Abstract
Modular multiplication and modular reduction are the atomic constituents of most public-key cryptosystems. Amongst the numerous algorithms for performing these operations, a particularly elegant method was proposed by Barrett. This method builds the operation \(a \,\text {mod}\,b\) from bit shifts, multiplications and additions in \(\mathbb {Z}\). This allows to build modular reduction at very marginal code or silicon costs by leveraging existing hardware or software multipliers.
This paper presents a method allowing to double the speed of Barrett’s algorithm by using specific composite moduli. This is particularly useful for lightweight devices where such an optimization can make a difference in terms of power consumption, cost and processing time. The generation of composite moduli with a predetermined portion is a well-known technique and the use of such moduli is considered, in statu scientiæ, as safe as using randomly generated composite moduli.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For the sake of clarity we remove all tests meant to enforce the condition \(\text{ GCD }(e,\phi (n))=1\).
- 2.
A few more complexity bits can be grabbed if the variant described in the note at the end of Sect. 3 is used.
References
Barrett, P.: Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 311–323. Springer, Heidelberg (1987)
Bernstein, R.: Multiplication by integer constants. Softw. Pract. Exp. 16(7), 641–652 (1986)
Bosselaers, A., Govaerts, R., Vandewalle, J.: Comparison of three modular reduction functions. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 175–186. Springer, Heidelberg (1994)
Brickell, E.F.: A Fast Modular Multiplication Algorithm with Applications to TwoKey Cryptography. Crypto 1982, pp. 51–60. Springer, New York (1983)
Douguet, M., Dupaquis, V.: Modular reduction using a special form of the modulus. U.S. Patent Application 12/033,512, filed February 19, Atmel Corporation (2008)
Joye, M.: RSA moduli with a predetermined portion: techniques and applications. In: Chen, L., Mu, Y., Susilo, W. (eds.) ISPEC 2008. LNCS, vol. 4991, pp. 116–130. Springer, Heidelberg (2008)
Knežević, M., Batina, L., Verbauwhede, I.: Modular reduction without precomputational phase. In: Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1389–1392. IEEE (2009)
Knobloch, H.-J.: A smart card implementation of the Fiat-Shamir identification scheme. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 87–95. Springer, Heidelberg (1988)
Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2, 2nd edn. Addison Wesley, Reading (1981)
Lenstra, A.K.: Generating RSA moduli with a predetermined portion. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 1–10. Springer, Heidelberg (1998)
Meister, G.: On an implementation of the Mohan-Adiga algorithm. In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 496–500. Springer, Heidelberg (1991)
Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)
National Institute of Standards and Technology (NIST): Digital Signature Standard. FIPS PUB 186–2 (2013)
National Institute of Standards and Technology (NIST): Digital Signature Standard. FIPS PUB 186–4 (2013)
Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. Assoc. Comput. Mach. 21(2), 120–126 (1978)
Shparlinski, I.E.: On RSA moduli with prescribed bit patterns. Des. Codes Cryptogr. 39(1), 113–122 (2006)
Vanstone, S.A., Zuccherato, R.J.: Short RSA keys and their generation. J. Cryptol. 8(2), 101–114 (1995)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Géraud, R., Maimuţ, D., Naccache, D. (2016). Double-Speed Barrett Moduli. In: Ryan, P., Naccache, D., Quisquater, JJ. (eds) The New Codebreakers. Lecture Notes in Computer Science(), vol 9100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49301-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-662-49301-4_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49300-7
Online ISBN: 978-3-662-49301-4
eBook Packages: Computer ScienceComputer Science (R0)