Editing Finite field arithmetic (section)

=== C programming example ===
Here is some [[C (programming language)|C]] code which will add and multiply numbers in the characteristic 2  finite field of order 2<sup>8</sup>, used for example by Rijndael algorithm or Reed–Solomon, using the [[Ancient Egyptian multiplication#Russian peasant multiplication|Russian peasant multiplication algorithm]]:

<syntaxhighlight lang="c">
/* Add two numbers in the GF(2^8) finite field */
uint8_t gadd(uint8_t a, uint8_t b) {
	return a ^ b;
}

/* Multiply two numbers in the GF(2^8) finite field defined 
 * by the modulo polynomial relation x^8 + x^4 + x^3 + x + 1 = 0
 * (the other way being to do carryless multiplication followed by a modular reduction)
 */
uint8_t gmul(uint8_t a, uint8_t b) {
	uint8_t p = 0; /* accumulator for the product of the multiplication */
	while (a != 0 && b != 0) {
        if (b & 1) /* if the polynomial for b has a constant term, add the corresponding a to p */
            p ^= a; /* addition in GF(2^m) is an XOR of the polynomial coefficients */

        if (a & 0x80) /* GF modulo: if a has a nonzero term x^7, then must be reduced when it becomes x^8 */
            a = (a << 1) ^ 0x11b; /* subtract (XOR) the primitive polynomial x^8 + x^4 + x^3 + x + 1 (0b1_0001_1011) – you can change it but it must be irreducible */
        else
            a <<= 1; /* equivalent to a*x */
        b >>= 1;
	}
	return p;
}
</syntaxhighlight>

This example has [[Timing attack|cache, timing, and branch prediction side-channel]] leaks, and is not suitable for use in cryptography.