made, must be '+', '*', '/' or /-'

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
135
+136
+137
+138
+139
+140
+141
+142
+143
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+149
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+161
+162
+163
+164
+165
+166
+167
+168
+169
+170
+171
+172
def check_closure_law(self, arithmetic):
+    """
+        This function check the closure law.
+        By definition, every element in the GF is an abelian group, which respect the closure law: for a and b belongs to G, a + b belongs to G, the operation is a binary operation
+
+        Args:
+            Arithmetics (str): contain the operation to be made, must be '+', '*', '/' or /-'
+    """
+    if arithmetic not in ['+', '*', '/', '-']:
+        raise Exception("The arithmetic need to be '+', '*', '/' or '-'")
+
+    if arithmetic == '+':
+        G = self._add
+    elif arithmetic == '*':
+        G = self._mul
+    elif arithmetic == '/':
+        G = self._div
+    else:
+        G = self._sub
+
+    start = 0
+    """
+        In case of multiplicative, we bypass the first line, because all elements are zero, otherwise the test fail
+    """
+    if arithmetic == '*' or arithmetic == '/':
+        start = 1
+
+    isClosure = True
+    for x in range(start, self._q):
+        gr = Group(self._q, G[x], self._operation)
+        if not gr.closure():
+            isClosure = False
+        del gr
+
+    if isClosure:
+        print(f"The group {arithmetic} respect closure law")
+    else:
+        print(f"The group {arithmetic} does not respect closure law")
+
+
+
+ +
+ +
+ + +

+ check_identity_add() + +

+ + +
+ +

This function check the identity element and must satisfy this condition: $a + 0 = a$ for each element in the GF(n) +In Group Theory, an identity element is an element in the group which do not change the value every element in the group

+

Returns:

+ + +
+ Source code in Cryptotools/Groups/galois.py + 
174
+175
+176
+177
+178
+179
+180
+181
+182
+183
+184
+185
+186
+187
def check_identity_add(self):
+    """
+        This function check the identity element and must satisfy this condition: $a + 0 = a$ for each element in the GF(n)
+        In Group Theory, an identity element is an element in the group which do not change the value every element in the group
+
+        Returns:
+
+    """
+    for x in self._F:
+        if not self._identityAdd + x == x:
+            raise Exception(
+                    f"The identity element {self._identityAdd} "\
+                     "do not satisfy $a + element = a$"
+            )
+
+
+
+ +
+ +
+ + +

+ check_identity_mul() + +

+ + +
+ +

This function check the identity element and must satisfy this condition: $a * 1 = a$ for each element in the GF(n) +In Group Theory, an identity element is an element in the group which do not change the value every element in the group

+

Returns:

+ + +
+ Source code in Cryptotools/Groups/galois.py + 
189
+190
+191
+192
+193
+194
+195
+196
+197
+198
+199
+200
+201
+202
def check_identity_mul(self):
+    """
+        This function check the identity element and must satisfy this condition: $a * 1 = a$ for each element in the GF(n)
+        In Group Theory, an identity element is an element in the group which do not change the value every element in the group
+
+        Returns:
+
+    """
+    for x in self._F:
+        if not self._identityMul * x == x:
+            raise Exception(
+                    f"The identity element {self._identityAdd} "\
+                     "do not satisfy $a * element = a$"
+            )
+
+
+
+ +
+ +
+ + +

+ div() + +

+ + +
+ +

This function do the operation / on the Galois Field

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a list of the group with the division operation

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
 97
+ 98
+ 99
+100
+101
+102
+103
+104
+105
+106
+107
+108
+109
def div(self):
+    """
+        This function do the operation / on the Galois Field
+
+        Returns:
+            Return a list of the group with the division operation
+    """
+    for x in range(1, self._q):
+        for y in range(1, self._q):
+            inv = self._inverseModular(y, self._q)
+
+            self._div[x][y] = (x * inv) % self._q
+    return self._div
+
+
+
+ +
+ +
+ + +

+ getPrimitiveRoot() + +

+ + +
+ +

Return the list of primitives root

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return the primitive root

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
51
+52
+53
+54
+55
+56
+57
+58
def getPrimitiveRoot(self):
+    """
+        Return the list of primitives root
+
+        Returns:
+            Return the primitive root
+    """
+    return self._primitiveRoot
+
+
+
+ +
+ +
+ + +

+ isPrimitiveRoot(z, length) + +

+ + +
+ +

Check if z is a primitive root

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + z + (list) + – +
    +

    check if z is a primitive root

    +
    +
    • +
  • + length + (Integer) + – +
    +

    Length of the GF(q)

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
def isPrimitiveRoot(self, z, length):
+    """
+        Check if z is a primitive root 
+
+        Args:
+            z (list): check if z is a primitive root
+            length (Integer): Length of the GF(q)
+    """
+    if len(z) == length:
+        return True
+    return False
+
+
+
+ +
+ +
+ + +

+ mul() + +

+ + +
+ +

This function do the operation * on the Galois Field

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a list of the group with the multiplication operation

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
111
+112
+113
+114
+115
+116
+117
+118
+119
+120
+121
def mul(self):
+    """
+        This function do the operation * on the Galois Field
+
+        Returns:
+            Return a list of the group with the multiplication operation
+    """
+    for x in range(0, self._q):
+        for y in range(0, self._q):
+            self._mul[x][y] = (x * y) % self._q
+    return self._mul
+
+
+
+ +
+ +
+ + +

+ primitiveRoot() + +

+ + +
+ +

In this function, we going to find the primitive root modulo n of the galois field

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return the list of primitive root of the GF(q)

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
def primitiveRoot(self):
+    """
+        In this function, we going to find the primitive root modulo n of the galois field
+
+        Returns:
+           Return the list of primitive root of the GF(q) 
+    """
+    for x in range(2, self._q):
+        z = list()
+        for entry in range(1, self._q):
+            res = self._operation(x, entry, self._q)
+            if res not in z:
+                z.append(res)
+
+        if self.isPrimitiveRoot(z, self._q - 1):
+            if x not in self._primitiveRoot: # It's dirty, need to find why we have duplicate entry
+                self._primitiveRoot.append(x)
+    return z
+
+
+
+ +
+ +
+ + +

+ printMatrice(m) + +

+ + +
+ +

This function print the GF(m)

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + m + (list) + – +
    +

    Matrix of the GF

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
204
+205
+206
+207
+208
+209
+210
+211
+212
+213
+214
+215
+216
+217
+218
+219
+220
+221
+222
+223
+224
+225
+226
+227
def printMatrice(self, m):
+    """
+        This function print the GF(m)
+
+        Args:
+            m (list): Matrix of the GF
+    """
+    header = str()
+    header = "    "
+    for x in range(0, self._q):
+        header += f"{x} "
+
+    header += "\n--|" + "-" * (len(header)- 3) +"\n"
+
+    s = str()
+
+    for x in range(0, self._q):
+        s += f"{x} | "
+        for y in range(0, self._q):
+            s += f"{m[x][y]} "
+        s += "\n"
+
+    s = header + s
+    print(s)
+
+
+
+ +
+ +
+ + +

+ sub() + +

+ + +
+ +

This function do the operation - on the Galois Field

+ + + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a list of the group with the subtraction operation

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Groups/galois.py + 
123
+124
+125
+126
+127
+128
+129
+130
+131
+132
+133
def sub(self):
+    """
+        This function do the operation - on the Galois Field
+
+        Returns:
+            Return a list of the group with the subtraction operation
+    """
+    for x in range(0, self._q):
+        for y in range(0, self._q):
+            self._sub[x][y] = (x - y) % self._q
+    return self._sub
+
+
+
+ +
+ + + +
+ +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+ +
+ +
+ +
+ + + + « Previous + + + Next » + + +
+ + + + + + + + diff --git a/site/img/favicon.ico b/site/img/favicon.ico new file mode 100644 index 0000000..e85006a Binary files /dev/null and b/site/img/favicon.ico differ diff --git a/site/index.html b/site/index.html new file mode 100644 index 0000000..b2600a2 --- /dev/null +++ b/site/index.html @@ -0,0 +1,148 @@ + + + + + + + + CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Welcome to CryptoTools
    • +
  • +
    • +
+
+
+
+
+ +

Welcome to CryptoTools

+ + +
+
+ +
+
+ +
+ +
+ +
+ + + + + +
+ + + + + + + + + + diff --git a/site/installation/index.html b/site/installation/index.html new file mode 100644 index 0000000..0af617d --- /dev/null +++ b/site/installation/index.html @@ -0,0 +1,149 @@ + + + + + + + + Installation - CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Installation
    • +
  • +
    • +
+
+
+
+
+ +

Installation

+

To install the project, you should install from my own Python repository but, you first need to deploy your virtual enviroment.

+

Installation from source

+

You can install from the source. Download the project and install it in your virtual environment:

+
$ virtualenv ~/venv/cryptotools
+$ source ~/venv/cryptotools/bin/activate
+
+
$ git clone https://gitea.bucchino.org/gbucchino/cryptotools.git
+$ cd cryptotools
+$ python3 setup.py install
+
+

The installation is completed. You can know use Cryptotools package into your project. In the directory examples you may find some examples scripts

+ +
+
+ +
+
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+ +
+ +
+ + + + « Previous + + + Next » + + +
+ + + + + + + + diff --git a/site/introduction/index.html b/site/introduction/index.html new file mode 100644 index 0000000..245b451 --- /dev/null +++ b/site/introduction/index.html @@ -0,0 +1,136 @@ + + + + + + + + Introduction - CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Introduction
    • +
  • +
    • +
+
+
+
+
+ +

CryptoTools

+

CryptoTools is a Python package that provides low-level cryptographic primitives for generating strong numbers. With this project, it's possible to generate public key cryptosystems such as RSA. +This project has a academic purpose and can not be used in production enviroment yet.

+

So far, my cryptographic modules are not compliant with FIPS 140-3 but in the future, that will be.

+ +
+
+ +
+
+ +
+ +
+ +
+ + + + + Next » + + +
+ + + + + + + + diff --git a/site/js/html5shiv.min.js b/site/js/html5shiv.min.js new file mode 100644 index 0000000..1a01c94 --- /dev/null +++ b/site/js/html5shiv.min.js @@ -0,0 +1,4 @@ +/** +* @preserve HTML5 Shiv 3.7.3 | @afarkas @jdalton @jon_neal @rem | MIT/GPL2 Licensed +*/ +!function(a,b){function c(a,b){var c=a.createElement("p"),d=a.getElementsByTagName("head")[0]||a.documentElement;return c.innerHTML="x",d.insertBefore(c.lastChild,d.firstChild)}function d(){var a=t.elements;return"string"==typeof a?a.split(" "):a}function e(a,b){var c=t.elements;"string"!=typeof c&&(c=c.join(" ")),"string"!=typeof a&&(a=a.join(" ")),t.elements=c+" "+a,j(b)}function f(a){var b=s[a[q]];return b||(b={},r++,a[q]=r,s[r]=b),b}function g(a,c,d){if(c||(c=b),l)return c.createElement(a);d||(d=f(c));var e;return e=d.cache[a]?d.cache[a].cloneNode():p.test(a)?(d.cache[a]=d.createElem(a)).cloneNode():d.createElem(a),!e.canHaveChildren||o.test(a)||e.tagUrn?e:d.frag.appendChild(e)}function 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+ + +
+ +
+
+
    +
    • +
  • Low-level cryptographic
    • +
  • Number theory
    • +
  • +
    • +
+
+
+
+
+ +

Number Theory

+

For generating keys, we need strong prime number, they are the basis. CryptoTools provides functions for generating these numbers.

+

prime numbers

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ are_coprime(p1, p2) + +

+ + +
+ +

This function check if p1 and p2 are coprime

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + p1 + (list) + – +
    +

    list of prime numbers of the first number

    +
    +
    • +
  • + p2 + (list) + – +
    +

    list of prime number of the second number

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a boolean result

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
153
+154
+155
+156
+157
+158
+159
+160
+161
+162
+163
+164
+165
+166
+167
+168
+169
def are_coprime(p1, p2):
+    """
+        This function check if p1 and p2 are coprime
+
+        Args:
+            p1 (list): list of prime numbers of the first number
+            p2 (list): list of prime number of the second number
+
+        Returns:
+            Return a  boolean result
+    """
+    r = True
+    for entry in p1:
+        if entry in p2:
+            r = False
+            break
+    return r
+
+
+
+ +
+ +
+ + +

+ getPrimeNumber(n, safe=True) + +

+ + +
+ +

This function generate a large prime number +based on "A method for Obtaining Digital Signatures +and Public-Key Cryptosystems" +Section B. How to Find Large Prime Numbers +https://dl.acm.org/doi/pdf/10.1145/359340.359342

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    The size of the prime number. Must be a multiple of 64

    +
    +
    • +
  • + safe + (Boolean, default: + True +) + – +
    +

    When generating the prime number, the function must find a safe prime number, based on the Germain primes (2p + 1 is also a prime number)

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return the prime number

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
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+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
def getPrimeNumber(n, safe=True):
+    """
+        This function generate a large prime number
+        based on "A method for Obtaining Digital Signatures
+        and Public-Key Cryptosystems"
+        Section B. How to Find Large Prime Numbers
+        https://dl.acm.org/doi/pdf/10.1145/359340.359342
+
+        Args:
+            n (Integer): The size of the prime number. Must be a multiple of 64
+            safe (Boolean): When generating the prime number, the function must find a safe prime number, based on the Germain primes (2p + 1 is also a prime number)
+
+        Returns:
+            Return the prime number
+    """
+    if n % 64 != 0:
+        print("Must be multiple of 64")
+        return
+    # from sys import getsizeof
+    upper = getrandbits(n) 
+    lower = upper >> 7
+    r = randint(lower, upper)
+
+    while r % 2 == 0:
+        r += 1
+
+    # Now, we are going to compute r as prime number
+    i = 100
+    while 1:
+        # Check if it's a prime number
+        if _millerRabinTest(r):
+            break
+
+        # TODO: it's dirty, need to change that
+        # i = int(log2(r))
+        # i *= randint(2, 50)
+
+        r += i
+        # print(f"{i} {r}")
+        i += 1
+
+
+    # print(_fermatLittleTheorem(r))
+    # print(getsizeof(r))
+    return r
+
+
+
+ +
+ +
+ + +

+ getSmallPrimeNumber(n) + +

+ + +
+ +

This function is deprecated

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    Get the small prime number until n

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + int + – +
    +

    Return the first prime number found

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
 81
+ 82
+ 83
+ 84
+ 85
+ 86
+ 87
+ 88
+ 89
+ 90
+ 91
+ 92
+ 93
+ 94
+ 95
+ 96
+ 97
+ 98
+ 99
+100
+101
+102
+103
+104
+105
+106
def getSmallPrimeNumber(n) -> int:
+    """
+        This function is deprecated
+
+        Args:
+            n (Integer): Get the small prime number until n
+
+        Returns:
+            Return the first prime number found
+    """
+    is_prime = True
+
+    while True:  
+        for i in range(2, n):
+            # If n is divisible by i, it's not a prime, we break the loop
+            if n % i == 0:
+                is_prime = False
+                break
+
+        if is_prime:
+            break
+        is_prime = True
+        n = n + 1
+
+    p = n
+    return p
+
+
+
+ +
+ +
+ + +

+ get_n_prime_numbers(n) + +

+ + +
+ +

This function return a list of n prime numbers

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    the range, must be an integer

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return a list of n prime numbers

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
124
+125
+126
+127
+128
+129
+130
+131
+132
+133
+134
+135
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+142
+143
+144
+145
+146
+147
+148
+149
+150
+151
def get_n_prime_numbers(n) -> list:
+    """
+        This function return a list of n prime numbers
+
+        Args:
+            n (Integer): the range, must be an integer
+
+        Returns:
+            Return a list of n prime numbers
+    """
+    l = list()
+
+    count = 2
+    index = 0
+    while index < n:
+        is_prime = True
+        for x in range(2, count):
+            if count % x == 0:
+                is_prime = False
+                break
+
+        if is_prime:
+            l.append(count)
+            index += 1
+
+        count += 1
+
+    return l
+
+
+
+ +
+ +
+ + +

+ get_prime_numbers(n) + +

+ + +
+ +

This function get all prime number of the n

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    find all prime number until n is achieves

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return a list of prime numbers

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
108
+109
+110
+111
+112
+113
+114
+115
+116
+117
+118
+119
+120
+121
+122
def get_prime_numbers(n) -> list:
+    """
+        This function get all prime number of the n
+
+        Args:
+            n (Integer): find all prime number until n is achieves
+
+        Returns:
+         Return a list of prime numbers
+    """
+    l = list()
+    for i in range(2, n):
+        if n % i == 0:
+            l.append(i)
+    return l
+
+
+
+ +
+ +
+ + +

+ isPrimeNumber(p) + +

+ + +
+ +

Check if the number p is a prime number or not. The function iterate until p is achieve.

+

This function is not the efficient way to determine if p is prime or a composite.

+

To identify if p is prime or not, the function check the result of the Euclidean Division has a remainder. If yes, it's means it's not a prime number

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + p + (Integer) + – +
    +

    number if possible prime or not

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + – +
    +

    Return a boolean if the number p is a prime number or not. True if yes

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
 8
+ 9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
def isPrimeNumber(p):
+    """
+        Check if the number p is a prime number or not. The function iterate until p is achieve.
+
+        This function is not the efficient way to determine if p is prime or a composite.
+
+        To identify if p is prime or not, the function check the result of the Euclidean Division has a remainder. If yes, it's means it's not a prime number
+
+        Args:
+            p (Integer): number if possible prime or not
+
+        Returns:
+            Return a boolean if the number p is a prime number or not. True if yes
+    """
+    for i in range(2, p):
+        if p % i == 0:
+            return False
+    return True
+
+
+
+ +
+ +
+ + +

+ isSafePrime(n) + +

+ + +
+ +

This function has not been implemented yet, but check if the number n is a safe prime number. This function will test different properties of the possible prime number n

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    the prime number to check

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return a Boolean if the prime number n is safe or not. True if yes, otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
272
+273
+274
+275
+276
+277
+278
+279
+280
+281
+282
+283
+284
+285
+286
+287
def isSafePrime(n) -> bool:
+    """
+        This function has not been implemented yet, but check if the number n is a safe prime number. This function will test different properties of the possible prime number n
+
+        Args:
+            n (Integer): the prime number to check
+
+        Returns:
+            Return a Boolean if the prime number n is safe or not. True if yes, otherwise it's False
+    """
+    if n.bit_length() >= 256:
+        return True
+
+    # Do Sophie Germain's test
+
+    return False
+
+
+
+ +
+ +
+ + +

+ sieveOfEratosthenes(n) + +

+ + +
+ +

This function build a list of prime number based on the Sieve of Erastosthenes

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (Integer) + – +
    +

    Interate until n is achives

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return a list of all prime numbers to n

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
171
+172
+173
+174
+175
+176
+177
+178
+179
+180
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+188
+189
+190
+191
+192
+193
+194
+195
+196
+197
+198
def sieveOfEratosthenes(n) -> list:
+    """
+        This function build a list of prime number based on the Sieve of Erastosthenes
+
+        Args:
+            n (Integer): Interate until n is achives
+
+        Returns:
+            Return a list of all prime numbers to n
+    """
+    if n < 1:
+        return list()
+
+    eratost = dict()
+    for i in range(2, n):
+        eratost[i] = True
+
+    for i in range(2, n):
+        if eratost[i]:
+            for j in range(i*i, n, i):
+                eratost[j] = False
+
+    sieve = list()
+    for i in range(2, n):
+        if eratost[i]:
+            sieve.append(i)
+
+    return sieve
+
+
+
+ +
+ +
+ + +

+ sophieGermainPrime(p) + +

+ + +
+ +

Check if the number p is a safe prime number: 2p + 1 is also a prime

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + p + (Integer) + – +
    +

    Possible prime number

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + bool + – +
    +

    Return True if p is a safe prime number, otherwise it's False

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/primeNumber.py + 
259
+260
+261
+262
+263
+264
+265
+266
+267
+268
+269
+270
def sophieGermainPrime(p) -> bool:
+    """
+        Check if the number p is a safe prime number: 2p + 1 is also a prime
+
+        Args:
+            p (Integer):  Possible prime number
+
+        Returns:
+            Return True if p is a safe prime number, otherwise it's False
+    """
+    pp = 2 * p + 1
+    return _millerRabinTest(pp)
+
+
+
+ +
+ + + +
+ +
+ +

Fibonacci sequence

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + + +
+ + +

+ fibonacci(n) + +

+ + +
+ +

This function generate the list of Fibonacci

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + n + (integer) + – +
    +

    it's maximum value of Fibonacci sequence

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return a list of n element in the Fibonacci sequence

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Numbers/numbers.py + 
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+ 5
+ 6
+ 7
+ 8
+ 9
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+17
+18
def fibonacci(n) -> list:
+    """
+        This function generate the list of Fibonacci
+
+        Args:
+            n (integer): it's maximum value of Fibonacci sequence
+
+        Returns:
+            Return a list of n element in the Fibonacci sequence
+    """
+    fibo = [0, 1]
+    fibo.append(fibo[0] + fibo[1])
+    for i in range(2, n):
+        fibo.append(fibo[i - 1] + fibo[i])
+    return fibo
+
+
+
+ +
+ + + +
+ +
+ +
+ +
+
+ +
+
+ +
+ +
+ +
+ + + + « Previous + + + Next » + + +
+ + + + + + + + diff --git a/site/objects.inv b/site/objects.inv new file mode 100644 index 0000000..429a915 Binary files /dev/null and b/site/objects.inv differ diff --git a/site/rsa/index.html b/site/rsa/index.html new file mode 100644 index 0000000..4ac228f --- /dev/null +++ b/site/rsa/index.html @@ -0,0 +1,1086 @@ + + + + + + + + RSA - CryptoTools documentation + + + + + + + + + + + + + + +
+ + +
+ +
+
+
    +
    • +
  • Public Keys
    • +
  • RSA
    • +
  • +
    • +
+
+
+
+
+ +

CryptoTools provides several functions for generating a RSA Keys

+ + +
+ + + + +
+ + + + + + + + + + +
+ + + + + + + + + +
+ + + +

+ RSA + + +

+ + +
+ + +

This class generate public key based on RSA algorithm

+ + + + + + + + + + + + + +
Attributes: +
    +
  • + p + – +
    +

    it's the prime number for generating the modulus

    +
    +
    • +
  • + q + – +
    +

    it's the second prime number for the modulus

    +
    +
    • +
  • + public + – +
    +

    Object of the RSAKey which is the public key

    +
    +
    • +
  • + private + – +
    +

    Object of the RSAKey for the private key

    +
    +
    • +
+
+ + + + + + + +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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class RSA:
+    """
+        This class generate public key based on RSA algorithm
+
+        Attributes:
+            p: it's the prime number for generating the modulus
+            q: it's the second prime number for the modulus
+            public: Object of the RSAKey which is the public key
+            private: Object of the RSAKey for the private key
+    """
+
+    def __init__(self):
+        """
+            Build a RSA Key
+        """
+        self._p = None
+        self._q = None
+        self._public = None
+        self._private = None
+
+    def generateKeys(self, size=512):
+        """
+            This function generate both public and private keys
+            Args:
+                size: It's the size of the key and must be multiple of 64
+        """
+        # p and q must be coprime
+        self._p = getPrimeNumber(size)
+        self._q = getPrimeNumber(size)
+
+        # compute n = pq
+        n = self._p * self._q
+
+        phin = (self._p - 1) * (self._q - 1)
+
+        # e must be coprime with phi(n)
+        # According to the FIPS 186-5, the public key exponent must be odd
+        # and the minimum size is 65536 (Cf. Section 5.4 PKCS #1)
+        # https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-5.pdf
+        e = 65535 # Works
+        for i in range(2, phin - 1):
+            if gcd(phin, e) == 1:
+                break
+            e += 1
+        # print(gcd(phin, e))
+        self._public = RSAKey(e, n, getsizeof(e))
+
+        # d is the reverse modulo of phi(n)
+        # d = self._inverseModular(e, phin)
+        d = pow(e, -1, phin) # Works in python 3.8
+        self._private = RSAKey(d, n, getsizeof(d))
+
+    @property
+    def e(self):
+        return self._public.key
+
+    @property
+    def d(self):
+        return self._private.key
+
+    @property
+    def n(self):
+        return self._public.modulus
+
+    @property
+    def p(self):
+        return self._p
+
+    @property
+    def q(self):
+        return self._q
+
+    def encrypt(self, data) -> list:
+        """
+            This function return a list of data encrypted with the public key
+
+            Args:
+                data (str): it's the plaintext which need to be encrypted
+
+            Returns:
+                return a list of the data encrypted, each entries contains the value encoded
+        """
+        dataEncoded = self._str2bin(data)
+        return list(
+            pow(int(x), self._public.key, self._public.modulus)
+            for x in dataEncoded
+        )
+
+    def decrypt(self, data) -> list:
+        """
+            This function return a list decrypted with the private key
+
+            Args:
+                data (str): It's the encrypted data which need to be decrypted
+
+            Returns:
+                Return the list of data uncrypted into plaintext
+        """
+        decrypted = list()
+        for x in data:
+            d = pow(x, self._private.key, self._private.modulus)
+            decrypted.append(chr(d))
+        return ''.join(decrypted)
+
+    def _str2bin(self, data) -> list:
+        """
+            This function convert a string into the unicode value
+
+            Args:
+                data: the string which need to be converted
+
+            Returns:
+                Return a list of unicode values of data
+        """
+        return list(ord(x) for x in data)
+
+    def _inverseModular(self, a, n):
+        """
+            This function compute the modular inverse for finding d, the decryption key
+
+            Args:
+                a (Integer): the base of the exponent
+                n (Integer): the modulus
+        """
+        for b in range(1, n):
+            #if pow(a, b, n) == 1:
+            if (a * b) % n == 1:
+                inv = b
+                break
+        return inv
+
+
+ + + +
+ + + + + + + + + + +
+ + +

+ __init__() + +

+ + +
+ +

Build a RSA Key

+ + +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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+63
def __init__(self):
+    """
+        Build a RSA Key
+    """
+    self._p = None
+    self._q = None
+    self._public = None
+    self._private = None
+
+
+
+ +
+ +
+ + +

+ decrypt(data) + +

+ + +
+ +

This function return a list decrypted with the private key

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + data + (str) + – +
    +

    It's the encrypted data which need to be decrypted

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    Return the list of data uncrypted into plaintext

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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def decrypt(self, data) -> list:
+    """
+        This function return a list decrypted with the private key
+
+        Args:
+            data (str): It's the encrypted data which need to be decrypted
+
+        Returns:
+            Return the list of data uncrypted into plaintext
+    """
+    decrypted = list()
+    for x in data:
+        d = pow(x, self._private.key, self._private.modulus)
+        decrypted.append(chr(d))
+    return ''.join(decrypted)
+
+
+
+ +
+ +
+ + +

+ encrypt(data) + +

+ + +
+ +

This function return a list of data encrypted with the public key

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + data + (str) + – +
    +

    it's the plaintext which need to be encrypted

    +
    +
    • +
+
+ + + + + + + + + + + + +
Returns: +
    +
  • + list + – +
    +

    return a list of the data encrypted, each entries contains the value encoded

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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def encrypt(self, data) -> list:
+    """
+        This function return a list of data encrypted with the public key
+
+        Args:
+            data (str): it's the plaintext which need to be encrypted
+
+        Returns:
+            return a list of the data encrypted, each entries contains the value encoded
+    """
+    dataEncoded = self._str2bin(data)
+    return list(
+        pow(int(x), self._public.key, self._public.modulus)
+        for x in dataEncoded
+    )
+
+
+
+ +
+ +
+ + +

+ generateKeys(size=512) + +

+ + +
+ +

This function generate both public and private keys +Args: + size: It's the size of the key and must be multiple of 64

+ + +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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def generateKeys(self, size=512):
+    """
+        This function generate both public and private keys
+        Args:
+            size: It's the size of the key and must be multiple of 64
+    """
+    # p and q must be coprime
+    self._p = getPrimeNumber(size)
+    self._q = getPrimeNumber(size)
+
+    # compute n = pq
+    n = self._p * self._q
+
+    phin = (self._p - 1) * (self._q - 1)
+
+    # e must be coprime with phi(n)
+    # According to the FIPS 186-5, the public key exponent must be odd
+    # and the minimum size is 65536 (Cf. Section 5.4 PKCS #1)
+    # https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-5.pdf
+    e = 65535 # Works
+    for i in range(2, phin - 1):
+        if gcd(phin, e) == 1:
+            break
+        e += 1
+    # print(gcd(phin, e))
+    self._public = RSAKey(e, n, getsizeof(e))
+
+    # d is the reverse modulo of phi(n)
+    # d = self._inverseModular(e, phin)
+    d = pow(e, -1, phin) # Works in python 3.8
+    self._private = RSAKey(d, n, getsizeof(d))
+
+
+
+ +
+ + + +
+ +
+ +
+ +
+ + + +

+ RSAKey + + +

+ + +
+ + +

This class store the RSA key with the modulus associated +The key is a tuple of the key and the modulus n

+ + + + + + + + + + + + + +
Attributes: +
    +
  • + key + – +
    +

    It's the exponent key, can be public or private

    +
    +
    • +
  • + modulus + – +
    +

    It's the public modulus

    +
    +
    • +
  • + length + – +
    +

    It's the key length

    +
    +
    • +
+
+ + + + + + + +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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class RSAKey:
+    """
+        This class store the RSA key with the modulus associated
+        The key is a tuple of the key and the modulus n
+
+        Attributes:
+            key: It's the exponent key, can be public or private
+            modulus: It's the public modulus
+            length: It's the key length
+    """
+    def __init__(self, key, modulus, length):
+        """
+            Contain the RSA Key. An object of RSAKey can be a public key or a private key
+
+            Args:
+                key (Integer): it's the exponent of the key
+                modulus (Integer): it's the public modulus of the key
+                length (Integer): length of the key
+        """
+
+        self._key     = key
+        self._modulus = modulus
+        self._length  = length
+
+    @property
+    def key(self):
+        return self._key
+
+    @property
+    def modulus(self):
+        return self._modulus
+
+    @property
+    def length(self):
+        return self.length
+
+
+ + + +
+ + + + + + + + + + +
+ + +

+ __init__(key, modulus, length) + +

+ + +
+ +

Contain the RSA Key. An object of RSAKey can be a public key or a private key

+ + + + + + + + + + + + + +
Parameters: +
    +
  • + key + (Integer) + – +
    +

    it's the exponent of the key

    +
    +
    • +
  • + modulus + (Integer) + – +
    +

    it's the public modulus of the key

    +
    +
    • +
  • + length + (Integer) + – +
    +

    length of the key

    +
    +
    • +
+
+ +
+ Source code in Cryptotools/Encryptions/RSA.py + 
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def __init__(self, key, modulus, length):
+    """
+        Contain the RSA Key. An object of RSAKey can be a public key or a private key
+
+        Args:
+            key (Integer): it's the exponent of the key
+            modulus (Integer): it's the public modulus of the key
+            length (Integer): length of the key
+    """
+
+    self._key     = key
+    self._modulus = modulus
+    self._length  = length
+
+
+
+ +
+ + + +
+ +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+ +
+ +
+ +
+ + + + « Previous + + + Next » + + +
+ + + + + + + + diff --git a/site/sitemap.xml b/site/sitemap.xml new file mode 100644 index 0000000..0f8724e --- /dev/null +++ b/site/sitemap.xml @@ -0,0 +1,3 @@ + + + \ No newline at end of file diff --git a/site/sitemap.xml.gz b/site/sitemap.xml.gz new file mode 100644 index 0000000..d0bfde9 Binary files /dev/null and b/site/sitemap.xml.gz differ diff --git a/tests/carmi_function_oeis b/tests/carmi_function_oeis new file mode 100644 index 0000000..d42e74e --- /dev/null +++ b/tests/carmi_function_oeis @@ -0,0 +1 @@ +1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 4, 16, 6, 18, 4, 6, 10, 22, 2, 20, 12, 18, 6, 28, 4, 30, 8, 10, 16, 12, 6, 36, 18, 12, 4, 40, 6, 42, 10, 12, 22, 46, 4, 42, 20, 16, 12, 52, 18, 20, 6, 18, 28, 58, 4, 60, 30, 6, 16, 12, 10, 66, 16, 22, 12, 70, 6, 72, 36, 20, 18, 30, 12, 78, 4, 54 diff --git a/tests/carmi_numbers_oeis b/tests/carmi_numbers_oeis new file mode 100644 index 0000000..578e3db --- /dev/null +++ b/tests/carmi_numbers_oeis @@ -0,0 +1 @@ +561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721 diff --git a/tests/execute_tests.py b/tests/execute_tests.py new file mode 100755 index 0000000..3b6afcc --- /dev/null +++ b/tests/execute_tests.py @@ -0,0 +1,8 @@ +#!/usr/bin/bash + +set -e +source /home/geoffrey/venv/forensic/bin/activate; +for test in `ls tests/test_*.py`; do + echo $test; + python3 $test -v; +done diff --git a/tests/fibonacci_oeis b/tests/fibonacci_oeis new file mode 100644 index 0000000..a5b2ef4 --- /dev/null +++ b/tests/fibonacci_oeis @@ -0,0 +1 @@ +0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155 diff --git a/tests/phi_oeis b/tests/phi_oeis new file mode 100644 index 0000000..52a3679 --- /dev/null +++ b/tests/phi_oeis @@ -0,0 +1 @@ +1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44 diff --git a/tests/prime_numbers_oeis b/tests/prime_numbers_oeis new file mode 100644 index 0000000..ecb82d0 --- /dev/null +++ b/tests/prime_numbers_oeis @@ -0,0 +1 @@ +2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 diff --git a/tests/test_breaking_rsa.py b/tests/test_breaking_rsa.py new file mode 100644 index 0000000..e85677a --- /dev/null +++ b/tests/test_breaking_rsa.py @@ -0,0 +1,59 @@ +#!/usr/bin/env python3 + +import unittest +from Cryptotools.Numbers.primeNumber import isPrimeNumber +from math import sqrt, isqrt, ceil +from random import choice + + +class TestBreakingRSA(unittest.TestCase): + def test_breaking_rsa(self): + # primes = self._list_primes() + #p = choice(primes) + #q = choice(primes) + self._breaking_rsa(7901, 7817) + self._breaking_rsa(7907, 7919) + self._breaking_rsa(7103, 7127) # Works + # self._breaking_rsa(7103, 7121) # Doesn't works + + def _breaking_rsa(self, p, q): + # print(f"p = {p}") + # print(f"q = {q}") + n = p * q + # print(f"n = {n}") + + # a = isqrt(n) + 1 + a = ceil(sqrt(n)) + iteration = 1 + while True: + b2 = (a ** 2) - n + sqb2 = sqrt(b2) + if b2 % 2 == 0.0: + b = sqrt(b2) + break + a = a + 1 + iteration += 1 + # print(f"Iteration: {iteration}") + # print(f"a = {a}") + # print(f"b2 = {b2}") + # print(f"b = {b}") + p = int((a + b)) + q = int((a - b)) + #N = (a + b) * (a - b) + N = p * q + # print(isPrimeNumber(p)) + # print(isPrimeNumber(q)) + # print(N) + # print() + self.assertTrue(N == n) + + def _list_primes(self): + l = list() + i = 100 # We start at 100 + while (len(l) < 1000): + if isPrimeNumber(i): + l.append(i) + i = i + 1 + return l + +unittest.main() diff --git a/tests/test_carmi.py b/tests/test_carmi.py new file mode 100644 index 0000000..22f8b94 --- /dev/null +++ b/tests/test_carmi.py @@ -0,0 +1,44 @@ +import unittest + +from Cryptotools.Numbers.carmi import carmi_numbers, is_carmichael, carmichael_lambda + + +class TestCarmichael(unittest.TestCase): + def _generate_carmi(self): + # Source: https://oeis.org/A002322 + with open("tests/carmi_function_oeis", "r") as f: + data = f.readlines() + carmi_s = data[0].split(",") + carmi = list() + for entry in carmi_s: + carmi.append(entry.strip()) + return carmi + + def _generate_carmi_numbers(self): + # Source: https://oeis.org/A002997 + with open("tests/carmi_numbers_oeis", "r") as f: + data = f.readlines() + carmi_s = data[0].split(",") + carmi = list() + for entry in carmi_s: + carmi.append(entry.strip()) + return carmi # len 69 + + def test_carmi_lambda(self): + carmi_list = self._generate_carmi() + index = 2 + for i in range(2, len(carmi_list)): + res = carmichael_lambda(index) + print(f"{res} {carmi_list[i]}") + index += 1 + # self.assertEqual(res, carmi_list[i], "Wrong value") + + def test_carmi_number(self): + pass + #carmi_list = self._generate_carmi_numbers() + #for i in range(0, len(carmi_list) - 10): + # value = int(carmi_list[i]) + # l = carmi_numbers(value) + # r = is_carmichael(value, l) + # self.assertTrue(r) +unittest.main() diff --git a/tests/test_numbers.py b/tests/test_numbers.py new file mode 100644 index 0000000..14e49dd --- /dev/null +++ b/tests/test_numbers.py @@ -0,0 +1,22 @@ +import unittest + +from Cryptotools.Numbers.numbers import fibonacci + +class TestNumbers(unittest.TestCase): + def _generate_fibonacci(self): + # Source https://oeis.org/A000045 + with open("tests/fibonacci_oeis", "r") as f: + data = f.readlines() + fibo_s = data[0].split(",") + fibo = list() + for entry in fibo_s: + fibo.append(int(entry.strip())) + return fibo + + def test_fibonacci(self): + fibo_oeis = self._generate_fibonacci() + fibo = fibonacci(40) + for i in range(0, len(fibo_oeis)): + self.assertEqual(fibo[i], fibo_oeis[i], "Wrong value") + +unittest.main() diff --git a/tests/test_phi.py b/tests/test_phi.py new file mode 100644 index 0000000..38cedb9 --- /dev/null +++ b/tests/test_phi.py @@ -0,0 +1,26 @@ +import unittest + +from Cryptotools.Numbers.coprime import phi + +class TestPhi(unittest.TestCase): + def _generate_phi(self): + # Source: https://oeis.org/A000010 + with open("tests/phi_oeis", "r") as f: + data = f.readlines() + phi_s = data[0].split(",") + phi = list() + for entry in phi_s: + phi.append(int(entry.strip())) + return phi # len 69 + + def test_phi(self): + phi_list = self._generate_phi() + index = 0 + for i in range(1, len(phi_list) + 1): + res = phi(i) + # print(f"{i} {res}") + value = phi_list[index] + # print(f"{res}: {value}") + self.assertEqual(res, value, "Wrong value") + index += 1 +unittest.main() diff --git a/tests/test_primes.py b/tests/test_primes.py new file mode 100644 index 0000000..f373cbd --- /dev/null +++ b/tests/test_primes.py @@ -0,0 +1,57 @@ +import unittest + +from Cryptotools.Numbers.primeNumber import getPrimeNumber, isPrimeNumber, sieveOfEratosthenes, _fermatLittleTheorem + +""" + To confirm if our algorithms for generating a list of prime numbers, + Our test check with the list generated by OEIS: + https://oeis.org/A000040 + +""" + +class TestPrime(unittest.TestCase): + """ + Lorsqu'on test, on genere une liste de prime number + Et une liste de non prime number + Nos tests vont verifier si les test de millerRabbin fonctionne et retourne bien que les non primes sont bien non primes + """ + def _generate_sieve(self): + # Source: https://oeis.org/A002322 + with open("tests/prime_numbers_oeis", "r") as f: + data = f.readlines() + p_s = data[0].split(",") + primes = list() + for entry in p_s: + primes.append(int(entry.strip())) + return primes + + def test_prime(self): + pass + #for i in range(25): + # n = getPrimeNumber(128) + + def test_is_prime(self): + pass + #for i in range(25): + # n = getPrimeNumber() + # # self.assertTrue(isPrimeNumber(n)) + + def test_sieve_eratost(self): + sieves = self._generate_sieve() + eratost = sieveOfEratosthenes(100) + + for i in range(len(eratost)): + self.assertEqual(eratost[i], sieves[i], "Wrong value") + + def test_fermat(self): + numbers = { + 5: True, + 19: True, + 20: False + } + + #for number in numbers: + # if numbers[number] != _fermatLittleTheorem(number): + # self.assertFalse(numbers[number]) + +unittest.main()