First commit

2026-01-11 09:19:22 +01:00 · 2026-01-11 09:19:22 +01:00 · 50f7b1159e
commit 50f7b1159e
108 changed files with 11791 additions and 0 deletions
--- a/.gitignore
+++ b/.gitignore
@ -0,0 +1,5 @@
+__pycache__/
+build/**
+Cryptotools.egg-info/**
+dist/**
+**.swp
--- a/Cryptotools/Cryptotools.egg-info/PKG-INFO
+++ 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
--- a/Cryptotools/Cryptotools.egg-info/SOURCES.txt
+++ 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
--- a/Cryptotools/Cryptotools.egg-info/dependency_links.txt
+++ b/Cryptotools/Cryptotools.egg-info/dependency_links.txt
@ -0,0 +1 @@
+
--- a/Cryptotools/Cryptotools.egg-info/top_level.txt
+++ b/Cryptotools/Cryptotools.egg-info/top_level.txt
@ -0,0 +1 @@
+Numbers
--- a/Cryptotools/Encryptions/RSA.py
+++ 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
--- a/Cryptotools/Encryptions/init.py
+++ b/Cryptotools/Encryptions/init.py
@ -0,0 +1 @@
+#!/usr/bin/env python3
--- a/Cryptotools/Groups/init.py
+++ b/Cryptotools/Groups/init.py
@ -0,0 +1 @@
+#!/usr/bin/env python3
--- a/Cryptotools/Groups/cyclic.py
+++ 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
--- a/Cryptotools/Groups/galois.py
+++ 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)
--- a/Cryptotools/Groups/group.py
+++ 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
--- a/Cryptotools/Numbers/init.py
+++ b/Cryptotools/Numbers/init.py
@ -0,0 +1 @@
+#!/usr/bin/env python3
--- a/Cryptotools/Numbers/pycache/init.cpython-310.pyc
+++ b/Cryptotools/Numbers/pycache/init.cpython-310.pyc
--- a/Cryptotools/Numbers/pycache/init.cpython-39.pyc
+++ b/Cryptotools/Numbers/pycache/init.cpython-39.pyc
--- a/Cryptotools/Numbers/pycache/carmi.cpython-310.pyc
+++ b/Cryptotools/Numbers/pycache/carmi.cpython-310.pyc
--- a/Cryptotools/Numbers/pycache/carmi.cpython-39.pyc
+++ b/Cryptotools/Numbers/pycache/carmi.cpython-39.pyc
--- a/Cryptotools/Numbers/carmi.py
+++ 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
--- a/Cryptotools/Numbers/coprime.py
+++ 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
+
--- a/Cryptotools/Numbers/factorization.py
+++ 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
--- a/Cryptotools/Numbers/numbers.py
+++ 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
+
--- a/Cryptotools/Numbers/primeNumber.py
+++ 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
+
--- a/Cryptotools/Utils/init.py
+++ b/Cryptotools/Utils/init.py
@ -0,0 +1 @@
+#!/usr/bin/env python3
--- a/Cryptotools/Utils/number.py
+++ 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
--- a/Cryptotools/Utils/utils.py
+++ 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
+    
--- a/Cryptotools/init.py
+++ b/Cryptotools/init.py
@ -0,0 +1,3 @@
+#!/usr/bin/env python3
+
+__all__ = ["Numbers", "Groups"]
--- a/Cryptotools/lcm.py
+++ 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
+
--- a/README.md
+++ 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
+```
--- a/docs/examples-rsa-keys.md
+++ 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}")
+```
--- a/docs/group-theory.md
+++ 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
--- a/docs/index.md
+++ 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)
+
+
--- a/docs/installation.md
+++ 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
--- a/docs/introduction.md
+++ 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.
--- a/docs/number-theory.md
+++ 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
+
--- a/docs/rsa.md
+++ b/docs/rsa.md
@ -0,0 +1,3 @@
+CryptoTools provides several functions for generating a RSA Keys
+
+::: Cryptotools.Encryptions.RSA
--- a/examples/carmi.py
+++ 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)}")
--- a/examples/coprime.py
+++ 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)}")
--- a/examples/cyclic.py
+++ 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")
--- a/examples/cyclic_dlp.py
+++ 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)}")
--- a/examples/cyclic_generator.py
+++ 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}")
--- a/examples/cyclic_subgroup.py
+++ 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}")
--- a/examples/cyclic_test.py
+++ 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)
--- a/examples/dh_cyclic.py
+++ 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}")
--- a/examples/dh_no_generator.py
+++ 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)
--- a/examples/fermat_factorization.py
+++ 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()
+
--- a/examples/fibonacci.py
+++ b/examples/fibonacci.py
@ -0,0 +1,5 @@
+#!/usr/bin/env python3
+
+from Cryptotools.Numbers.numbers import fibonacci
+
+print(fibonacci(40))
--- a/examples/galois.py
+++ 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()
--- a/examples/group.py
+++ 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()
+
--- a/examples/main.py
+++ b/examples/main.py
@ -0,0 +1,6 @@
+#!/usr/bin/env python3
+
+from utils.primeNumber import primeNumber
+
+print(primeNumber(19))
+print(primeNumber(20))
--- a/examples/plotlib.py
+++ 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()
+
--- a/examples/pollard.py
+++ 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))
--- a/examples/prime.py
+++ 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")
--- a/examples/pseudoprimes.py
+++ 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))
--- a/examples/rsa.py
+++ 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}")
--- a/examples/rsa_problem.py
+++ 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))
+
--- a/examples/sieveOfEratosthenes.py
+++ b/examples/sieveOfEratosthenes.py
@ -0,0 +1,6 @@
+#!/usr/bin/env python3
+
+from Cryptotools.Numbers.primeNumber import sieveOfEratosthenes
+
+print(sieveOfEratosthenes(100))
+print()
--- a/examples/test_gcd.py
+++ 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))
--- a/mkdocs.yml
+++ 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 
--- a/pyproject.toml
+++ 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"
--- a/requirements.txt
+++ b/requirements.txt
@ -0,0 +1,5 @@
+sympy
+mkdocs
+mkdocstrings[python]
+mkdocs-readthedocs
+mkdocs-material
--- a/setup.py
+++ 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"}
+)
--- a/sieves_base.py
+++ 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(")")
--- a/site/404.html
+++ b/site/404.html
@ -0,0 +1,122 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+    <meta charset="utf-8" />
+    <meta http-equiv="X-UA-Compatible" content="IE=edge" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+      <link rel="shortcut icon" href="/img/favicon.ico" />
+    <title>CryptoTools documentation</title>
+    <link rel="stylesheet" href="/css/theme.css" />
+    <link rel="stylesheet" href="/css/theme_extra.css" />
+        <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/styles/github.min.css" />
+        <link href="/assets/_mkdocstrings.css" rel="stylesheet" />
+    
+    <!--[if lt IE 9]>
+      <script src="/js/html5shiv.min.js"></script>
+    <![endif]-->
+      <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min.js"></script>
+      <script>hljs.highlightAll();</script> 
+</head>
+
+<body class="wy-body-for-nav" role="document">
+
+  <div class="wy-grid-for-nav">
+    <nav data-toggle="wy-nav-shift" class="wy-nav-side stickynav">
+    <div class="wy-side-scroll">
+      <div class="wy-side-nav-search">
+          <a href="/." class="icon icon-home"> CryptoTools documentation
+        </a>
+      </div>
+
+      <div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">
+              <ul>
+                <li class="toctree-l1"><a class="reference internal" href="/introduction/">Introduction</a>
+                </li>
+              </ul>
+              <ul>
+                <li class="toctree-l1"><a class="reference internal" href="/installation/">Installation</a>
+                </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Low-level cryptographic</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="/number-theory/">Number theory</a>
+                  </li>
+                  <li class="toctree-l1"><a class="reference internal" href="/group-theory/">Group theory</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Public Keys</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="/rsa/">RSA</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Examples</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="/examples-rsa-keys/">Generating RSA Keys</a>
+                  </li>
+              </ul>
+      </div>
+    </div>
+    </nav>
+
+    <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">
+      <nav class="wy-nav-top" role="navigation" aria-label="Mobile navigation menu">
+          <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
+          <a href="/.">CryptoTools documentation</a>
+        
+      </nav>
+      <div class="wy-nav-content">
+        <div class="rst-content"><div role="navigation" aria-label="breadcrumbs navigation">
+  <ul class="wy-breadcrumbs">
+    <li><a href="/." class="icon icon-home" aria-label="Docs"></a></li>
+    <li class="wy-breadcrumbs-aside">
+    </li>
+  </ul>
+  <hr/>
+</div>
+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+            <div class="section" itemprop="articleBody">
+              
+
+  <h1 id="404-page-not-found">404</h1>
+
+  <p><strong>Page not found</strong></p>
+
+
+            </div>
+          </div><footer>
+
+  <hr/>
+
+  <div role="contentinfo">
+    <!-- Copyright etc -->
+  </div>
+
+  Built with <a href="https://www.mkdocs.org/">MkDocs</a> using a <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
+</footer>
+          
+        </div>
+      </div>
+
+    </section>
+
+  </div>
+
+  <div class="rst-versions" role="note" aria-label="Versions">
+  <span class="rst-current-version" data-toggle="rst-current-version">
+    
+    
+    
+  </span>
+</div>
+    <script src="/js/jquery-3.6.0.min.js"></script>
+    <script>var base_url = "/";</script>
+    <script src="/js/theme_extra.js"></script>
+    <script src="/js/theme.js"></script>
+    <script>
+        jQuery(function () {
+            SphinxRtdTheme.Navigation.enable(true);
+        });
+    </script>
+
+</body>
+</html>
--- a/site/assets/_mkdocstrings.css
+++ b/site/assets/_mkdocstrings.css
@ -0,0 +1,57 @@
+
+/* Avoid breaking parameters name, etc. in table cells. */
+.doc-contents td code {
+  word-break: normal !important;
+}
+
+/* No line break before first paragraph of descriptions. */
+.doc-md-description,
+.doc-md-description>p:first-child {
+  display: inline;
+}
+
+/* Avoid breaking code headings. */
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+  white-space: normal;
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+
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+
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+  margin: 1em 0 1em;
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+  padding: 0 !important;
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+  margin-bottom: 0 !important;
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+.doc-contents td.code {
+  padding: 0 !important;
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+
+.doc-contents td.linenos {
+  padding: 0 8px !important;
+}
+
+.doc-children,
+footer {
+  margin-top: 20px;
+}
--- a/site/css/fonts/Roboto-Slab-Bold.woff
+++ b/site/css/fonts/Roboto-Slab-Bold.woff
--- a/site/css/fonts/Roboto-Slab-Bold.woff2
+++ b/site/css/fonts/Roboto-Slab-Bold.woff2
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+++ b/site/css/fonts/Roboto-Slab-Regular.woff
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+++ b/site/css/fonts/Roboto-Slab-Regular.woff2
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+++ b/site/css/fonts/fontawesome-webfont.eot
--- a/site/css/fonts/fontawesome-webfont.svg
+++ b/site/css/fonts/fontawesome-webfont.svg
--- a/site/css/fonts/fontawesome-webfont.ttf
+++ b/site/css/fonts/fontawesome-webfont.ttf
--- a/site/css/fonts/fontawesome-webfont.woff
+++ b/site/css/fonts/fontawesome-webfont.woff
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+++ b/site/css/theme.css
--- a/site/css/theme_extra.css
+++ b/site/css/theme_extra.css
@ -0,0 +1,197 @@
+/*
+ * Wrap inline code samples otherwise they shoot of the side and
+ * can't be read at all.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/313
+ * https://github.com/mkdocs/mkdocs/issues/233
+ * https://github.com/mkdocs/mkdocs/issues/834
+ */
+.rst-content code {
+    white-space: pre-wrap;
+    word-wrap: break-word;
+    padding: 2px 5px;
+}
+
+/**
+ * Make code blocks display as blocks and give them the appropriate
+ * font size and padding.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/855
+ * https://github.com/mkdocs/mkdocs/issues/834
+ * https://github.com/mkdocs/mkdocs/issues/233
+ */
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+    white-space: pre;
+    word-wrap: normal;
+    display: block;
+    padding: 12px;
+    font-size: 12px;
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+
+/**
+ * Fix code colors
+ *
+ * https://github.com/mkdocs/mkdocs/issues/2027
+ */
+.rst-content code {
+    color: #E74C3C;
+}
+
+.rst-content pre code {
+    color: #000;
+    background: #f8f8f8;
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+
+/*
+ * Fix link colors when the link text is inline code.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/718
+ */
+a code {
+    color: #2980B9;
+}
+a:hover code {
+    color: #3091d1;
+}
+a:visited code {
+    color: #9B59B6;
+}
+
+/*
+ * The CSS classes from highlight.js seem to clash with the
+ * ReadTheDocs theme causing some code to be incorrectly made
+ * bold and italic.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/411
+ */
+pre .cs, pre .c {
+    font-weight: inherit;
+    font-style: inherit;
+}
+
+/*
+ * Fix some issues with the theme and non-highlighted code
+ * samples. Without and highlighting styles attached the
+ * formatting is broken.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/319
+ */
+.rst-content .no-highlight {
+    display: block;
+    padding: 0.5em;
+    color: #333;
+}
+
+
+/*
+ * Additions specific to the search functionality provided by MkDocs
+ */
+
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+    margin-top: 23px;
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+
+.search-results article:first-child {
+    border-top: none;
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+
+form .search-query {
+    width: 100%;
+    border-radius: 50px;
+    padding: 6px 12px;
+    border-color: #D1D4D5;
+}
+
+/*
+ * Improve inline code blocks within admonitions.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/656
+ */
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+    color: #404040;
+    border: 1px solid #c7c9cb;
+    border: 1px solid rgba(0, 0, 0, 0.2);
+    background: #f8fbfd;
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+ * 90 % on the .rst-current-version container, combined with the padding of 12px
+ * above and below. These amendments fix this in two steps: First, make sure the
+ * .rst-current-version container has a fixed height of 40px, achieved using
+ * line-height, and then applying a padding-bottom of 40px to this container. In
+ * a second step, the items within that container are re-aligned using flexbox.
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+ * https://github.com/mkdocs/mkdocs/issues/2012
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+ * [No repo_name]
+ *         Next >>                    // On the first page
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+ * https://github.com/mkdocs/mkdocs/issues/2012
+ */
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--- a/site/examples-rsa-keys/index.html
+++ b/site/examples-rsa-keys/index.html
@ -0,0 +1,158 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+    <meta charset="utf-8" />
+    <meta http-equiv="X-UA-Compatible" content="IE=edge" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+      <link rel="shortcut icon" href="../img/favicon.ico" />
+    <title>Generating RSA Keys - CryptoTools documentation</title>
+    <link rel="stylesheet" href="../css/theme.css" />
+    <link rel="stylesheet" href="../css/theme_extra.css" />
+        <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/styles/github.min.css" />
+        <link href="../assets/_mkdocstrings.css" rel="stylesheet" />
+    
+      <script>
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+      </script>
+    
+    <!--[if lt IE 9]>
+      <script src="../js/html5shiv.min.js"></script>
+    <![endif]-->
+      <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min.js"></script>
+      <script>hljs.highlightAll();</script> 
+</head>
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+                </li>
+              </ul>
+              <ul>
+                <li class="toctree-l1"><a class="reference internal" href="../installation/">Installation</a>
+                </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Low-level cryptographic</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../number-theory/">Number theory</a>
+                  </li>
+                  <li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Public Keys</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Examples</span></p>
+              <ul class="current">
+                  <li class="toctree-l1 current"><a class="reference internal current" href="#">Generating RSA Keys</a>
+    <ul class="current">
+    </ul>
+                  </li>
+              </ul>
+      </div>
+    </div>
+    </nav>
+
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+      <nav class="wy-nav-top" role="navigation" aria-label="Mobile navigation menu">
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+        
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+          <li class="breadcrumb-item">Examples</li>
+      <li class="breadcrumb-item active">Generating RSA Keys</li>
+    <li class="wy-breadcrumbs-aside">
+    </li>
+  </ul>
+  <hr/>
+</div>
+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+            <div class="section" itemprop="articleBody">
+              
+                <h1 id="generating-rsa-keys">Generating RSA Keys</h1>
+<p>In the following section, we are going to see how to generate RSA Keys and how to encrypt data with these keys.</p>
+<pre><code>#!/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 = &quot;I am encrypted with RSA&quot;
+print(f&quot;plaintext: {s}&quot;)
+encrypted = rsa.encrypt(s)
+
+# Encrypt data
+print(f&quot;ciphertext: {encrypted}&quot;)
+
+# We decrypt
+plaintext = rsa.decrypt(encrypted)
+print(f&quot;Plaintext: {plaintext}&quot;)
+</code></pre>
+              
+            </div>
+          </div><footer>
+    <div class="rst-footer-buttons" role="navigation" aria-label="Footer Navigation">
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+                <li class="toctree-l1"><a class="reference internal" href="installation/">Installation</a>
+                </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Low-level cryptographic</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="number-theory/">Number theory</a>
+                  </li>
+                  <li class="toctree-l1"><a class="reference internal" href="group-theory/">Group theory</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Public Keys</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="rsa/">RSA</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Examples</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="examples-rsa-keys/">Generating RSA Keys</a>
+                  </li>
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+<ul>
+<li><a href="/introduction">Introduction</a></li>
+<li><a href="/installation">Installation</a></li>
+<li>Low-Level Cryptographic<ul>
+<li><a href="/number-theory">Number Theory</a></li>
+<li><a href="/group-theory">Group Theory</a></li>
+</ul>
+</li>
+<li>Public Keys:<ul>
+<li><a href="/rsa">RSA</a></li>
+</ul>
+</li>
+<li>Examples:<ul>
+<li><a href="/examples-rsa-keys">Generating RSA keys</a></li>
+</ul>
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+              </ul>
+              <ul class="current">
+                <li class="toctree-l1 current"><a class="reference internal current" href="#">Installation</a>
+    <ul class="current">
+    <li class="toctree-l2"><a class="reference internal" href="#installation-from-source">Installation from source</a>
+    </li>
+    </ul>
+                </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Low-level cryptographic</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../number-theory/">Number theory</a>
+                  </li>
+                  <li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Public Keys</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Examples</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
+                  </li>
+              </ul>
+      </div>
+    </div>
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+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+            <div class="section" itemprop="articleBody">
+              
+                <h1 id="installation">Installation</h1>
+<p>To install the project, you should install from my own Python repository but, you first need to deploy your virtual enviroment.</p>
+<h2 id="installation-from-source">Installation from source</h2>
+<p>You can install from the source. Download the <a href="https://gitea.bucchino.org/gbucchino/cryptotools.git">project</a> and install it in your virtual environment:</p>
+<pre><code>$ virtualenv ~/venv/cryptotools
+$ source ~/venv/cryptotools/bin/activate
+</code></pre>
+<pre><code>$ git clone https://gitea.bucchino.org/gbucchino/cryptotools.git
+$ cd cryptotools
+$ python3 setup.py install
+</code></pre>
+<p>The installation is completed. You can know use Cryptotools package into your project. In the directory <code>examples</code> you may find some examples scripts</p>
+              
+            </div>
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+    <ul class="current">
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+              <ul>
+                <li class="toctree-l1"><a class="reference internal" href="../installation/">Installation</a>
+                </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Low-level cryptographic</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../number-theory/">Number theory</a>
+                  </li>
+                  <li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Public Keys</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
+                  </li>
+              </ul>
+              <p class="caption"><span class="caption-text">Examples</span></p>
+              <ul>
+                  <li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
+                  </li>
+              </ul>
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+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
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+              
+                <h1 id="cryptotools">CryptoTools</h1>
+<p>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.</p>
+<p>So far, my cryptographic modules are not compliant with <a href="https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.140-3.pdf">FIPS 140-3</a> but in the future, that will be.</p>
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+* @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<style>"+b+"</style>",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 h(a,c){if(a||(a=b),l)return a.createDocumentFragment();c=c||f(a);for(var e=c.frag.cloneNode(),g=0,h=d(),i=h.length;i>g;g++)e.createElement(h[g]);return e}function i(a,b){b.cache||(b.cache={},b.createElem=a.createElement,b.createFrag=a.createDocumentFragment,b.frag=b.createFrag()),a.createElement=function(c){return t.shivMethods?g(c,a,b):b.createElem(c)},a.createDocumentFragment=Function("h,f","return function(){var n=f.cloneNode(),c=n.createElement;h.shivMethods&&("+d().join().replace(/[\w\-:]+/g,function(a){return b.createElem(a),b.frag.createElement(a),'c("'+a+'")'})+");return n}")(t,b.frag)}function j(a){a||(a=b);var d=f(a);return!t.shivCSS||k||d.hasCSS||(d.hasCSS=!!c(a,"article,aside,dialog,figcaption,figure,footer,header,hgroup,main,nav,section{display:block}mark{background:#FF0;color:#000}template{display:none}")),l||i(a,d),a}var k,l,m="3.7.3",n=a.html5||{},o=/^<|^(?:button|map|select|textarea|object|iframe|option|optgroup)$/i,p=/^(?:a|b|code|div|fieldset|h1|h2|h3|h4|h5|h6|i|label|li|ol|p|q|span|strong|style|table|tbody|td|th|tr|ul)$/i,q="_html5shiv",r=0,s={};!function(){try{var a=b.createElement("a");a.innerHTML="<xyz></xyz>",k="hidden"in a,l=1==a.childNodes.length||function(){b.createElement("a");var a=b.createDocumentFragment();return"undefined"==typeof a.cloneNode||"undefined"==typeof a.createDocumentFragment||"undefined"==typeof a.createElement}()}catch(c){k=!0,l=!0}}();var t={elements:n.elements||"abbr article aside audio bdi canvas data datalist details dialog figcaption figure footer header hgroup main mark meter nav output picture progress section summary template time video",version:m,shivCSS:n.shivCSS!==!1,supportsUnknownElements:l,shivMethods:n.shivMethods!==!1,type:"default",shivDocument:j,createElement:g,createDocumentFragment:h,addElements:e};a.html5=t,j(b),"object"==typeof module&&module.exports&&(module.exports=t)}("undefined"!=typeof window?window:this,document);
--- a/site/js/jquery-3.6.0.min.js
+++ b/site/js/jquery-3.6.0.min.js
--- a/site/js/theme.js
+++ b/site/js/theme.js
--- a/site/js/theme_extra.js
+++ b/site/js/theme_extra.js
@ -0,0 +1,8 @@
+/*
+ * Assign 'docutils' class to tables so styling and
+ * JavaScript behavior is applied.
+ *
+ * https://github.com/mkdocs/mkdocs/issues/2028
+ */
+
+$('div.rst-content table').addClass('docutils');
--- a/site/number-theory/index.html
+++ b/site/number-theory/index.html
--- a/site/objects.inv
+++ b/site/objects.inv
--- a/site/rsa/index.html
+++ b/site/rsa/index.html
--- a/site/sitemap.xml
+++ b/site/sitemap.xml
@ -0,0 +1,3 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
+</urlset>
--- a/site/sitemap.xml.gz
+++ b/site/sitemap.xml.gz
--- a/tests/carmi_function_oeis
+++ 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
--- a/tests/carmi_numbers_oeis
+++ 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
--- a/tests/execute_tests.py
+++ 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
--- a/Show More
+++ b/Show More
				`@ -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`
				`@ -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`