Abstract
In AfricaCrypt 2012, several algorithms are proposed for the batch verification of ECDSA signatures. In this paper, we propose three randomization methods for these batch-verification algorithms. Our first proposal is based on Montgomery ladders, and the second on computing square-roots in the underlying field. Both these techniques use numeric arithmetic only. Our third proposal exploits symbolic computations leading to a seminumeric algorithm. We theoretically and experimentally establish that for standard ECDSA signatures, our seminumeric randomization algorithm in tandem with the batch-verification algorithm S2′ gives the best speedup over individual verification. If each ECDSA signature contains an extra bit to uniquely identify the correct y-coordinate of the elliptic-curve point appearing in the signature, then the second numeric randomization algorithm followed by the naive batch-verification algorithm N′ yields the best performance gains. We detail our study for NIST prime and Koblitz curves.
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References
Karati, S., Das, A., Roychowdhury, D., Bellur, B., Bhattacharya, D., Iyer, A.: Batch verification of ECDSA signatures. In: Mitrokotsa, A., Vaudenay, S. (eds.) AFRICACRYPT 2012. LNCS, vol. 7374, pp. 1–18. Springer, Heidelberg (2012)
Bernstein, D.J., Doumen, J., Lange, T., Oosterwijk, J.-J.: Faster batch forgery identification. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 454–473. Springer, Heidelberg (2012)
Naccache, D., M’Raïhi, D., Vaudenay, S., Raphaeli, D.: Can D.S.A. be improved?: Complexity trade-offs with the digital signature standard. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 77–85. Springer, Heidelberg (1995)
Antipa, A., Brown, D., Gallant, R., Lambert, R., Struik, R., Vanstone, S.: Accelerated verification of ECDSA signatures. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 307–318. Springer, Heidelberg (2006)
Cheon, J.H., Yi, J.H.: Fast batch verification of multiple signatures. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 442–457. Springer, Heidelberg (2007)
Montgomery, P.L.: Speeding up Pollard and elliptic curve methods of factorization. In: Mathematics of Computation, vol. 48(177), pp. 243–264 (1987)
Joye, M.: Security analysis of RSA-type cryptosystems. Phd thesis, UCL Crypto Group, Belgium (1997)
NIST: Recommended elliptic curves for federal government use (1999), http://csrc.nist.gov/encryption
Montgomery, P.L.: Evaluating recurrences of form X m + n = f(X m ,X n ,X m − n ) via Lucas chains. Microsoft research article, 582 (1992)
Stam, M.: On Montgomery-like representations for elliptic curves over GF(2k). In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 240–253. Springer, Heidelberg (2002)
Stam, M.: Speeding up subgroup cryptosystems. PhD thesis, Technische Universiteit Eindhoven (2003)
Bellare, M., Garay, J.A., Rabin, T.: Fast batch verification for modular exponentiation and digital signatures. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 236–250. Springer, Heidelberg (1998)
Brier, E., Joye, M.: Weierstraß elliptic curves and side-channel attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)
López, J., Dahab, R.: Fast multiplication on elliptic curves over GF(2m) without precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)
Fischer, W., Giraud, C., Knudsen, E.W., Seifert, J.P.: Parallel scalar multiplication on general elliptic curves over F p hedged against non-differential side-channel attacks. IACR Cryptology ePrint Archive 2002/007 (2002)
Bernstein, D.J., Lange, T.: Explicit-Formulas Database (2007), http://www.hyperelliptic.org/EFD/index.html
Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography, 2nd edn. Chapman & Hall/CRC (2012)
Hankerson, D., Menezes, A.J., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer-Verlag New York, Inc., Secaucus (2003)
PARI Group: PARI/GP home (2008), http://pari.math.u-bordeaux.fr/
Lange, T.: A note on López-Dahab coordinates. IACR Cryptology ePrint Archive 2004/323 (2004)
Solinas, J.A.: Improved algorithms for arithmetic on anomalous binary curves. Technical report, Originally presented in Advances in Cryptography, Crypto 1997 (1997)
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Karati, S., Das, A., Roychoudhury, D. (2014). Randomized Batch Verification of Standard ECDSA Signatures. In: Chakraborty, R.S., Matyas, V., Schaumont, P. (eds) Security, Privacy, and Applied Cryptography Engineering. SPACE 2014. Lecture Notes in Computer Science, vol 8804. Springer, Cham. https://doi.org/10.1007/978-3-319-12060-7_16
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DOI: https://doi.org/10.1007/978-3-319-12060-7_16
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