Comments on efficient batch verification test for digital signatures based on elliptic curves
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Keisuke Hakuta
,
Hiroyuki Ochiai
Abstract
Batch verification for digital signature scheme is a method to verify multiple signatures simultaneously. The complex exponent test (CE test for short) proposed by Cheon and Lee is one of the most efficient batch verification tests for several digital signature schemes on certain types of elliptic curves (including Koblitz curves). The security of the CE test relies essentially on the cardinality of a subset of a residue ring of the endomorphism ring of an elliptic curve over an ideal. They have evaluated the cardinality of the above subset, and have illustrated the effectiveness of the CE test by using the evaluation. The aim of this paper is to point out that their evaluation contains a flaw. The flaw is generally related to two roots of a quadratic equation which is used in their argument. We mend the flaw of their evaluation. Our correct evaluation shows that the CE test can achieve the same security as the underlying signature scheme on Koblitz curves. As a result, the CE test is a secure batch verification when the underlying signature scheme uses Koblitz curves.
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(Communicated by István Gaál)
Acknowledgement
The authors would like to thank the anonymous reviewers for their valuable comments.
References
[1] Antipa, A.—Brown, D.—Gallant, R.—Lambert, R.—Struik, R.—Vanstone, S.: Accelerated verification of ECDSA signatures. In: Selected Areas in Cryptography 2005, Lecture Notes in Comput. Sci. 3897, Springer, Berlin, 2006, pp. 307–318.10.1007/11693383_21Suche in Google Scholar
[2] Artin, M.: Algebra, Addison Wesley, 2010.10.1007/978-3-0348-9124-0Suche in Google Scholar
[3] Bellare, M.—Garay, J.—Rabin, T.: Fast Batch Verification for Modular Exponentiation and Digital Signatures, In: Advances in Cryptology – EUROCRYPT ’98. Lecture Notes in Comput. Sci. 1403, Springer, Berlin, 1998, pp. 236–250.10.1007/BFb0054130Suche in Google Scholar
[4] Blake, I.—Seroussi, G.—Smart, N.: Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series, Vol. 265, Cambridge University Press, 1999.10.1017/CBO9781107360211Suche in Google Scholar
[5] Cheon, J. H.—Lee, D. H.: Use of sparse and/or complex exponents in Batch verification of exponentiations, IEEE Trans. Comput. 55 (2006), 1536–1542.10.1109/TC.2006.207Suche in Google Scholar
[6] Galbraith, S. D.—Smart, N. P.: Evaluation Report for CRYPTREC: Security Level of Cryptography – ECDLP Mathematical Problem, CRYPTREC 2001, Tech. Report, No. 1029, 2001; available from http://www.ipa.go.jp/security/enc/CRYPTREC/fy15/doc/1029_report.pdfSuche in Google Scholar
[7] Hakuta, K.—Sato, H.—Takagi, T.: Efficient arithmetic on subfield elliptic curves over small finite fields of odd characteristic, J. Math. Cryptol. 4 (2010), 199–238.10.1007/978-3-540-79104-1_22Suche in Google Scholar
[8] Hakuta, K.—Sato, H.—Takagi, T.: Explicit lower bound for the length of minimal weight τ-adic expansions on Koblitz curves, J. Math-for-Ind. 2A (2010), 75–83.Suche in Google Scholar
[9] Hakuta, K.—Sato, H.—Takagi, T.: Some properties of τ-adic expansions on hyperelliptic Koblitz curves, J. Appl. Math. Comput. 58 (2018), 367–388.10.1007/s12190-017-1149-5Suche in Google Scholar
[10] Hakuta, K.—Katoh, Y.—Sato, H.—Takagi, T.: Batch verification suitable for efficiently verifying a limited number of signatures. In: International Conference on Information Security and Cryptology 2012, Lecture Notes in Comput. Sci. 7839, Springer, Berlin, 2013, pp. 425–440.10.1007/978-3-642-37682-5_30Suche in Google Scholar
[11] Koblitz, N.: Hyperelliptic cryptosystems, J. Cryptology 1 (1989), 139–150.10.1007/BF02252872Suche in Google Scholar
[12] Koblitz, N.: CM-curves with good cryptographic properties. In: Advances in Cryptology – CRYPTO 1991, Lecture Notes in Comput. Sci. 576, Springer, Berlin, 1992, pp. 279–287.10.1007/3-540-46766-1_22Suche in Google Scholar
[13] Lang, S.: Algebra. Graduate Texts in Math. 211, Springer-Verlag, New York, 2002.10.1007/978-1-4613-0041-0Suche in Google Scholar
[14] National Institute of Standards and Technology, Digital Signature Standard (DSS), Federal Information Processing Standards Publication 186-4, July 2013; available from http://csrc.nist.gov/publications/PubsFIPS.html.Suche in Google Scholar
[15] Schnorr, C. P.: Efficient signature generation by smart cards, J. Cryptology 4 (1991), 161–174.10.1007/BF00196725Suche in Google Scholar
[16] Silverman, J. H.: The Arithmetic of Elliptic Curves. Graduate Texts in Math. 106, Springer-Verlag, New York, 2009.10.1007/978-0-387-09494-6Suche in Google Scholar
[17] Smart, N. P.: Elliptic curve cryptosystems over small fields of odd characteristic, J. Cryptology 12 (1999), 141–151.10.1007/PL00003820Suche in Google Scholar
[18] Solinas, J. A.: Efficient Arithmetic on Koblitz Curves, Des. Codes Cryptogr. 19 (2000), 195–249.10.1007/978-1-4757-6856-5_6Suche in Google Scholar
© 2022 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
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Artikel in diesem Heft
- Regular Papers
- The poset of morphism-extension classes of countable graphs
- Characterization of monadic BL-algebras by state operators
- Comments on efficient batch verification test for digital signatures based on elliptic curves
- On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
- An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers
- Some results on certain types of Putcha semigroups
- Inner functions in QK spaces and multipliers
- Certain subclasses of meromorphic multivalent q-starlike and q-convex functions
- Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes
- Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type
- Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group
- Intuitionistic fuzzy Tribonacci I-convergent sequence spaces
- Lipschitz class functions and their general Fourier coefficients
-
Spectra and fine spectra of the generalized upper difference operator with triple repetition
- A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere
- An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
- On the Unit-Chen distribution with associated quantile regression and applications
- A note on Lévy subordinators in cones of fuzzy sets in Banach spaces
- Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors
- Upper bound for variance of finite mixtures of power exponential distributions
- The dimension Dind of finite topological T0-spaces
