Faster Residue Multiplication Modulo 521-bit Mersenne Prime and an Application to ECC

Date

2018-08-01

Author

Ali, Shoukat
Cenk, Murat

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

229
views

0
downloads

We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz matrix-vector product. The total arithmetic cost of our proposed algorithms is less than that of existing algorithms, with algorithms for 64- and 32-bit residue multiplication giving the best timing results on our test machine. The transition from 64- to 32-bit implementation is full of challenges because the number of limbs doubles and the limbs' bitlengths are cut in half. Without using any intrinsics or SIMD/assembly instructions in our implementation on an Intel(R) Core i5 - 6402P CPU @ 2.80GHz, we find 136 and 550 cycles for our 64- and 32-bit residue multiplications, respectively. In addition, we implement constant-time variable- and fixed-base scalar multiplication for the standard NIST curve P-521 and Edwards curve E-521.

Subject Keywords

Residue multiplication, Toeplitz matrix-vector product, Mersenne prime elliptic curve cryptography, Variable- and fixed-base scalar multiplication, 32-and 64-bit platforms

URI

https://hdl.handle.net/11511/31401

Journal

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS

DOI

https://doi.org/10.1109/tcsi.2018.2791285

Collections

Graduate School of Applied Mathematics, Article

Suggestions

OpenMETU
Core

Faster residue multiplication modulo 521-bit mersenne prime and application to ECC Ali, Shoukat; Cenk, Murat; Department of Cryptography (2017) We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz Matrix-Vector Product (TMVP). The total arithmetic cost of our proposed algorithms is less than the existing algorithms and we select the ones, 32- and 64-bit residue multiplication, with the best timing results on our testing machine(s). For the 64-bit residue multiplication we have presented three versions of our algorithm along with their arithmetic cost and from impleme...
A New Algorithm for Residue Multiplication Modulo 2(521)-1 Ali, Shoukat; Cenk, Murat (2016-12-02) We present a new algorithm for residue multiplication modulo the Mersenne prime p = 2(521) - 1 based on the Toeplitz matrix-vector product. For this modulus, our algorithm yields better result in terms of the total number of operations than the previously known best algorithm of Granger and Scott presented in Public Key Cryptography (PKC) 2015. We have implemented three versions of our algorithm to provide an extensive comparison - according to the best of our knowledge with respect to the well-known algori...
Speeding up Curve25519 using Toeplitz Matrix-vector Multiplication Taskin, Halil Kemal; Cenk, Murat (2018-01-24) This paper proposes a new multiplication algorithm over F-2(255)-19 where the de-facto standard Curve25519 [2] algorithm is based on. Our algorithm for the underlying finite field multiplication exploits the Toeplitz matrix-vector multiplication and achieves salient results. We have used a new radix representation that is infeasible when used with schoolbook multiplication techniques but has notable advantages when used with Toeplitz matrix-vector multiplication methods. We present the new algorithm and dis...
Faster Montgomery modular multiplication without pre-computational phase for some classes of finite fields Akleylek, Sedat; Cenk, Murat; Özbudak, Ferruh (2010-09-24) In this paper, we give faster versions of Montgomery modular multiplication algorithm without pre-computational phase for GF(p) and GF(2 m ) which can be considered as a generalization of [3], [4] and [5]. We propose sets of moduli different than [3], [4] and [5] which can be used in PKC applications. We show that one can obtain efficient Montgomery modular multiplication architecture in view of the number of AND gates and XOR gates by choosing proposed sets of moduli. We eliminate precomputational phase wi...
Improved three-way split formulas for binary polynomial multiplication Cenk, Murat; Hasan, M. Anwar (2011-08-12) In this paper we deal with 3-way split formulas for binary field multiplication with five recursive multiplications of smaller sizes. We first recall the formula proposed by Bernstein at CRYPTO 2009 and derive the complexity of a parallel multiplier based on this formula. We then propose a new set of 3-way split formulas with five recursive multiplications based on field extension. We evaluate their complexities and provide a comparison.

Citation Formats

S. Ali and M. Cenk, “Faster Residue Multiplication Modulo 521-bit Mersenne Prime and an Application to ECC,” IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS, pp. 2477–2490, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/31401.