Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases
Information and Computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A survey of fast exponentiation methods
Journal of Algorithms
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Software Implementation of the NIST Elliptic Curves Over Prime Fields
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
PKC '01 Proceedings of the 4th International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
CRYPTO '89 Proceedings of the 9th Annual International Cryptology Conference on Advances in Cryptology
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Many standard elliptic curves (e.g. NIST, SECG, ANSI X9.62, WTLS, ...) over the finite field $\mathbb{F}_p$ have pa 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.