Low space complexity CRT-based bit-parallel [formula omitted] polynomial basis multipliers for irreducible trinomials

Abstract

This paper presents new space complexity records for the fastest parallel GF(2n) multipliers for about 22% values of n such that a degree-n irreducible trinomial f=un+uk+1 exists over GF(2). By selecting the largest possible value of k∈(n/2,2n/3], we further reduce the space complexities of the Chinese remainder theorem (CRT)-based hybrid polynomial basis multipliers. Our experimental results show that among the 539 values of n∈[5,999] such that f is irreducible for some k∈[2,n−2], there are 317 values of n such that k∈(n/2,2n/3]. For these irreducible trinomials, the space complexities of the CRT-based hybrid multipliers are reduced by 14.3% on average. As a comparison, the previous CRT-based multipliers considered the case k∈[2,n/2], and the improvement rate is 8.4% on average for only 290 values of n among these 539 values of n.