Abstract
In this paper, we study some computational security assumptions involved in two cryptographic applications related to the RSA cryptosystem. To this end, we use exponential sums to bound the statistical distances between these distributions and the uniform distribution. We are interested in studying the k least (or most) significant bits of \(x^e \bmod N\), where N is an RSA modulus and x only belongs to a small interval of [0,N).
First of all, we provide the first rigorous evidence that the cryptographic pseudo-random generator proposed by Micali and Schnorr is based on firm foundations. This proof is missing in the original paper and does not cover the parameters chosen by the authors. Consequently, we extend the proof to get a new result closer to these parameters using recently new exponential sums results and we show some limitations of our technique. Finally, we look at the semantic security of the RSA padding scheme called PKCS#1 v1.5 which is still used a lot in practice. We show that parts of the ciphertext are indistinguisable from uniform bitstrings.
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Fouque, PA., Zapalowicz, JC. (2014). Statistical Properties of Short RSA Distribution and Their Cryptographic Applications. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_45
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DOI: https://doi.org/10.1007/978-3-319-08783-2_45
Publisher Name: Springer, Cham
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