DOCSLIB.ORG

Area-Time Efficient Hardware Architecture for Signature Based on Ed448 2943

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Area-Time Efficient Hardware Architecture for Signature Based on Ed448 2943 2942 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 68, NO. 8, AUGUST 2021 Area-Time Efficient Hardware Architecture for Signature Based on Ed448 Mojtaba Bisheh-Niasar , Reza Azarderakhsh , and Mehran Mozaffari Kermani , Senior Member, IEEE Abstract—In this brief, we proposed a highly-optimized FPGA- 521-bit, employing a hardware/software co-design approach. based implementation of the Ed448 digital signature algorithm. Additionally, a multi-core FPGA scheme was presented in [4] Despite significant progress in elliptic curve cryptography (ECC) with an arbitrary prime modulus up to 528-bit. It was applied implementations, Ed448 hardware architecture, to the best of our over different curves, including NIST curves, Brainpool P512 knowledge, has not been investigated in the literature. In this r1, and SEC P256 k1. However, leakage exploitation using work, we demonstrate a high throughput while maintaining low resource architecture for Ed448 by employing a new combined horizontal attacks was not studied in these works. algorithm for refined Karatsuba-based multiplier with precise Recently, few works implement Curve448 scalar multiplica- scheduling. Furthermore, a compact distributed memory unit is tion on FPGA. The authors in [5] proposed a high-performance developed to increase speed while keeping the area low. Our architecture employing schoolbook multiplication. It has been variable-base-point Ed448 architecture performs 327 signatures extended in [6] by adding some countermeasures to pro- and 189 verifications per second at a notably higher security level tect against side-channel analysis (SCA) attacks. The work of 224 bits, using not more than 6,617 Slices and 16 DSPs on a of [7] proposed a compact and fast point multiplication Xilinx Artix-7 FPGA. We also proposed possible countermeasures targeting area-constrained and time-constrained applications. and extensions to Ed448 to counter the physical attacks. Furthermore, [8] presented a LUT-based architecture employ- Index Terms—Elliptic curve cryptography, EdDSA, Ed448, ing the most significant digit multiplier. However, FPGA-based FPGA, side-channel. implementations of Curve448 cannot be directly compared to ours because Ed448 requires more computation units, e.g., hash core and modulo L reduction, and it takes more clock I. INTRODUCTION cycles. In particular, some dedicated optimizations cannot be applied to the Ed448 scheme since the architecture is reused to DWARDS-CURVES Digital Signature Algorithm perform non-modular multiplication and modulo L reduction. (EdDSA) is a secure digital signature algorithm (DSA) E The work of [9] presented an implementation of AVR supporting unified addition formulas without exception. and MSP embedded processors for Ed448. It employed the EdDSA was first proposed by Bernstein et al. [1], includ- subtractive Karatsuba approach to implementing three-level ing two efficient Ed25519 and Ed448 schemes offering Karatsuba multiplication and the two-level Karatsuba squaring 127 and 224-bit security, respectively. EdDSA satisfies high- on low-end Internet-of-Thing devices. performance and constant-time requirements for implementing This brief proposes, to the best of our knowledge, the applications at a higher security level as a part of the TLS and first hardware implementation of the Ed448, while previous OpenSSH protocols. Moreover, Safe-Curve policies are con- works implemented ECDSA. The EdDSA provides a high- sidered in designing EdDSA to address existing weaknesses secure scheme employing a hash function to generate a random in many ECDSA curves [2]. Since hybrid cryptosystems are number in a secretly deterministic way which copes with the crucial to transition to post-quantum cryptography (PQC), well-known pitfall in implementing DSA and ECDSA. We implementing a highly optimized EdDSA scheme is an active propose a new approach for implementing variable-base-point research area. Nevertheless, Ed448 has not got sufficient Ed448 architecture employing refined Karatsuba multiplication study, especially in the field of hardware implementation. combined with computation of the restricted-X coordinates of Most existing implementations of DSA bring into focus a point over its isogenous map to reduce the computation com- on at most 128-bit security level curves, such as NIST P256 plexity. Our proposed pure hardware implementation presents and Ed25519. However, [3] proposed a scalable architecture a novel design of the Ed448 algorithm, with higher efficiency, in prime fields over NIST recommended field sizes up to enhanced maximum frequency, and lower area costs, which is an essential aspect of the embedded system’s architecture. Manuscript received February 22, 2021; revised March 12, 2021; accepted March 19, 2021. Date of publication March 23, 2021; date of current version The previous works [3], [4] implemented a significant num- July 30, 2021. This work was supported in part by NSF under Grant 1801341, ber of DSP and BRAM cores; this could seriously reduce and in part by NIST under Grant 60NANB16D246. This brief was recom- the efficiency. Our work utilizes only a few DSPs, employing mended by Associate Editor C. W. Sham. (Corresponding author: Mojtaba pipeline architecture with comparable cycle counts, advancing Bisheh-Niasar.) Mojtaba Bisheh-Niasar and Reza Azarderakhsh are with CEECS the hardware implementation efficiency. Department, Florida Atlantic University, Boca Raton, FL 33431 USA (e-mail: The compact structure of the proposed design over other [email protected]; [email protected]). competitive designs can be used in embedded systems and Mehran Mozaffari Kermani is with the CSE Department, University of Internet-of-Things (IoT) devices where energy consumption is South Florida, Tampa, FL 33620 USA (e-mail: address: [email protected]). Color versions of one or more figures in this article are available at critical as a hardware accelerator to offload computations from https://doi.org/10.1109/TCSII.2021.3068136. the microcontroller units (MCU). In particular, our objective Digital Object Identifier 10.1109/TCSII.2021.3068136 is to create a system-on-chip (SoC) crypo-accelerator with an 1549-7747 c 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: University of South Florida. Downloaded on August 02,2021 at 22:52:27 UTC from IEEE Xplore. Restrictions apply. BISHEH-NIASAR et al.: AREA-TIME EFFICIENT HARDWARE ARCHITECTURE FOR SIGNATURE BASED ON ED448 2943 MCU such as ARM processors that achieve high area-time Algorithm 1 Proposed Point Multiplication Over Ed448 efficiency, rather than creating a very low area or ultra-high- Input: k, PEd_Base = (xEd_Base, yEd_Base) performance implementations at the high cost of the other. Require: QEd = k · PEd_Base Our implementation can also be integrated as an off-chip solu- Initial step: convert the base point from Edwards tion; however, other criteria, such as performance, are often curve to its isogenous Montgomery curve: PMont_Base = as important or more important than efficiency in the exter- isogenous_map(PEd_Base), kMont = k 2, P0 = 0, P1 = nal crypto-chip design, which is beyond of this work. Not only PMont_Base does our architecture inherently provide protection against tim- ing and SPA attacks, but also advanced security mechanisms 1: for i from 445 downto 0 do //Montgomery Ladder 2: if k = 0 then to avoid DPA attacks are included, which is missing in the Monti 3: P1 = P0 + P1 literature. We implement the proposed architecture in a Xilinx P = × P Artix-7 FPGA and provide performance evaluations. 4: 0 2 0 The rest of this brief is organized as follows: Section II 5: else 6: P0 = P0 + P1 reviews the preliminaries. Our proposed architecture is P = × P presented in Section III. Section IV presents the results and 7: 1 2 1 comparison with previous work. Finally, we conclude this brief 8: end if in Section V. 9: end for 10: YMont = y_recovery(P0 = (XMont, ZMont)) 11: (XEd, YEd, ZEd) = dual_isogenous_map(XMont, YMont, ZMont) II. PRELIMINARIES 12: (xEd, yEd) = (XEd/ZEd, YEd/ZEd) 2 2 Twisted Edwards curve E is defined by E : ax + y = 13: return QEd = (xEd, yEd) 2 2 1 + dx y where P = (x, y) ∈ Fp × Fp. Twisted Edwards curve offers fast execution of high-security elliptic curve cryptography employing a unified addition formula between In [12], the authors introduced an efficient unified point P3 = (x3, y3) = (x1, y1) + (x2, y2) such that: addition and a dedicated point doubling formula at the cost × + × × − × × of 10M + 1S + 1k + 7A and 3M + 4S + 5A, where M, S, k, ( , ) = x1 y2 x2 y1 , y1 y2 a x1 x2 x3 y3 (1) and A are a multiplication, a squaring, a multiplication by a 1+d×x1×x2×y1×y2 1−d×x1×x2×y1×y2 constant, and an addition cost, respectively. EdDSA provides a high-performance digital signature algo- rithm, which is more resistant to side-channel attacks [10]. The detailed schedule of the EdDSA algorithm can be described A. Side-Channel Protection into three major functions as follows: Since ECDSA is vulnerable against SCA shown in [14], • KeyGen(s): This algorithm takes a security parameter s as implementing the SCA-protected EdDSA algorithm gained input and outputs a secret (signing) key sk and a public more attention. Implementing a resistant scheme for Ed448 (verification) key pk with associated message space M. signatures can be achieved easier due to the unified addition • Sign(sk, m): This algorithm takes a secret key sk and a formula. On the other hand, although ECDSA can be broken message m ∈ M as input and outputs a signature (R, S). in case of reusing nonce in only a few signatures, Ed448 does • Verify(pk, m, R, S): This algorithm takes a public key pk, not employ a unique random number for each signature. a message m ∈ M, and a signature (R, S) as input and A constant-time and secret-independent computation should outputs an acceptance validation b ∈{0, 1}. be performed to protect architecture against timing and simple For correctness, we require that for all s ∈ Fp, for all power analysis (SPA) attacks.
Load more

Recommended publications
  • Elliptic Curve Arithmetic for Cryptography
    Elliptic Curve Arithmetic for Cryptography Srinivasa Rao Subramanya Rao August 2017 A Thesis submitted for the degree of Doctor of Philosophy of The Australian National University c Copyright by Srinivasa Rao Subramanya Rao 2017 All Rights Reserved Declaration I confirm that this thesis is a result of my own work, except as cited in the references. I also confirm that I have not previously submitted any part of this work for the purposes of an award of any degree or diploma from any University. Srinivasa Rao Subramanya Rao March 2017. Previously Published Content During the course of research contributing to this thesis, the following papers were published by the author of this thesis. This PhD thesis contains material from these papers: 1. Srinivasa Rao Subramanya Rao, A Note on Schoenmakers Algorithm for Multi Exponentiation, 12th International Conference on Security and Cryptography (Colmar, France), Proceedings of SECRYPT 2015, pages 384-391. 2. Srinivasa Rao Subramanya Rao, Interesting Results Arising from Karatsuba Multiplication - Montgomery family of formulae, in Proceedings of Sixth International Conference on Computer and Communication Technology 2015 (Allahabad, India), pages 317-322. 3. Srinivasa Rao Subramanya Rao, Three Dimensional Montgomery Ladder, Differential Point Tripling on Montgomery Curves and Point Quintupling on Weierstrass' and Edwards Curves, 8th International Conference on Cryptology in Africa, Proceedings of AFRICACRYPT 2016, (Fes, Morocco), LNCS 9646, Springer, pages 84-106. 4. Srinivasa Rao Subramanya Rao, Differential Addition in Edwards Coordinates Revisited and a Short Note on Doubling in Twisted Edwards Form, 13th International Conference on Security and Cryptography (Lisbon, Portugal), Proceedings of SECRYPT 2016, pages 336-343 and 5.
    [Show full text]
  • The Complete Cost of Cofactor H = 1
    The complete cost of cofactor h = 1 Peter Schwabe and Daan Sprenkels Radboud University, Digital Security Group, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands [email protected], [email protected] Abstract. This paper presents optimized software for constant-time variable-base scalar multiplication on prime-order Weierstraß curves us- ing the complete addition and doubling formulas presented by Renes, Costello, and Batina in 2016. Our software targets three different mi- croarchitectures: Intel Sandy Bridge, Intel Haswell, and ARM Cortex- M4. We use a 255-bit elliptic curve over F2255−19 that was proposed by Barreto in 2017. The reason for choosing this curve in our software is that it allows most meaningful comparison of our results with optimized software for Curve25519. The goal of this comparison is to get an under- standing of the cost of using cofactor-one curves with complete formulas when compared to widely used Montgomery (or twisted Edwards) curves that inherently have a non-trivial cofactor. Keywords: Elliptic Curve Cryptography · SIMD · Curve25519 · scalar multiplication · prime-field arithmetic · cofactor security 1 Introduction Since its invention in 1985, independently by Koblitz [34] and Miller [38], elliptic- curve cryptography (ECC) has widely been accepted as the state of the art in asymmetric cryptography. This success story is mainly due to the fact that attacks have not substantially improved: with a choice of elliptic curve that was in the 80s already considered conservative, the best attack for solving the elliptic- curve discrete-logarithm problem is still the generic Pollard-rho algorithm (with minor speedups, for example by exploiting the efficiently computable negation map in elliptic-curve groups [14]).
    [Show full text]
  • Montgomery Curves and the Montgomery Ladder
    Montgomery curves and the Montgomery ladder Daniel J. Bernstein and Tanja Lange Technische Universiteit Eindhoven, The Netherlands University of Illinois at Chicago, USA Abstract. The Montgomery ladder is a remarkably simple method of computing scalar multiples of points on a broad class of elliptic curves. This article surveys a wide range of topics related to the Montgomery ladder, both from the historical perspective of Weierstrass curves and from the modern perspective of Edwards curves. New material includes a full proof of a complete constant-time ladder algorithm suitable for cryptographic applications. This article is planned to appear as Chapter 4 of the book Topics in Com- putational Number Theory inspired by Peter L. Montgomery, edited by Joppe W. Bos and Arjen K. Lenstra. Two cross-references are to de- scriptions of ECM in FFT extension for algebraic-group factorization algorithms, by Richard P. Brent, Alexander Kruppa, and Paul Zimmer- mann, which is planned to appear as Chapter 9 of the same book. Author list in alphabetical order; see https://www.ams.org/profession/ leaders/culture/CultureStatement04.pdf. This work was supported by the Commission of the European Communities through the Horizon 2020 program under project ICT-645421 (ECRYPT-CSA), by the U.S. National Sci- ence Foundation under grants 1018836 and 1314919, and by the Netherlands Organisation for Scientific Research (NWO) under grant 639.073.005. “Any opinions, findings, and conclusions or recommendations expressed in this ma- terial are those of the author(s) and do not necessarily reflect the views of the National Science Foundation” (or other funding agencies).
    [Show full text]
  • 18.783 S17 Elliptic Curves Lecture 11: Index Calculus, Smooth Numbers, Factoring Integers
    18.783 Elliptic Curves Spring 2017 Lecture #11 03/15/2017 11 Index calculus, smooth numbers, and factoring integers Having explored generic algorithms for the discrete logarithm problem in some detail, we now consider a non-generic algorithm that uses index calculus. 1 This algorithm depends critically on the distribution of smooth numbers (integers with small prime factors), which naturally leads to a discussion of two algorithms for factoring integers that also depend on smooth numbers: the Pollard p − 1 method and the elliptic curve method (ECM). 11.1 Index calculus Index calculus is a method for computing discrete logarithms in the multiplicative group of a finite field. This might not seem directly relevant to the elliptic curve discrete logarithm problem, but aswe shall see whenwe discuss pairing-based cryptography, thesetwo problems are not unrelated. Moreover, the same approach can be applied to elliptic curves over non- prime finite fields, as well as abelian varieties of higher dimension [7, 8, 9].2 We will restrict our attention to the simplest case, a finite field of prime order p. Let us identify Fp ' Z=pZ with our usual choice of representatives for Z=pZ, integers in the interval [0;N], with N = p − 1. This innocent looking statement is precisely what will allow us to “cheat”, that is, to do something that a generic algorithm cannot. It lets us lift elements of Fp ' Z=pZ to the ring Z where we may consider their prime factorizations, something that makes no sense in a multiplicative group (every element is a unit) and is not available to a generic algorithm.
    [Show full text]
  • Twisted Edwards Curves
    Twisted Edwards Curves Daniel J. Bernstein1, Peter Birkner2, Marc Joye3, Tanja Lange2, and Christiane Peters2 1 Department of Mathematics, Statistics, and Computer Science (M/C 249) University of Illinois at Chicago, Chicago, IL 60607–7045, USA [email protected] 2 Department of Mathematics and Computer Science Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands [email protected], [email protected], [email protected] 3 Thomson R&D France Technology Group, Corporate Research, Security Laboratory 1 avenue de Belle Fontaine, 35576 Cesson-S´evign´eCedex, France [email protected] Abstract. This paper introduces “twisted Edwards curves,” a general- ization of the recently introduced Edwards curves; shows that twisted Edwards curves include more curves over finite fields, and in particular every elliptic curve in Montgomery form; shows how to cover even more curves via isogenies; presents fast explicit formulas for twisted Edwards curves in projective and inverted coordinates; and shows that twisted Edwards curves save time for many curves that were already expressible as Edwards curves. Keywords: Elliptic curves, Edwards curves, twisted Edwards curves, Montgomery curves, isogenies. 1 Introduction Edwards in [13], generalizing an example from Euler and Gauss, introduced an addition law for the curves x2 + y2 = c2(1 + x2y2) over a non-binary field k. Edwards showed that every elliptic curve over k can be expressed in the form x2 + y2 = c2(1 + x2y2) if k is algebraically closed. However, over a finite field, only a small fraction of elliptic curves can be expressed in this form. Bernstein and Lange in [4] presented fast explicit formulas for addition and doubling in coordinates (X : Y : Z) representing (x, y) = (X/Z, Y/Z) on an Edwards curve, and showed that these explicit formulas save time in elliptic- curve cryptography.
    [Show full text]
  • Isomorphism Classes of Edwards Curves Over Finite Fields
    Isomorphism classes of Edwards curves over finite fields Reza Rezaeian Farashahi Department of Computing Macquarie University Sydney, NSW 2109, Australia and Department of Mathematical Sciences Isfahan University of Technology P.O. Box 85145 Isfahan, Iran [email protected] Dustin Moody National Institute of Standards and Technology (NIST) 100 Bureau Drive, Gaithersburg, MD, 20899, USA [email protected] Hongfeng Wu College of Sciences North China University of Technology Beijing 100144, P.R. China [email protected] Abstract Edwards curves are an alternate model for elliptic curves, which have attracted notice in cryptography. We give exact formulas for the number of Fq-isomorphism classes of Edwards curves and twisted Ed­ wards curves. This answers a question recently asked by R. Farashahi and I. Shparlinski. 1 1 Introduction Elliptic curves have been an object of much study in mathematics. Recall that an elliptic curve is a smooth projective genus 1 curve, with a given rational point. The traditional model for an elliptic curve has been the Weierstrass equation 2 3 2 Y + a1XY + a3Y = X + a2X + a4X + a6; where the ai are elements in some field F. While other models for elliptic curves have long been known, in the past few years there has been renewed interest in these alternate models. This attention has primarily come from the cryptographic community. In 2007, Edwards proposed a new model for elliptic curves [10]. Let F be a field with characteristic p 6= 2. These original Edwards curves, defined over F, are given by the equation 2 2 2 2 2 EE;c : X + Y = c (1 + X Y ); (1) with c 2 F and c5 6= c.
    [Show full text]
  • The Elliptic Curve Method and Other Integer Factorization Algorithms
    The Elliptic Curve Method and Other Integer Factorization Algorithms John Wright April 12, 2012 Contents 1 Introduction 2 2 Preliminaries 3 2.1 Greatest common divisors and modular arithmetic . 3 2.2 Basic definitions and theorems . 6 2.3 RSACryptosystem........................ 9 3 Factorization Algorithms 11 3.1 TheSieveofEratosthenes . 11 3.2 TrialDivision ........................... 12 3.3 Fermat’sLittleTheorem . 13 3.4 PseudoprimeTest......................... 14 3.5 StrongPseudoprimeTest . 16 3.6 AmethodofFermat ....................... 17 3.7 TheQuadraticSieve . 20 3.8 Pollard Rho Factorization Method . 22 4 Elliptic curves 24 4.1 Additiononanellipticcurve . 28 4.2 Reduction of elliptic curves defined modulo N ......... 31 4.3 Reduction of curves modulo p .................. 33 4.4 Lenstra’s Elliptic Curve Integer Factorization Method . 34 4.5 TheECMintheprojectiveplane . 36 5 Improving the Elliptic Curve Method 38 5.1 MontgomeryCurves . 38 5.2 Addition for elliptic curves in Montgomery form . 42 5.3 Montgomery multiplication . 47 5.4 Recentdevelopments . 49 5.5 Conclusion ............................ 50 1 Chapter 1 Introduction The Fundamental Theorem of Arithmetic, first proved by Gauss [2], states that every positive integer has a unique factorization into primes. That is, for every positive integer N, a1 a2 ak N = p1 p2 ...pk where the pi’s are distinct primes and each ai is a positive integer. This paper is motivated by a computational question: given an arbitrary integer N, how might we find a non-trivial factor of N? That is, a factor of N other than N and 1. ± ± While it is computationally easy to multiply numbers together, factor- ing a general integer is “generally believed to be hard” [4].
    [Show full text]
  • Elliptic Curve Factorization Method (ECM), Introduced by Hendrik Lenstra [4], a Heuristically Subexponential Running
    18.783 Elliptic Curves Spring 2013 Lecture #12 03/19/2013 We now consider our first practical application of elliptic curves: factoring integers. Before presenting the elliptic curve method (ECM) for factoring integers, we first present an older algorithm of Pollard that motivates the ECM approach. 12.1 Pollard p − 1 method In 1974, Pollard introduced a randomized (Monte Carlo) algorithm for factoring integers [6]. It makes use of a smoothness parameter B. Algorithm 12.1 (Pollard p − 1 factorization). Input: An integer N to be factored and a smoothness bound B. Output: A proper divisor of N or failure. 1. Pick a random integer a 2 [1;N − 1]. 2. If d = gcd(a; N) is not 1 then return d. 3. Set b = a and for each prime ` ≤ B: a. Set b = b`e mod N, where `e−1 < N ≤ `e. b. If d = gcd(b − 1;N) is not 1 then return d if d < N or failure if d = N. 4. Return failure Rather than using a fixed bound B, we could simply let the algorithm keep running through primes ` until it either succeeds or fails in step 3b. But in practice one typically uses a very small smoothness bound B and switches to a different algorithm if the p − 1 method fail. In any case, it is convenient to have B fixed for the purposes of analysis. Theorem 12.2. Let p and q be prime divisors of N, and let `p and `q be the largest prime divisors of p − 1 and q − 1, respectively.
    [Show full text]
  • Montgomery Curves and Their Arithmetic the Case of Large Characteristic fields
    Montgomery curves and their arithmetic The case of large characteristic fields Craig Costello · Benjamin Smith A survey in tribute to Peter L. Montgomery Abstract Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery's x-only arithmetic and Ladder algorithm, x- only Diffie–Hellman, y-coordinate recovery, and 2-dimensional and Euclidean differential addition chains such as Montgomery's PRAC algorithm. Keywords Montgomery curve · Montgomery ladder · elliptic curve cryptog- raphy · scalar multiplication 1 Introduction Peter L. Montgomery's landmark 1987 paper Speeding the Pollard and elliptic curve methods of factorization [38] introduced what became known as Mont- gomery curves and the Montgomery ladder as a way of accelerating Lenstra's ECM factorization method [33]. However, they have gone on to have a far broader impact: while remaining a crucial component of modern factoring software, they have also become central to elliptic curve cryptography. Consider the following situation: let q be a prime power and let E be an elliptic curve over Fq, with group law ⊕, identity point O, and negation map . We have scalar multiplication endomorphisms on E defined by [k]: P 7−! P ⊕ · · · ⊕ P | {z } k times Craig Costello Microsoft Research, USA E-mail: [email protected] Benjamin Smith INRIA and Laboratoire d'Informatique de l'Ecole´ polytechnique (LIX), France E-mail: [email protected] 2 Craig Costello, Benjamin Smith for every k in Z (with [−k]P := [k]( P )).
    [Show full text]
  • CRYPTREC Review of Eddsa CRYPTREC EX-3003-2020
    CRYPTREC EX-3003-2020 CRYPTREC Review of EdDSA Steven D. Galbraith Mathematics Department, University of Auckland, Auckland, New Zealand. [email protected] 1 Executive summary The EdDSA signature scheme is a digital signature based on the Elliptic Curve Discrete Logarithm Problem (ECDLP). It was proposed by Bernstein, Duif, Lange, Schwabe and Yang in 2012 [18]. It builds on a long line of discrete logarithm based signature schemes, including Elgamal, Schnorr, DSA, and ECDSA. The security of signature schemes of this type is well-understood, and elliptic curve cryptography is a mature field. My conclusions and opinions: 1. EdDSA is a good design for a signature scheme (except perhaps for the key clamping, see Section 6.1, which seems to cause more difficulties than it provides benefits). 2. EdDSA is more closely related to Schnorr signatures than ECDSA, and so enjoys many of the rigorous security guarantees that are known for Schnorr signatures, including recent work on tight security proofs. 3. Deterministic signatures solve some of the security problems of discrete log signatures, but constant time imple- mentation is still critical in many settings. 4. EdDSA is superior to ECDSA when doing batch verification of a large number of signatures. 5. Curve 25519 provides a high level of security for the next 10-20 years, and 448-bit keys (such as in Ed448) are over-conservative and not recommended. 6. It is unlikely that quantum computers capable of solving 256-bit ECDLP instances can be built within the next 10 years. 7. I am confident that EdDSA using Curve25519 is a good signature scheme for use up to 2030.
    [Show full text]
  • Parallel Approaches for Efficient Scalar Multiplication Over Elliptic
    Parallel Approaches for Efficient Scalar Multiplication over Elliptic Curve Christophe Negre, Jean-Marc Robert To cite this version: Christophe Negre, Jean-Marc Robert. Parallel Approaches for Efficient Scalar Multiplication over Elliptic Curve. SECRYPT, Jul 2015, Colmar, France. pp.202-209, 10.5220/0005512502020209. hal- 01206530 HAL Id: hal-01206530 https://hal.archives-ouvertes.fr/hal-01206530 Submitted on 29 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Parallel Approaches for Efficient Scalar Multiplication over Elliptic Curve Christophe Negre1 and Jean-Marc Robert1 1 Equipe DALI (Universit´ede Perpignan) and LIRMM (Univ. de Montpellier, CNRS), Perpignan, France fchristophe.negre, [email protected] Keywords: Elliptic curve, parallel implementation, scalar multiplication, prime field, binary field. Abstract: This paper deals with parallel implementation of scalar multiplication over an elliptic curve. We present parallel approaches which split the scalar into two parts for E(Fp) or three parts for E(F2m ) and perform in parallel the scalar multiplication with each part of the scalar. We present timing results of these approaches implemented over an Intel Core i7 for NIST binary curves B233, B409 and for the twisted Edwards curve Curve25519 (Bernstein, 2006).
    [Show full text]
  • Edwards Curve Points Counting Method and Supersingular Edwards and Montgomery Curves III a PREPRINT
    EDWARDS CURVE POINTS COUNTING METHOD AND SUPERSINGULAR EDWARDS AND MONTGOMERY CURVES III A PREPRINT Ruslan Skuratovskii1 [email protected], [email protected] National Technical University of Ukraine ”KPI im. Igor Sikorsky ”, Institute of Cybernetics of NASU. April 23, 2020 ABSTRACT We consider algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn . It is known that many modern cryptosystems [11] can be naturally transformed into elliptic curves [5]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with correspondingfield characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve Ed[Fp] is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over Fpn in a finite field is inves- tigated and the field characteristic, where this degree is minimal, is found. A birational isomor- phism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over Fpn .
    [Show full text]