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November  2020, 14(4): 703-726. doi: 10.3934/amc.2020090

Speeding up regular elliptic curve scalar multiplication without precomputation

1. 

Institute of Mathematics, State Academy of Sciences and, PGItech Corp., Pyongyang, Democratic People's Republic of Korea

2. 

Institute of Mathematics, State Academy of Sciences, Pyongyang, Democratic People's Republic of Korea

3. 

PGItech Corp., Pyongyang, Democratic People's Republic of Korea

Received  August 2019 Revised  March 2020 Published  July 2020

This paper presents a series of Montgomery scalar multiplication algorithms on general short Weierstrass curves over fields with characteristic greater than 3, which need only 12 field multiplications per scalar bit using 8 $ \sim $ 9 field registers, thus outperform the binary NAF method on average. Over binary fields, the Montgomery scalar multiplication algorithm which was presented at the first CHES workshop by López and Dahab has been a favorite of ECC implementors, due to its nice properties such as high efficiency (outperforming the binary NAF), natural SPA-resistance, generality (coping with all ordinary curves) and implementation easiness. Over odd characteristic fields, the new scalar multiplication algorithms are the first ones featuring all these properties. Building-blocks of our contribution are new efficient differential addition-and-doubling formulae and a novel conception of on-the-fly adaptive coordinates which varies in accordance with not only the base point but also the bits of the given scalar.

Citation: Kwang Ho Kim, Junyop Choe, Song Yun Kim, Namsu Kim, Sekung Hong. Speeding up regular elliptic curve scalar multiplication without precomputation. Advances in Mathematics of Communications, 2020, 14 (4) : 703-726. doi: 10.3934/amc.2020090
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show all references

References:
[1]

T. Akishita and T. Takagi, Zero-value point attacks on elliptic curve cryptosystem, in ISC 2003 (eds. C. Boyd and W. Mao), Springer, (2003), 218–233. doi: 10.1007/10958513_17.  Google Scholar

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[3]

A. BauerE. JaulmesE. ProuffJ. R. Reinhard and J. Wild, Horizontal collision correlation attack on elliptic curves - extended version, Cryptography and Communications, 7 (2015), 91-119.  doi: 10.1007/s12095-014-0111-8.  Google Scholar

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[5]

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D. J. Bernstein and T. Lange, Faster addition and doubling on elliptic curves, in ASIACRYPT 2007 (ed. K. Kurosawa), Springer, (2007), 29–50. doi: 10.1007/978-3-540-76900-2_3.  Google Scholar

[9]

O. Billet and M. Joye, The jacobi model of an elliptic curve and side-channel analysis, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (eds. M. Fossorier, T. Hoeholdt and A. Poli), Springer, (2003), 34–42. doi: 10.1007/3-540-44828-4_5.  Google Scholar

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[11]

J. W. BosC. CostelloP. Longa and M. Naehrig, Selecting elliptic curves for cryptography: An efficiency and security analysis, J. Cryptogr. Eng., 6 (2016), 259-286.  doi: 10.1007/s13389-015-0097-y.  Google Scholar

[12]

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[13]

D. R. L. Brown, Multi-Dimensional Montgomery Ladders for Elliptic Curves, Cryptology ePrint Archive, Report 2006/220, 2006. Available from: http://eprint.iacr.org/2006/220. Google Scholar

[14]

C. Research, Standards for Efficient Cryptography, SEC 2: Recommended Elliptic Curve Domain Parameters, Version 2.0, 2010. Available from: http://www.secg.org/. Google Scholar

[15]

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[16]

P. N. Chung, C. Costello and B. Smith, Fast, Uniform, and Compact Scalar Multiplication for Elliptic Curves and Genus 2 Jacobians with Applications to Signature Schemes, Cryptology ePrint Archive, Report 2015/983, 2015. Available from: http://eprint.iacr.org/2015/983.  Google Scholar

[17]

C. Clavier, B. Feix, G. Gagnerot, M. Roussellet and V. Verneuil, Horizontal correlation analysis on exponentiation, in ICISC 2010 (eds. M. Soriano, S. Qing and J. López), Springer, (2010), 46–61. Google Scholar

[18]

J. Coron, Resistance against differential power analysis for elliptic curve cryptosystems, in CHES 1999 (eds. Ç. K. Koç and C. Paar), Springer, (1999), 292–302. doi: 10.1007/3-540-48059-5_25.  Google Scholar

[19]

C. Costello and P. Longa, FourQ: Four-dimensional decompositions on a Q-curve over the mersenne prime, in ASIACRYPT 2015 (eds. T. Iwata and J. H. Cheon), Springer, (2015), 214–235. doi: 10.1007/978-3-662-48797-6_10.  Google Scholar

[20]

J. -C. Courrège, B. Feix and M. Roussellet, Simple power analysis on exponentiation revisited, in CARDIS 2010 (eds. D. Gollmann, J. -L. Lanet and J. Iguchi-Cartigny), Springer, (2010), 65–79. Google Scholar

[21]

J. -L. Danger, S. Guilley, P. Hoogvorst, C. Murdica and D. Naccache, Improving the big mac attack on elliptic curve cryptography, The New Codebreakers, Lecture Notes in Comput. Sci., 9100, Springer, Berlin, 2016,374–386. Available from: http://eprint.iacr.org/2015/819. doi: 10.1007/978-3-662-49301-4_23.  Google Scholar

[22]

C. Doche, T. Icart and D. R. Kohel, Efficient scalar multiplication by isogeny decompositions, in PKC 2006 (eds. M. Yung, Y. Dodis, A. Kiayias and T. Malkin), Springer, (2006), 191–206. doi: 10.1007/11745853_13.  Google Scholar

[23]

E. Brainpool, ECC Brainpool Standard Curves and Curve Generation, 2005. Available from: http://www.ecc-brainpool.org/download/Domain-parameters.pdf. Google Scholar

[24]

J. Fan, X. Guo, E. D. Mulder, P. Schaumont, B. Preneel and I. Verbauwhede, State-of-the-art of secure ECC implementations: A survey on known side-channel attacks and countermeasures, in HOST 2010, (2010), 76–87. doi: 10.1109/HST.2010.5513110.  Google Scholar

[25]

B. Fay, Double-and-Add with Relative Jacobian Coordinates, Cryptology ePrint Archive, Report 2014/1014, 2014. Available from: http://eprint.iacr.org/2014/1014. Google Scholar

[26]

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Table 1.  Comparison of costs per bit and memory usages between regular scalar multiplication algorithms for short Weierstrass curves. Algorithms specific to a special curve family are marked by (*). $ E $ is the cost of a secured on-the-fly NAF scalar recoding. $ E' $ equals $ E $ plus the precomputation cost per scalar bit. A field negation is assumed to cost $ 0.5A $
Algorithm Cost per scalar bit #field regs.
Classic
Double-addition-always [18,70] $ 11M+9S+M_\mathfrak{a}+18A $ 12
Window algorithm based on Joye-Tunstall recoding
Rivain [70] $ \frac{(4M+4S+11A)\omega+12M+4S+8A}{\omega}+E' $ $ 3\cdot2^{\omega-1}+5 $
Rivain* [70] $ \frac{(4M+4S+12A)\omega+12M+4S+8A}{\omega}+E' $ $ 3\cdot2^{\omega-1}+4 $
Bos et al.* [11] $ \frac{(4M+4S+8A)\omega+12M+1S+5A}{\omega}+E' $ $ 5\cdot2^{\omega-1}+8 $
Indistinguishable formulae
Unified formulae [12] $ 16M+\frac{20}{3}S+\frac{4}{3}M_\mathfrak{a}+\frac{40}{3}A+\frac{4}{3}E $ N/A
Complete formulae* [69,59] $ 16M+4M_\mathfrak{a}+\frac{8}{3}M_\mathfrak{b}+36A+\frac{4}{3}E $ N/A
Atomicity
Chevallier et al. [15] $ \frac{41}{3}M+\frac{205}{6}A+\frac{41}{3}E $ 10
Longa* [56] $ 6M+6S+24A+6E $ 12
Giraud-Verneuil [30] $ 7M+7S+28A+7E $ 11
Giraud-Verneuil* [30] $ 10M+\frac{10}{3}S+\frac{50}{3}A+\frac{5}{3}E $ 11
Rondepierre [72] $ \frac{32}{3}M+4S+12A+\frac{4}{3}E $ 11
Rondepierre* [72,71] $ \frac{32}{3}M+\frac{8}{3}S+\frac{40}{3}A+\frac{4}{3}E $ 11
Montgomery ladder and Joye's double-add
Brier-Joye [12] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+15A $ N/A
Izu-Takagi [45] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+15A $ 18
Fischer et al. [27] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+14A $ 9
Izu et al. [44] $ 10M+4S+2M_\mathfrak{a}+1M_\mathfrak{b}+18A $ 8
Goundar et al. Alg.13 [32,31] $ 9M+7S+27A $ 9
Goundar et al. Alg.14 [32] $ 8M+7S+3M_\mathfrak{a}+1M_\mathfrak{b}+27A $ 8
Venelli-Dassance [73] $ 9M+5S+ $(N/A)$ A $ N/A
Hutter et al. Alg.5 [43] $ 9M+5S+1M_\mathfrak{a}+1M_\mathfrak{b}+16A $ 8
Hutter et al. Alg.6 [43] $ 10M+5S+13A $ 10
Goundar et al. [33] $ 9M+5S+19A $ 10
Rivain Alg.13+Alg.14 [70] $ 9M+5S+18A $ 8
Rivain suggestion [70] $ 8M+6S+26A $ 8
Fay [25] $ 13M+5S+14A $ 8
Chung et al. [16] $ 8M+7S+2M_\mathfrak{a}+3M_\mathfrak{b}+12A $ N/A
This work (Sec. 3.2) 8M+4S+15.5A 8
This work (Sec. 3.3) 8M+4S+12.5A 9
Hamburg [35] 8M+4S+9A 8
This work (Sec. 3.4) 6M+6S+20A 9
This work (Sec. 3.4) 6M+6S+18A 9
Binary NAF method vulnerable to SPA: for the purpose of comparison only
[37,44] $ \frac{20}{3}M+7S+\frac{40}{3}A+E $ 8
Algorithm Cost per scalar bit #field regs.
Classic
Double-addition-always [18,70] $ 11M+9S+M_\mathfrak{a}+18A $ 12
Window algorithm based on Joye-Tunstall recoding
Rivain [70] $ \frac{(4M+4S+11A)\omega+12M+4S+8A}{\omega}+E' $ $ 3\cdot2^{\omega-1}+5 $
Rivain* [70] $ \frac{(4M+4S+12A)\omega+12M+4S+8A}{\omega}+E' $ $ 3\cdot2^{\omega-1}+4 $
Bos et al.* [11] $ \frac{(4M+4S+8A)\omega+12M+1S+5A}{\omega}+E' $ $ 5\cdot2^{\omega-1}+8 $
Indistinguishable formulae
Unified formulae [12] $ 16M+\frac{20}{3}S+\frac{4}{3}M_\mathfrak{a}+\frac{40}{3}A+\frac{4}{3}E $ N/A
Complete formulae* [69,59] $ 16M+4M_\mathfrak{a}+\frac{8}{3}M_\mathfrak{b}+36A+\frac{4}{3}E $ N/A
Atomicity
Chevallier et al. [15] $ \frac{41}{3}M+\frac{205}{6}A+\frac{41}{3}E $ 10
Longa* [56] $ 6M+6S+24A+6E $ 12
Giraud-Verneuil [30] $ 7M+7S+28A+7E $ 11
Giraud-Verneuil* [30] $ 10M+\frac{10}{3}S+\frac{50}{3}A+\frac{5}{3}E $ 11
Rondepierre [72] $ \frac{32}{3}M+4S+12A+\frac{4}{3}E $ 11
Rondepierre* [72,71] $ \frac{32}{3}M+\frac{8}{3}S+\frac{40}{3}A+\frac{4}{3}E $ 11
Montgomery ladder and Joye's double-add
Brier-Joye [12] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+15A $ N/A
Izu-Takagi [45] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+15A $ 18
Fischer et al. [27] $ 10M+5S+2M_\mathfrak{a}+2M_\mathfrak{b}+14A $ 9
Izu et al. [44] $ 10M+4S+2M_\mathfrak{a}+1M_\mathfrak{b}+18A $ 8
Goundar et al. Alg.13 [32,31] $ 9M+7S+27A $ 9
Goundar et al. Alg.14 [32] $ 8M+7S+3M_\mathfrak{a}+1M_\mathfrak{b}+27A $ 8
Venelli-Dassance [73] $ 9M+5S+ $(N/A)$ A $ N/A
Hutter et al. Alg.5 [43] $ 9M+5S+1M_\mathfrak{a}+1M_\mathfrak{b}+16A $ 8
Hutter et al. Alg.6 [43] $ 10M+5S+13A $ 10
Goundar et al. [33] $ 9M+5S+19A $ 10
Rivain Alg.13+Alg.14 [70] $ 9M+5S+18A $ 8
Rivain suggestion [70] $ 8M+6S+26A $ 8
Fay [25] $ 13M+5S+14A $ 8
Chung et al. [16] $ 8M+7S+2M_\mathfrak{a}+3M_\mathfrak{b}+12A $ N/A
This work (Sec. 3.2) 8M+4S+15.5A 8
This work (Sec. 3.3) 8M+4S+12.5A 9
Hamburg [35] 8M+4S+9A 8
This work (Sec. 3.4) 6M+6S+20A 9
This work (Sec. 3.4) 6M+6S+18A 9
Binary NAF method vulnerable to SPA: for the purpose of comparison only
[37,44] $ \frac{20}{3}M+7S+\frac{40}{3}A+E $ 8
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