Abstract
The Montgomery multiplication is often used for efficient implementations of public-key cryptosystems. This algorithm occasionally needs an extra subtraction in the final step, and the correlation of these subtractions can be considered as an invariant of the algorithm. Some side channel attacks on cryptosystems using Montgomery Multiplication has been proposed applying the correlation estimated heuristically. In this paper, we theoretically analyze the properties of the final subtraction in Montgomery multiplication. We investigate the distribution of the outputs of multiplications in the fixed length interval included between 0 and the underlying modulus. Integrating these distributions, we present some proofs with a reasonable assumption for the appearance ratio of the final subtraction, which have been heuristically estimated by previous papers. Moreover, we present a new invariant of the final subtraction: x · y with y = 3x mod m, where m is the underlying modulus. Finally we show a possible attack on elliptic curve cryptosystems using this invariant.
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
Preview
Unable to display preview. Download preview PDF.
References
Akishita, T., Takagi, T.: Zero-Value Point Attacks on Elliptic Curve Cryptosystem. In: Boyd, C., Mao, W. (eds.) ISC 2003. LNCS, vol. 2851, pp. 218–233. Springer, Heidelberg (2003)
Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)
Coron, J.-S.: Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)
Dhem, J.-F., Koeune, F., Leroux, P.-A., Mestré, P., Quisquater, J.-J., Willems, J.-L.: A Practical Implementation of the Timing Attack. In: Schneier, B., Quisquater, J.-J. (eds.) CARDIS 1998. LNCS, vol. 1820, pp. 167–182. Springer, Heidelberg (2000)
Goubin, L.: A Refined Power-Analysis Attack on Elliptic Curve Cryptosystems. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 199–211. Springer, Heidelberg (2002)
Koeune, F., Schindler, W., Quisqater, J.-J.: Improving Divide and Conquer Attacks against Cryptosystems by Better Error Detection / Correction Strategies. In: Honary, B. (ed.) Cryptography and Coding 2001. LNCS, vol. 2260, pp. 245–267. Springer, Heidelberg (2001)
Kocher, C.: Timing attacks on Implementations of Diffie-Hellman, RSA, DSS, and other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)
Kocher, C., Jaffe, J., Jun, B.: Differential Power Analysis. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)
Montgomery, P.L.: Speeding the Pollard and Elliptic Curve Methods of Factorization. Mathematics of Computation 48, 243–264 (1987)
Schindler, W.: A Timing Attack against RSA with the Chinese Remainder Theorem. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 109–124. Springer, Heidelberg (2000)
Schindler, W.: Optimized Timing Attacks against Public Key Cryptosystems. In: Statistic & Decisions, vol. 20, pp. 191–210. R. Oldenbourg Verlag, Munich (2002)
Schindler, W., Walter, C.: More Detail for a Combined Timing and Power Attack against Implementations of RSA. In: Paterson, K.G. (ed.) Cryptography and Coding 2003. LNCS, vol. 2898, pp. 245–263. Springer, Heidelberg (2003)
Standards for Efficient Cryptography Group (SECG), Specification of Standards for Efficient Cryptography. Available from: http://www.secg.org
Walter, C.: Precise Bounds for Montgomery Modular Multiplication and Some Potentially Insecure RSA Moduli. In: Preneel, B. (ed.) CT-RSA 2002. LNCS, vol. 2271, pp. 30–39. Springer, Heidelberg (2002)
Walter, C., Thompson, S.: Distinguishing Exponent Digits by Observing Modular Subtractions. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 192–207. Springer, Heidelberg (2001)
Hachez, G., Quisquater, J.-J.: Montgomery Exponentiation with no Final Subtraction: Improved Results. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 293–301. Springer, Heidelberg (2000)
Walter, C.D.: Montgomery’s Multiplication Technique: How to Make It Smaller and Faster. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 80–93. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sato, H., Schepers, D., Takagi, T. (2004). Exact Analysis of Montgomery Multiplication. In: Canteaut, A., Viswanathan, K. (eds) Progress in Cryptology - INDOCRYPT 2004. INDOCRYPT 2004. Lecture Notes in Computer Science, vol 3348. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30556-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-30556-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24130-0
Online ISBN: 978-3-540-30556-9
eBook Packages: Computer ScienceComputer Science (R0)