On inversion modulo pseudo-Mersenne primes

It is well established that the method of choice for implementing a side-channel secure modular inversion, is to use Fermat\u27s little theorem. So $1/x = x^{p-2} \bmod p$. This can be calculated using any multiply-and-square method safe in the knowledge that no branching or indexing with potentially secret data (such as $x$) will be required. However in the case where the modulus $p$ is a pseudo-Mersenne, or Mersenne, prime of the form $p=2^n-c$, where $c$ is small, this process can be optimized to greatly reduce the number of multiplications required. Unfortunately an optimal solution must it appears be tailored specifically depending on $n$ and $c$. What appears to be missing from the literature is a near-optimal heuristic method that works reasonably well in all cases

Low-Weight Primes for Lightweight Elliptic Curve Cryptography on 8-bit AVR Processors

Small 8-bit RISC processors and micro-controllers based on the AVR instruction set architecture are widely used in the embedded domain with applications ranging from smartcards over control systems to wireless sensor nodes. Many of these applications require asymmetric encryption or authentication, which has spurred a body of research into implementation aspects of Elliptic Curve Cryptography (ECC) on the AVR platform. In this paper, we study the suitability of a special class of finite fields, the so-called Optimal Prime Fields (OPFs), for a "lightweight" implementation of ECC with a view towards high performance and security. An OPF is a finite field Fp defined by a prime of the form p = u*2^k + v, whereby both u and v are "small" (in relation to 2^k) so that they fit into one or two registers of an AVR processor. OPFs have a low Hamming weight, which allows for a very efficient implementation of the modular reduction since only the non-zero words of p need to be processed. We describe a special variant of Montgomery multiplication for OPFs that does not execute any input-dependent conditional statements (e.g. branch instructions) and is, hence, resistant against certain side-channel attacks. When executed on an Atmel ATmega processor, a multiplication in a 160-bit OPF takes just 3237 cycles, which compares favorably with other implementations of 160-bit modular multiplication on an 8-bit processor. We also describe a performance-optimized and a security-optimized implementation of elliptic curve scalar multiplication over OPFs. The former uses a GLV curve and executes in 4.19M cycles (over a 160-bit OPF), while the latter is based on a Montgomery curve and has an execution time of approximately 5.93M cycles. Both results improve the state-of-the-art in lightweight ECC on 8-bit processors

A comparison of different finite fields for elliptic curve cryptosystems

AbstractWe examine the relative efficiency of four methods for finite field representation in the context of elliptic curve cryptography (ECC). We conclude that a set of fields called the optimized extension fields (OEFs) give greater performance, even when used with affine coordinates, when compared against the type of fields recommended in the emerging ECC standards. Although this performance advantage is only marginal, and hence, there is probably no need to change the current standards to allow OEF fields in standards compliant implementations

Efficient Arithmetic In (Pseudo-)Mersenne Prime Order Fields

Elliptic curve cryptography requires efficient arithmetic over the underlying field. In particular, fast implementation of multiplication and squaring over the finite field is required for efficient projective coordinate based scalar multiplication as well as for inversion using Fermat’s little theorem. In the present work we consider the problem of obtaining efficient algorithms for field multiplication and squaring. From a theoretical point of view, we present a number of algorithms for multiplication/squaring and reduction which are appropriate for different settings. Our algorithms collect together and generalise ideas which are scattered across various papers and codes. At the same time, we also introduce new ideas to improve upon existing works. A key theoretical feature of our work, which is not present in previous works, is that we provide formal statements and detailed proofs of correctness of the different reduction algorithms that we describe. On the implementation aspect, a total of fourteen primes are considered, covering all previously proposed cryptographically relevant (pseudo-)Mersenne prime order fields at various security levels. For each of these fields, we provide 64-bit assembly implementations of the relevant multiplication and squaring algorithms targeted towards two different modern Intel architectures. We were able to find previous 64-bit implementations for six of the fourteen primes considered in this work. On the Haswell and Skylake processors of Intel, for all the six primes where previous implementations are available, our implementations outperform such previous implementations

Fast implementation of Curve25519 using AVX2

AVX2 is the newest instruction set on the Intel Haswell processor that provides simultaneous execution of operations over vectors of 256 bits. This work presents the advances on the applicability of AVX2 on the development of an efficient software implementation of the elliptic curve Diffie-Hellman protocol using the Curve25519 elliptic curve. Also, we will discuss some advantages that vector instructions offer as an alternative method to accelerate prime field and elliptic curve arithmetic. The performance of our implementation shows a slight improvement against the fastest state-of-the-art implementations.AVX2 is the newest instruction set on the Intel Haswell processor that provides simultaneous execution of operations over vectors of 256 bits. This work presents the advances on the applicability of AVX2 on the development of an efficient software impleme9230329345FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOSEM INFOMAÇÃO4th International Conference on Cryptology and Information Security in Latin AmericaThe authors would like to thank the anonymous reviewers for their helpful suggestions and comments. Additionally, they would like to show their gratitude to J´er´emie Detrey for his valuable comments on an earlier version of the manuscrip

Fast and compact elliptic-curve cryptography


Elliptic curve cryptosystems have improved greatly in speed over the past few years. In this paper we outline a new elliptic curve signature and key agreement implementation which achieves record speeds while remaining relatively compact. For example, on Intel Sandy Bridge, a curve with about $2^{250}$ points produces a signature in just under 60k clock cycles, verifies in under 169k clock cycles, and computes a Diffie-Hellman shared secret in under 153k clock cycles. Our implementation has a small footprint: the library is under 55kB. We also post competitive timings on ARM processors, verifying a signature in under 626k Tegra-2 cycles. We introduce faster field arithmetic, a new point compression algorithm, an improved fixed-base scalar multiplication algorithm and a new way to verify signatures without inversions or coordinate recovery. Some of these improvements should be applicable to other systems

Aspects of hardware methodologies for the NTRU public-key cryptosystem

Cryptographic algorithms which take into account requirements for varying levels of security and reduced power consumption in embedded devices are now receiving additional attention. The NTRUEncrypt algorithm has been shown to provide certain advantages when designing low power and resource constrained systems, while still providing comparable security levels to higher complexity algorithms. The research presented in this thesis starts with an examination of the general NTRUEncrypt system, followed by a more practical examination with respect to the IEEE 1363.1 draft standard. In contrast to previous research, the focus is shifted away from specific optimizations but rather provides a study of many of the recommended practices and suggested optimizations with particular emphasis on polynomial arithmetic and parameter selection. Various methods are examined for storing, inverting and multiplying polynomials used in the system. Recommendations for algorithm and parameter selection are made regarding implementation in software and hardware with respect to the resources available. Although the underlying mathematical principles have not been significantly questioned, stable recommended practices are still being developed for the NTRUEncrypt system. As a further complication, recommended optimizations have come from various researchers and have been split between hardware and software implementations. In this thesis, a generic VHDL model is presented, based on the IEEE 1363.1 draft standard, which is designed for adaptation to software or hardware implementation while providing flexibility for changes in recommended practices

Customisable arithmetic hardware designs

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