Abstract
A large set of moduli, for which the speed of bipartite modular multiplication considerably increases, is proposed in this work. By considering state of the art attacks on public-key cryptosystems, we show that the proposed set is safe to use in practice for both elliptic curve cryptography and RSA cryptosystems. We propose a hardware architecture for the modular multiplier that is based on our method. The results show that, concerning the speed, our proposed architecture outperforms the modular multiplier based on standard bipartite modular multiplication. Additionally, our design consumes less area compared to the standard solutions.
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References
ANSI. ANSI X9.62 The Elliptic Curve Digital Signature Algorithm (ECDSA), http://www.ansi.org
Avanzi, R.M., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton (2005)
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)
Boneh, D., Durfee, G., Howgrave-Graham, N.: Factoring N = p r q for Large r. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 326–337. Springer, Heidelberg (1999)
Coppersmith, D.: Factoring with a Hint, IBM Research Report RC 19905 (1995)
Coppersmith, D.: Finding a small root of a bivariate integer equation; factoring with high bits known. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 178–189. Springer, Heidelberg (1996)
Coppersmith, D.: Small Solutions to Polynomial Equations, and Low Exponent Vulnerabilities. Journal of Cryptology 10(4), 233–260 (1996)
Dhem, J.-F.: Design of an Efficient Public-Key Cryptographic Library for RISC-based Smart Cards. PhD Thesis (1998)
Diffie, W., Hellman, M.E.: New Directions in Cryptography. IEEE Transactions on Information Theory 22, 644–654 (1976)
Joye, M.: RSA Moduli with a Predetermined Portion: Techniques and Applications. Information Security Practice and Experience, 116–130 (2008)
Kaihara, M.E., Takagi, N.: Bipartite Modular Multiplication. In: Rao, J.R., Sunar, B. (eds.) CHES 2005. LNCS, vol. 3659, pp. 201–210. Springer, Heidelberg (2005)
Lenstra, A.: Generating RSA Moduli with a Predetermined Portion. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 1–10. Springer, Heidelberg (1998)
May, A.: Using LLL-Reduction for Solving RSA and Factorization Problems: A Survey (2007), http://www.informatik.tu-darmstadt.de/KP/publications/07/lll.pdf
Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)
Montgomery, P.: Modular Multiplication without Trial Division. Mathematics of Computation 44(170), 519–521 (1985)
National Institute of Standards and Technology. FIPS 186-2: Digital Signature Standard (January 2000)
Potgieter, M.J., van Dyk, B.J.: Two Hardware Implementations of the Group Operations Necessary for Implementing an Elliptic Curve Cryptosystem over a Characteristic Two Finite Field. In: IEEE AFRICON. 6th Africon Conference in Africa, pp. 187–192 (2002)
Quisquater, J.-J.: Encoding System According to the So-Called RSA Method, by Means of a Microcontroller and Arrangement Implementing this System, US Patent #5,166,978 (1992)
Rivest, R.L., Shamir, A.: Efficient Factoring Based on Partial Information. In: Pichler, F. (ed.) EUROCRYPT 1985. LNCS, vol. 219, pp. 31–34. Springer, Heidelberg (1986)
Rivest, R.L., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM 21(2), 120–126 (1978)
Standards for Efficient Cryptography. SEC2: Recommended Elliptic Curve Domain Parameters (2010), http://www.secg.org
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Knežević, M., Vercauteren, F., Verbauwhede, I. (2010). Speeding Up Bipartite Modular Multiplication. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_12
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DOI: https://doi.org/10.1007/978-3-642-13797-6_12
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