Lookup table based multiplication technique for GF(2m) with cryptographic significance

Abstract

Finite field multiplication is one of the most important arithmetic operations in the binary finite field GF(2m). Multiplication is required for many cryptographic techniques based on the discrete logarithm problem (DLP) in the multiplicative group of a finite field or additive group of points on an elliptic curve defined over a finite field. A look up table (LUT) based multiplication technique for GF(2m) in polynomial basis is presented, which is applicable to any field size m. Unlike older LUT techniques, this LUT is calculated only once, and for all. The LUT is of constant size and the technique is valid for any m, whereas many older LUT techniques worked only for composite m. It is shown that the new technique results in the lowest number of word-level operations in software environments on contemporary computing platforms. This GF(2m) multiplication method results in significant performance gains for elliptic curve cryptography in software environments.