A New Algorithm for Residue Multiplication Modulo 2(521)-1

Date

2016-12-02

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

248
views

0
downloads

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 algorithms and to show the robustness of our algorithm for this 521-bit Mersenne prime modulus. Each version is having less number of operations than its counterpart. On our machine, Intel Pentium CPU G2010 @ 2.80 GHz machine with gcc 5.3.1 compiler, we find that for each version of our algorithm modulus p is more efficient than modulus 2p. Hence, by using Granger and Scott code, constant-time variable-base scalar multiplication, for modulus p we find 1, 251, 502 clock cycles for P-521 (NIST and SECG curve) and 1, 055, 105 cycles for E-521 (Edwards curve). While, on the same machine the clock cycles counts of Granger-Scott code (modulus 2p) for P-521 and E-521 are 1, 332, 165 and 1, 148, 871 respectively.

Subject Keywords

Residue multiplication, Toeplitz matrix-vector product, Mersenne prime, Elliptic curve cryptography

URI

https://hdl.handle.net/11511/32521

DOI

https://doi.org/10.1007/978-3-319-53177-9_9

Collections

Graduate School of Applied Mathematics, Conference / Seminar

Suggestions

OpenMETU
Core

Faster Residue Multiplication Modulo 521-bit Mersenne Prime and an Application to ECC Ali, Shoukat; Cenk, Murat (2018-08-01) 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...
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 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...
Some new results on binary polynomial multiplication Cenk, Murat (2015-11-01) This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein's 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein's 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials ...
An improved algorithm for iterative matrix-vector multiplications over finite fields Mangır, Ceyda; Cenk, Murat; Manguoğlu, Murat (2018-11-09) Cryptographic computations such as factoring integers and computing discrete logarithms over finite fields require solving a large system of linear equations. When dealing with such systems iterative approaches such as Wiedemann or Lanczos are used. Both methods are based on the computation of a Krylov subspace in which the computational cost is often dominated by successive matrix-vector products. We introduce a new algorithm for computing iterative matrix-vector multiplications over finite fields. The pro...

Citation Formats

S. Ali and M. Cenk, “A New Algorithm for Residue Multiplication Modulo 2(521)-1,” 2016, vol. 10157, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32521.