High-performance implementation approach of elliptic curve cryptosystem for wireless network applications

Elliptic curve cryptosystems (ECCs) are becoming more popular because of the reduced number of key bits required in comparison to other cryptosystems (e.g. a 160 bit ECC has roughly the same security as 1024 bit RSA). ECCs are especially suited to smart cards because of the limited memory and computational power available on these devices. However, the side-channel attacks which have recently been proposed can obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time. This information may be used to break implementations that have not incorporated defences against these attacks. This paper presents a new defence against Simple Power Analysis (SPA). This new defence is based on the NAF (non-adjacent form) representation of a scalar and requires 44% fewer additions and 11% extra doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm.

In this paper, we research the operation rules and scalar multiplication of Twisted Edwards elliptic curve. Based on the properties of point addition and double point operations, we propose the fast continuous double point algorithm, continuous point addition algorithm and continuous double point-point addition algorithm on Twisted Edwards elliptic curve. These algorithms reduce the amount of operations significantly by reducing the inverse operations. For comparison purposes, the top-level operations of the new algorithm use the non-adjacent form(NAF) and double base number system(DBNS) scalar multiplication algorithms. In contrast, it effectively improves the computational efficiency of the Twisted Edwards elliptic curve scalar multiplication algorithm, in which it is more than 50% faster than the underlying operation using Horner's rule, continuous multiplier algorithm, and about 10% to 35% or more faster than the base operation using the 4-NNAF algorithm.

The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms—including Gallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) multiplication, as well as higher-dimensional and higher-genus constructions—is to produce a short basis of a certain integer lattice involving the eigenvalues of the endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar coefficients, and the faster the resulting scalar multiplication. Typically, we use the eigenvalues to write down a long basis, which we then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more specialized algorithm. In this work, we use elementary facts about quadratic rings to immediately write down a short basis for the elliptic curve GLV, GLS, GLV+GLS, and Q \mathbb {Q} -curve constructions, and for genus 2 real multiplication constructions. We do not pretend that this represents a significant optimization in scalar multiplication, since the lattice reduction step is always an offline precomputation—but it does give a better insight into the structure of scalar decompositions. In any case, it is always more convenient to use a ready-made short basis than it is to compute a new one.

This work presents an evaluation of different software implementations of algorithms to compute the most demanding operation in cryptographic schemes based on Elliptic Curve Cryptography (ECC), the scalar multiplication. Five different methods were studied, including traditional and more sophisticated methods for elliptic curve cryptography defined over prime and binary fields. For evaluation, the key generation algorithm in ECC consisting on a single scalar multiplication was implemented on a P500h LG smarthphone, which includes an ARM processor running at 600MHz. Both execution time and memory usage was evaluated. It was found that in general, scalar multiplication runs 8 times faster for ECC defined over prime fields, being the NAF the best performer method. For ECC over binary fields, the best performer method was wNAF. The results presented in this work could help a designer to select the most appropriate method when implementing ECC-based cryptographic schemes such as encryption or digital signatures on mobile devices like smartphones, meeting implementation requirements in terms of execution time and memory usage.

Parallelization of scalable elliptic curve cryptosystem processors in GF(2m)

The work presents an elliptic curve crypto accelerator to implement scalar multiplication on standardized NIST curve over Galois binary fields by using polynomial basis. The proposed design provides efficient hardware utilization and low power consumption. For this purpose, hybrid finite field multiplier is proposed using standard Karatsuba multiplier and shlft-and-add multiplication algorithms. The proposed hybrid finite field multiplier can perform one finite field multiplication in $\frac{m}{2}$ clock cycles, with m being the key-length. Montgomery algorithm with projective coordinates (LopezDahab) is used for the computation of scalar multiplication. An FSM based control unit is developed, governing the overall operations of proposed accelerator. The proposed architecture has been modeled in Verilog-HDL and implemented up to place and route level using the Xilinx Vivado and ISE design software for various FPGA devices. The achieved results demonstrate the convenience of proposed ECC accelerator for resource constrained embedded systems applications requiring low area and low power hardware.

We present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time 2M + 5S + 2m c and for addition of points in time 7M + 2S, where M is the cost of multiplication, S the cost of squaring, and m c the cost of multiplication by a constant. By a study of the Kummer curves \(\mathcal{K}\) = E/{[±1]}, we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with 4M + 4S + 2m c + m t per bit where m t is multiplication by a constant that depends on P.

We present a new, misuse-resistant scheme for online authenticated encryption, following the framework set forth by Fleischmann et al. (FSE 2012). Our scheme, COBRA, is roughly as efficient as the GCM mode of operation for nonce-based authenticated encryption, performing one block cipher call plus one finite field multiplication per message block in a parallelizable way. The major difference from GCM is that COBRA preserves privacy up to prefix under nonce repetition. However, COBRA only provides authenticity against nonce-respecting adversaries. As compared to COPA (ASIACRYPT 2013), our new scheme requires no block cipher inverse and hence enjoys provable security under a weaker assumption about the underlying block cipher. In addition, COBRA can possibly perform better than COPA on platforms where finite field multiplication can be implemented faster than the block cipher in use, since COBRA essentially replaces half of the block cipher calls in COPA with finite field multiplications.

In this paper we introduce scalar multiplication algorithms for several classes of elliptic and hyperelliptic curves. The methods are variations on Yao's scalar multiplication algorithm where independent group operations are shown in an explicit way. We can thusmerge several group operations and reduce the number of field operations by means of Montgomery's trick. The results are that scalar multiplication on elliptic curves in even characteristic based on point halving can be improved by at least 10% and the performance of Koblitz curves by 25% to 32%.

An area-efficient bit-serial sequential polynomial basis finite field GF(2m) multiplier

We propose to represent a scalar in the form of d (1/2)(superscript a)3(superscript b), where a and b are integers and d is an odd integer belonging to a given set. This representation is a combination of the extended double-base number system (DBNS) and the double-base chain representation using powers of 1/2 and 3. Experimental results show that our approach leads to a shorter DBNS expansion and a lower complexity in elliptic curve scalar multiplication, with the cost of only a few pre-computations and storages. This contributes to the efficient implementation of elliptic curve cryptography (ECC).

Extension of Montgomery multiplication algorithms in GF(p) are studied and analyzed. The time and space requirements of various state-of-the-art algorithms are presented. We propose Modified Montgomery Modular Multiplication Algorithms that reduces the number of computational operations such as number of additions, memory reads and writes involved in the existing algorithms, thereby, saving considerable time and area for execution. Many design examples has been solved to prove the theoretical correctness of the proposed algorithms. Complexity analysis shows that Modified Coarsely Integrated Scanning (MCIOS) consume less space and time compared to other modified Montgomery Algorithms. To verify the logical correctness, the proposed MCIOS algorithm was implemented in Xilinx Spartan3E FPGA. The total memory for execution of 64-bit operand is 135484 KB for MCIOS and 140496 KB for existing Coarsely Integrated Scanning (CIOS) method. The proposed algorithm can be changed to be suitable for any arbitrary Galois field size with little modifications. Also the proposed algorithm can be developed as architecture suitable for System on Chip (SoC) implementations of Elliptic curve cryptosystem. Subsequently, the system can be developed as a 3D chip.

Embedded systems are increasingly providing secure functionalities, which often rely on some dedicated hardware for symmetric and public-key cryptography. When resources are limited, elliptic curve cryptography (ECC) may be chosen instead of the more widely known RSA, which needs much longer keys for the same security level. However, ECC may be vulnerable, as any other cryptographic implementation, to side channel analysis, which may reveal secret information by analyzing collateral sources of information, such as power consumption. Countermeasures must be thus adopted at the design level, in order to ensure robust and secure operation of the device. We propose here a new scalar multiplication algorithm on an elliptic curve, based on a novel randomized window method. This design is protected against side channel attacks (Timing, Simple and Differential Power Analysis) and it is implemented over prime fields, but it can be applied to binary fields as well. In order to evaluate this countermeasure, we provide its costs, and an estimation of the additional entropy added to the computation against side channels attacks.

In this paper, techniques for fast implementations of Montgomery's modular multiplication algorithm are described. These use non-redundant multi-bit recoding. These are compared regarding the area and computation requirements and compared with conventional implementation.

This paper presents a high performance algorithm for modular multiplication on a graphics processing unit (GPU) implemented in assembler. The proposed algorithm carries out finite field multiplication over the NIST prime fields of size 192, 224, 256 and 384 bits. Included is a detailed explanation of our algorithm, an instruction count analysis, and a comparison to recently published work; compared to the next fastest design, the proposed algorithm's execution time is 27 to 71 times faster.