Abstract: We present an algorithm that by using the τ and τ −1 Frobenius operators concurrently allows us to obtain a parallelized version of the classical τ-and-add scalar multiplication algorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to efficient implementations of both τ and τ −1, thus showing that our algorithm can be effectively applied on all the NIST-recommended curves. We also present design details of software and hardware implementations of our procedure. In a two-processor workstation software implementation, we report experimental data showing that our parallel algorithm is able to achieve a speedup factor of almost 2 when compared with the standard sequential point multiplication. In our hardware implementation, the parallel version yields a more modest acceleration of 17 % when compared with the traditional point multiplication algorithm. Although the focus is on Koblitz curves, analogous strategies are discussed for other curves, in particular for random curves over binary fields.

Abstract. Fast arithmetic for characteristic three finite fields F3 m is desirable in pairing-based cryptography because there is a suitable family of elliptic curves over F3 m having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of F3 m, and use the results to devise faster multiplication algorithms. We carefully compare multiplication in F3 m using polynomial bases and Gaussian normal bases. Finally, we compare the speed of encryption and decryption for the Boneh-Franklin and Sakai-Kasahara identity-based encryption schemes at the 128-bit security level, in the case where supersingular elliptic curves with embedding degrees 2, 4 and 6 are employed. 1.

Abstract. Finitefieldarithmetichasreceivedaconsiderableattentioninthecurrentcryptographic applications. Many researchers have focused on finite field multiplication due to its importance in various cryptographic operations. Moreover, finite field multiplication can be considered as a cornerstone for elliptic curve cryptosystems. Fan and Hasan [1] introduced a new sub-quadratic computational complexity approach for finite field multiplication. It is based on Toeplitz matrix-vector products. In this paper we consider efficient implementation of this approach on general purpose processors usingType II Optimal Normal Basis (ONB II). To this end, a memory and time efficient implementation scheme is proposed for the Fan and Hasan approach. Also, in this paper we provide a modified version of the best quadratic complexity multiplication algorithm due to Reyhani-Masoleh [2]. The proposed modification reduces the number of OR and SHIFT instructions by 50% and the number of AND instructions by about 25%. We simulate the implementation on three different architectures and present the results. Furthermore, we present an idea to fully parallelize the implementation of the Fan and Hasan scheme.

Efficient and high-performance implementation of point multiplication is crucial for elliptic curve cryptosystems. In this paper, we present a new double point multiplication algorithm based on differential addition chains. We use our scheme to implement single point multiplication on binary elliptic curves with efficiently computable endomorphisms. Our proposed scheme has a uniform structure and has some degree of built-in resistance against side channel analysis attacks. We design a crypto-processor based on the proposed algorithm for double point multiplication and evaluate its area and time efficiency on FPGA. To the best of the authors ’ knowledge, this is the first hardware implementation of single point multiplication (using double point multiplication) on elliptic curves with efficiently computable endomorphisms. Our analysis and timing results show that the expected acceleration in point multiplication is considerable. Prototypes of the proposed architectures are implemented and experimental results are presented.