├── README.md ├── utils.py ├── elliptic_group.py ├── elliptic_cryptography.py ├── elliptic_element.py └── .gitignore /README.md: -------------------------------------------------------------------------------- 1 | # Elliptic Curve Cryptography 2 | 3 | The library can be run by changing the file `elliptic_cryptography.py`. -------------------------------------------------------------------------------- /utils.py: -------------------------------------------------------------------------------- 1 | def modular_inverse(a, m): 2 | # return pow(a, -1, m) 3 | def egcd(a, b): 4 | if a == 0: 5 | return (b, 0, 1) 6 | else: 7 | g, y, x = egcd(b % a, a) 8 | return (g, x - (b // a) * y, y) 9 | 10 | def modinv(a, m): 11 | g, x, y = egcd(a, m) 12 | if g != 1: 13 | raise Exception(f"Modular inverse doesn't exist for {a} under modulo {m}") 14 | else: 15 | return x % m 16 | 17 | return modinv(a, m) 18 | 19 | def modular_exp(a, x, m): 20 | if (a == 0 or a == 1): 21 | return a 22 | 23 | res = 1 24 | a %= m 25 | while (x > 0): 26 | if (x % 2 == 1): 27 | res = (res * a) % m 28 | x >>= 1 29 | a = (a * a) % m 30 | 31 | return res -------------------------------------------------------------------------------- /elliptic_group.py: -------------------------------------------------------------------------------- 1 | import math 2 | from elliptic_element import EllipticGroupElement 3 | 4 | class EllipticGroup: 5 | def __init__(self, a, b, m): 6 | self.a, self.b = a, b 7 | # check for singularity of the elliptic curve 8 | if (4 * self.a**3 + 27 * self.b**2) % m == 0: 9 | raise ValueError(f"{self.a} and {self.b} will result in a singular elliptic curve") 10 | 11 | self.m = m 12 | 13 | # creating group elements 14 | self.elems = [EllipticGroupElement(math.inf, math.inf, self.a, self.b, self.m)] 15 | for i in range(1, self.m): 16 | temp = (i ** 3 + self.a * i + self.b) % self.m 17 | # loop through all possible values for y 18 | for j in range(1, self.m): 19 | # if j leaves a quadratic residue append the (i, j) pair to the elem 20 | if (j * j) % self.m == temp: 21 | self.elems.append(EllipticGroupElement(i, j, self.a, self.b, self.m)) 22 | 23 | # check for closure property of group, if not make the elems empty 24 | for i in self.elems: 25 | for j in self.elems: 26 | if i + j not in self.elems: 27 | self.elems = [] 28 | 29 | def __len__(self): 30 | return len(self.elems) 31 | 32 | def __repr__(self): 33 | return ( 34 | f"a = {self.a}, b = {self.b}, m = {self.m}\n" 35 | f"Group Elements: \n{repr(self.elems)}" 36 | ) -------------------------------------------------------------------------------- /elliptic_cryptography.py: -------------------------------------------------------------------------------- 1 | from elliptic_element import EllipticGroupElement 2 | from elliptic_group import EllipticGroup 3 | 4 | class EllipticCryptography: 5 | def __init__(self, alpha, sk, a, b, m): 6 | self.a, self.b, self.m = a, b, m 7 | self.alpha = EllipticGroupElement(*alpha, self.a, self.b, self.m) 8 | self.sk = sk 9 | self.pk = self.alpha * sk 10 | self.group = EllipticGroup(a, b, m) 11 | if not len(self.group): 12 | raise ValueError(f"a={self.a}, b={self.b}, m={self.m} doesn't form a Elliptic group") 13 | 14 | def encipher(self, text, k=3): 15 | text = text.lower() 16 | ciphertext = "" 17 | self.y1 = self.alpha * k 18 | for char in text: 19 | m = self.group.elems[ord(char) - ord('a')] 20 | y = m + (self.pk * k) 21 | ciphertext += chr(self.group.elems.index(y) + ord('a')) 22 | 23 | return ciphertext, self.y1 24 | 25 | def decipher(self, text): 26 | plaintext = "" 27 | for char in text: 28 | x = self.group.elems[ord(char) - ord('a')] 29 | c = x - (self.y1 * self.sk) 30 | plaintext += chr(self.group.elems.index(c) + ord('a')) 31 | 32 | return plaintext 33 | 34 | if __name__ == '__main__': 35 | # message = [(10, 9)] 36 | message = "sivaram" 37 | ecc = EllipticCryptography(alpha=(2, 2), sk=7, a=1, b=17, m=23) 38 | ciphertext = ecc.encipher(message) 39 | plaintext = ecc.decipher(ciphertext[0]) 40 | 41 | print("Original text:", message) 42 | print("Ciphertext:", ciphertext) 43 | print("Deciphered:", plaintext) -------------------------------------------------------------------------------- /elliptic_element.py: -------------------------------------------------------------------------------- 1 | import math 2 | import copy 3 | 4 | from utils import modular_inverse 5 | 6 | class EllipticGroupElement: 7 | def __init__(self, x, y, a, b, m): 8 | self.x, self.y = x, y 9 | self.a, self.b = a, b 10 | self.m = m 11 | 12 | def __add__(self, Q): 13 | # if either of the point is O (point at infinity) then return the same 14 | if self.x == math.inf: 15 | return Q 16 | if Q.x == math.inf: 17 | return self 18 | 19 | # If P and Q are same, find slope of the tangent then find R 20 | if self == Q: 21 | slope = (3 * self.x**2 + self.a) * \ 22 | modular_inverse((2 * self.y) % self.m, self.m) 23 | xr = slope**2 - 2 * self.x 24 | yr = slope*(self.x - xr) - self.y 25 | 26 | # x's are same and y's are opposite on the elliptic curve 27 | elif self.x == Q.x and (self.y + Q.y) % self.m == 0: 28 | return EllipticGroupElement(math.inf, math.inf, self.a, self.b, self.m) 29 | 30 | else: 31 | slope = (self.y - Q.y) * \ 32 | modular_inverse((self.x - Q.x) % self.m, self.m) 33 | xr = slope**2 - self.x - Q.x 34 | yr = slope*(self.x - xr) - self.y 35 | 36 | xr %= self.m 37 | yr %= self.m 38 | return EllipticGroupElement(xr, yr, self.a, self.b, self.m) 39 | 40 | def __sub__(self, Q): 41 | if Q.x == math.inf: 42 | return self 43 | 44 | Qinv = EllipticGroupElement(Q.x, self.m - Q.y, self.a, self.b, self.m) 45 | return self + Qinv 46 | 47 | def __mul__(self, scalar : int): 48 | temp = copy.deepcopy(self) 49 | ans = copy.deepcopy(self) 50 | for _ in range(scalar - 1): 51 | ans = ans + temp 52 | return ans 53 | 54 | def __eq__(self, Q): 55 | return self.x == Q.x and self.y == Q.y 56 | 57 | def __ne__(self, Q): 58 | return not self == Q 59 | 60 | def __repr__(self): 61 | return f"Elem: ({self.x}, {self.y})" -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # 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