Fast software implementation of binary elliptic curve cryptography

Abstract

Abstract. This paper presents an efficient and side channel protected software implementation of point multiplication for the standard NIST and SECG binary elliptic curves. The enhanced performance is achieved by improving the Lòpez-Dahab/Montgomery method at the algorithmic level, and by leveraging Intel's AVX architecture and the pclmulqdq processor instruction at the coding level. The fast carry-less multiplication is further used to speed up the reduction on the newest Haswell platforms. For the five NIST curves over GF (2 m ) with m ∈ {163, 233, 283, 409, 571}, the resulting point multiplication implementation is about 6 to 12 times faster than that of OpenSSL-1.0.1e, enhancing the ECDHE and ECDSA algorithms significantly.

Elliptic Curve Multiset Hash

“…Using a combination of C++ templates and code generation, we were able to write generic code to support many different configurations without sacrificing runtime performance; only for modular reduction was a custom implementation required for each supported field. We incorporated existing fast x86/x86-64 polynomial multiplication, squaring, and modular reduction routines for F 2 163 , F 2 193 , F 2 233 , F 2 239 , F 2 283 , F 2 409 , F 2 571 [33] and for F 2 127 [28]. We implemented field inversion using a polynomial-basis Itoh-Tsujii inversion method making use of multi-squaring tables [34,29,28,14].…”

“…As our test platforms we used an Intel Westmere i7-970 3.2 GHz CPU (with 12 MiB L3 cache) and an Intel Haswell i7-4790K 4.0 GHz CPU (with 8 MiB L3 cache). Both of these processors support the PCLMULQDQ instruction for carry-less multiplication, Westmere being the first Intel architecture to support it; on the much more recent Haswell architecture, where this instruction has significantly lower cost, alternative modular reduction routines based on it are used for F 2 163 , F 2 283 , and F 2 571 for a modest gain in performance [33]. Our implementation used a word size of W = 128 bits and a block size of B = 8 bits for all half trace and multi-squaring tables.…”

Faster 64-bit universal hashing using carry-less multiplications

“…Elliptic curve cryptography benefits from the pclmulqdq instruction [32,33,39]. Bluhm and Gueron pointed out that the benefits are increased on the Haswell microarchitecture due to the higher throughput and lower latency of the instruction [8].…”

Differential Fault Attacks and Countermeasures in Elliptic Curve Cryptography

“…Thus, modular arithmetic operations should be well studied in order to find an efficient implementation, authors in [2] and [3] studied arithmetic operation over finite fields. Many software implementation was ben proposed [4,5,6]. In this paper, a software implementation of scalar multiplication over prime elliptic curves using GMP Library will be firstly presented, which is a portable library written in C, with some optimized routines written in assembly code, specialized for different processors, more information is given in [7].…”