Multi-Base Chains for Faster Elliptic Curve Cryptography

This research addresses a multi-base number system (MBNS) for faster elliptic curve cryptography (ECC). The emphasis is on speeding up the main operation of ECC: scalar multiplication (tP). Mainly, it addresses the two issues of using the MBNS with ECC: deriving optimized formulas and choosing fast methods. To address the first issue, this research studies the optimized formulas (e.g., 3P, 5P) in different elliptic curve coordinate systems over prime and binary fields. For elliptic curves over prime fields, affine Weierstrass, Jacobian Weierstrass, and standard twisted Edwards coordinate systems are reviewed. For binary elliptic curves, affine, Lambda-projective, and twisted mu4-normal coordinate systems are reviewed. Additionally, whenever possible, this research derives several optimized formulas for these coordinate systems. To address the second issue, this research theoretically and experimentally studies the MBNS methods with respect to the average chain length, the average chain cost, and the average conversion cost. The reviewed MBNS methods are greedy, ternary/binary, multi-base NAF, tree-based, and rDAG-based. The emphasis is on these methods\u27 techniques to convert integer t to multi-base chains. Additionally, this research develops bucket methods that advance the MBNS methods. The experimental results show that the MBNS methods with the optimized formulas, in general, have good improvements on the performance of scalar multiplication, compared to the single-base number system methods

Efficient and Secure ECDSA Algorithm and its Applications: A Survey

Public-key cryptography algorithms, especially elliptic curve cryptography (ECC)and elliptic curve digital signature algorithm (ECDSA) have been attracting attention frommany researchers in different institutions because these algorithms provide security andhigh performance when being used in many areas such as electronic-healthcare, electronicbanking,electronic-commerce, electronic-vehicular, and electronic-governance. These algorithmsheighten security against various attacks and the same time improve performanceto obtain efficiencies (time, memory, reduced computation complexity, and energy saving)in an environment of constrained source and large systems. This paper presents detailedand a comprehensive survey of an update of the ECDSA algorithm in terms of performance,security, and applications

Efficient Arithmetic for the Implementation of Elliptic Curve Cryptography

The technology of elliptic curve cryptography is now an important branch in public-key based crypto-system. Cryptographic mechanisms based on elliptic curves depend on the arithmetic of points on the curve. The most important arithmetic is multiplying a point on the curve by an integer. This operation is known as elliptic curve scalar (or point) multiplication operation. A cryptographic device is supposed to perform this operation efficiently and securely. The elliptic curve scalar multiplication operation is performed by combining the elliptic curve point routines that are defined in terms of the underlying finite field arithmetic operations. This thesis focuses on hardware architecture designs of elliptic curve operations. In the first part, we aim at finding new architectures to implement the finite field arithmetic multiplication operation more efficiently. In this regard, we propose novel schemes for the serial-out bit-level (SOBL) arithmetic multiplication operation in the polynomial basis over F_2^m. We show that the smallest SOBL scheme presented here can provide about 26-30\% reduction in area-complexity cost and about 22-24\% reduction in power consumptions for F_2^{163} compared to the current state-of-the-art bit-level multiplier schemes. Then, we employ the proposed SOBL schemes to present new hybrid-double multiplication architectures that perform two multiplications with latency comparable to the latency of a single multiplication. Then, in the second part of this thesis, we investigate the different algorithms for the implementation of elliptic curve scalar multiplication operation. We focus our interest in three aspects, namely, the finite field arithmetic cost, the critical path delay, and the protection strength from side-channel attacks (SCAs) based on simple power analysis. In this regard, we propose a novel scheme for the scalar multiplication operation that is based on processing three bits of the scalar in the exact same sequence of five point arithmetic operations. We analyse the security of our scheme and show that its security holds against both SCAs and safe-error fault attacks. In addition, we show how the properties of the proposed elliptic curve scalar multiplication scheme yields an efficient hardware design for the implementation of a single scalar multiplication on a prime extended twisted Edwards curve incorporating 8 parallel multiplication operations. Our comparison results show that the proposed hardware architecture for the twisted Edwards curve model implemented using the proposed scalar multiplication scheme is the fastest secure SCA protected scalar multiplication scheme over prime field reported in the literature

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a mod-ified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as Pn i=1 2 bi3ti where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we pro-pose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups

A Survey of Fast Scalar Multiplication on Elliptic Curve Cryptography for Lightweight Embedded Devices

Elliptic curve cryptography (ECC) is one of the most famous asymmetric cryptographic schemes which offers the same level of security with much shorter keys than the other widely used asymmetric cryptographic algorithm, Rivest, Shamir, and Adleman (RSA). In ECC, the main and most heavily used operation is the scalar multiplication kP, where the scalar value k is a private integer and must be secured. Various methods for fast scalar multiplication are based on the binary/ternary representation of the scalar. In this chapter, we present various methods to make fast scalar multiplication on ECC over prime field for lightweight embedded devices like wireless sensor network (WSN) and Internet of Things (IoT)

Optimizing double-base elliptic-curve single-scalar multiplication

This paper analyzes the best speeds that can be obtained for single-scalar multiplication with variable base point by combining a huge range of options: • many choices of coordinate systems and formulas for individual group operations, including new formulas for tripling on Edwards curves; • double-base chains with many different doubling/tripling ratios, including standard base-2 chains as an extreme case; • many precomputation strategies, going beyond Dimitrov, Imbert, Mishra (Asiacrypt 2005) and Doche and Imbert (Indocrypt 2006). The analysis takes account of speedups such as S – M tradeoffs and includes recent advances such as inverted Edwards coordinates. The main conclusions are as follows. Optimized precomputations and triplings save time for single-scalar multiplication in Jacobian coordinates, Hessian curves, and tripling-oriented Doche/Icart/Kohel curves. However, even faster single-scalar multiplication is possible in Jacobi intersections, Edwards curves, extended Jacobi-quartic coordinates, and inverted Edwards coordinates, thanks to extremely fast doublings and additions; there is no evidence that double-base chains are worthwhile for the fastest curves. Inverted Edwards coordinates are the speed leader

Faster computation of the Tate pairing

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in the addition and doubling. Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also speed up pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.Comment: 15 pages, 2 figures. Final version accepted for publication in Journal of Number Theor