Fast Point Decompression for Standard Elliptic Curves

Abstract

Many standard elliptic curves (e.g. NIST, SECG, ANSI X9.62, WTLS, ...) over the finite field $\mathbb{F}_p$ have p a prime of Mersenne-like form—this yields faster field arithmetic. Point compression cuts the storage requirement for points (public keys) in half and is hence desirable. Point decompression in turn involves a square root computation. Given the special Mersenne-like form of a prime, in this paper we examine the problem of efficiently computing square roots in the base field. Although the motivation comes from standard curves, our analysis is for fast square roots in any arbitrary Mersenne-like prime field satisfying $p \equiv 3 \pmod 4$. Using well-known methods from number theory, we present a general strategy for fast square root computation in these base fields. Significant speedup in the exponentiation is achieved compared to general methods for exponentiation. Both software and hardware implementation results are given, with a focus on standard elliptic curves.