Speeding up Curve25519 using Toeplitz Matrix-vector Multiplication

2018-01-24
Taskin, Halil Kemal
Cenk, Murat
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 discuss the comparison and implementation details. In addition, we evaluate the delay complexity of four-core almost embarrassingly parallel implementation of our algorithm when computations are performed using multi-core systems.

Suggestions

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...
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...
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...
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...
On the generalisation of special moduli for faster interleaved montgomery modular multiplication
AKLEYLEK, SEDAT; Cenk, Murat; Özbudak, Ferruh (2013-09-01)
In this study, the authors give a generalisation of special moduli for faster interleaved Montgomery modular multiplication algorithm with simplified pre-computational phase for GF(p(n)), where p 2 is a prime number and n is a positive integer. The authors propose different sets of moduli that can be used in elliptic curve crytographic applications and pairing-based cryptography. Moreover, this method also leads to efficient implementations for the elliptic curve parameters given in standards. It is shown t...
Citation Formats
H. K. Taskin and M. Cenk, “Speeding up Curve25519 using Toeplitz Matrix-vector Multiplication,” 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/31681.