This library implements some arithmetic operations in finite fields.

Implemented operations

galois-lib implements not so many algorithms, because it is designed to be a helper library for cryptographic libraries. Hence it contains only few necessary operations.

Multiplication in $GF(2 ^ {8})$

Multiplication in finite field of size $2 ^ {8}$ is implemented using precomputed logarithm and exponents tables. It works according to the following obvious formula:

$$ a \cdot b = \exp \left( \log a + \log b \right), $$

where all operations are assumed to be in the finite field.

multiplication in $GF(2 ^ {128})$

Multiplication in $GF(2 ^ {128})$ is implemented using some kind of Karatsuba algorithm with help of pclmulqdq instruction. More information here: Intel® Carry-Less Multiplication Instruction and its Usage for Computing the GCM Mode.