from functools import reduce # constants used in the multGF2 function mask1 = mask2 = polyred = None def setGF2(degree, irPoly): """Define parameters of binary finite field GF(2^m)/g(x) - degree: extension degree of binary field - irPoly: coefficients of irreducible polynomial g(x) """ def i2P(sInt): """Convert an integer into a polynomial""" return [(sInt >> i) & 1 for i in reversed(range(sInt.bit_length()))] global mask1, mask2, polyred mask1 = mask2 = 1 << degree mask2 -= 1 polyred = reduce(lambda x, y: (x << 1) + y, i2P(irPoly)[1:]) def multGF2(p1, p2): """Multiply two polynomials in GF(2^m)/g(x)""" p = 0 while p2: if p2 & 1: p ^= p1 p1 <<= 1 if p1 & mask1: p1 ^= polyred p2 >>= 1 return p & mask2 if __name__ == "__main__": # Define binary field GF(2^3)/x^3 + x + 1 setGF2(3, 0b1011) # Evaluate the product (x^2 + x + 1)(x^2 + 1) print("{:02x}".format(multGF2(0b111, 0b101))) # Define binary field GF(2^8)/x^8 + x^4 + x^3 + x + 1 # (used in the Advanced Encryption Standard-AES) setGF2(8, 0b100011011) # Evaluate the product (x^7)(x^7 + x + 1) print("{:02x}".format(multGF2(0b10000000, 0b10000011)))
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