Forget that old maths … Modulo 2 is coming to town
There’s a new show in town and it’s part of the next generation of cybersecurity… Modulo 2 or GF(2) — Galois field of two elements. It is used in many areas including with checksums and ciphers, and now is being applied into quantum robust cryptography. The multiplication function involves multiplying the binary values and ignoring the remainder. They are easy to implement and fast in their operation, especially in cryptography and checksum functions. It basically involves some bit shifts and an EX-OR function, which makes it fast in computing the multiplication.
For example 111 x 101 gives:
111
x101
------
111
000
111
-----
11011
=====We can represent 101 as x²+1 and 110 as x²+x. Next we multiple them together to give:
(x²+1)×(x²+x)
and which is:
(x⁴+x³+x²+x)
which can be represented as 11110.
For example if we use the example on Page 6 [here] of 84 x 13 (and where ** represents “to the power of”):
84x13 = ((2**6+ 2**4+ 2**2)x(2**3+ 2**2+ 2**0)) (mod 2)
= (2**9+ 2**8+ 2**7+2**1+ 2**6+ 2**5+ 2**1 + 2**4+ 2**2) (mod 2)
= (2**9+ 2**8+ 2**7+ 2**5+ 2**2) (mod 2)which is 1110100100 [Calc]
The working out in its proper form is:
84x13 = ((2⁶+2⁴+2²)×(2³+2²+2⁰))(mod 2)
