Handbook of Applied Cryptography
Handbook of Applied Cryptography
Distinguishing Exponent Digits by Observing Modular Subtractions
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
Precise Bounds for Montgomery Modular Multiplication and Some Potentially Insecure RSA Moduli
CT-RSA '02 Proceedings of the The Cryptographer's Track at the RSA Conference on Topics in Cryptology
A Practical Implementation of the Timing Attack
CARDIS '98 Proceedings of the The International Conference on Smart Card Research and Applications
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems
CRYPTO '96 Proceedings of the 16th Annual International Cryptology Conference on Advances in Cryptology
Efficient Elliptic Curve Exponentiation Using Mixed Coordinates
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Proceedings of the 8th IMA International Conference on Cryptography and Coding
A Refined Power-Analysis Attack on Elliptic Curve Cryptosystems
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Montgomery's Multiplication Technique: How to Make It Smaller and Faster
CHES '99 Proceedings of the First International Workshop on Cryptographic Hardware and Embedded Systems
Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems
CHES '99 Proceedings of the First International Workshop on Cryptographic Hardware and Embedded Systems
A Timing Attack against RSA with the Chinese Remainder Theorem
CHES '00 Proceedings of the Second International Workshop on Cryptographic Hardware and Embedded Systems
Montgomery Exponentiation with no Final Subtractions: Improved Results
CHES '00 Proceedings of the Second International Workshop on Cryptographic Hardware and Embedded Systems
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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.