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#!/usr/bin/env python3
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from Cryptotools.Groups.group import Group
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class Galois:
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"""
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This class contain the Galois Field (Finite Field)
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Attributes:
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q (Integer): it's the number of the GF
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operation (Function): Function for generating the group
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identityAdd (Integer): it's the identity element in the GF(n) for the addition operation
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identityMul (Integer): it's the identity element in the GF(n) for the multiplicative operation
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"""
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def __init__(self, q, operation):
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self._q = q
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self._operation = operation
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self._identityAdd = 0
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self._identityMul = 1
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self._F = [x for x in range(q)]
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self._add = [[0 for x in range(q)] for y in range(q)]
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# TODO: May do we do a deep copy between all groups ?
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self._div = [[0 for x in range(q)] for y in range(q)]
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self._mul = [[0 for x in range(q)] for y in range(q)]
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self._sub = [[0 for x in range(q)] for y in range(q)]
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self._primitiveRoot = list()
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def primitiveRoot(self):
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"""
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In this function, we going to find the primitive root modulo n of the galois field
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Returns:
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Return the list of primitive root of the GF(q)
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"""
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for x in range(2, self._q):
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z = list()
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for entry in range(1, self._q):
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res = self._operation(x, entry, self._q)
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if res not in z:
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z.append(res)
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if self.isPrimitiveRoot(z, self._q - 1):
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if x not in self._primitiveRoot: # It's dirty, need to find why we have duplicate entry
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self._primitiveRoot.append(x)
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return z
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def getPrimitiveRoot(self):
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"""
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Return the list of primitives root
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Returns:
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Return the primitive root
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"""
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return self._primitiveRoot
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def isPrimitiveRoot(self, z, length):
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"""
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Check if z is a primitive root
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Args:
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z (list): check if z is a primitive root
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length (Integer): Length of the GF(q)
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"""
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if len(z) == length:
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return True
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return False
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def add(self):
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"""
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This function do the operation + on the Galois Field
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Returns:
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Return a list of the group with the addition operation
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"""
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for x in range(0, self._q):
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for y in range(0, self._q):
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self._add[x][y] = (x + y) % self._q
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return self._add
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def _inverseModular(self, a, n):
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"""
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This function find the reverse modular of a by n
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Returns:
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Return the reverse modular
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"""
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for b in range(1, n):
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if (a * b) % n == 1:
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inv = b
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break
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return inv
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def div(self):
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"""
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This function do the operation / on the Galois Field
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Returns:
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Return a list of the group with the division operation
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"""
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for x in range(1, self._q):
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for y in range(1, self._q):
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inv = self._inverseModular(y, self._q)
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self._div[x][y] = (x * inv) % self._q
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return self._div
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def mul(self):
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"""
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This function do the operation * on the Galois Field
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Returns:
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Return a list of the group with the multiplication operation
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"""
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for x in range(0, self._q):
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for y in range(0, self._q):
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self._mul[x][y] = (x * y) % self._q
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return self._mul
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def sub(self):
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"""
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This function do the operation - on the Galois Field
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Returns:
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Return a list of the group with the subtraction operation
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"""
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for x in range(0, self._q):
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for y in range(0, self._q):
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self._sub[x][y] = (x - y) % self._q
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return self._sub
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def check_closure_law(self, arithmetic):
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"""
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This function check the closure law.
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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
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Args:
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Arithmetics (str): contain the operation to be made, must be '+', '*', '/' or /-'
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"""
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if arithmetic not in ['+', '*', '/', '-']:
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raise Exception("The arithmetic need to be '+', '*', '/' or '-'")
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if arithmetic == '+':
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G = self._add
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elif arithmetic == '*':
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G = self._mul
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elif arithmetic == '/':
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G = self._div
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else:
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G = self._sub
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start = 0
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"""
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In case of multiplicative, we bypass the first line, because all elements are zero, otherwise the test fail
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"""
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if arithmetic == '*' or arithmetic == '/':
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start = 1
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isClosure = True
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for x in range(start, self._q):
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gr = Group(self._q, G[x], self._operation)
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if not gr.closure():
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isClosure = False
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del gr
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if isClosure:
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print(f"The group {arithmetic} respect closure law")
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else:
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print(f"The group {arithmetic} does not respect closure law")
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def check_identity_add(self):
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"""
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This function check the identity element and must satisfy this condition: $a + 0 = a$ for each element in the GF(n)
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In Group Theory, an identity element is an element in the group which do not change the value every element in the group
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Returns:
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"""
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for x in self._F:
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if not self._identityAdd + x == x:
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raise Exception(
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f"The identity element {self._identityAdd} "\
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"do not satisfy $a + element = a$"
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)
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def check_identity_mul(self):
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"""
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This function check the identity element and must satisfy this condition: $a * 1 = a$ for each element in the GF(n)
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In Group Theory, an identity element is an element in the group which do not change the value every element in the group
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Returns:
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"""
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for x in self._F:
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if not self._identityMul * x == x:
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raise Exception(
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f"The identity element {self._identityAdd} "\
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"do not satisfy $a * element = a$"
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)
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def printMatrice(self, m):
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"""
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This function print the GF(m)
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Args:
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m (list): Matrix of the GF
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"""
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header = str()
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header = " "
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for x in range(0, self._q):
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header += f"{x} "
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header += "\n--|" + "-" * (len(header)- 3) +"\n"
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s = str()
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for x in range(0, self._q):
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s += f"{x} | "
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for y in range(0, self._q):
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s += f"{m[x][y]} "
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s += "\n"
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s = header + s
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print(s)
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