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Small Private Key Attack Against a Family of RSA-Like Cryptosystems

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Secure IT Systems (NordSec 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14324))

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Abstract

Let \(N=pq\) be the product of two balanced prime numbers p and q. Elkamchouchi, Elshenawy and Shaban presented in 2002 an interesting RSA-like cryptosystem that uses the key equation \(ed - k (p^2-1)(q^2-1) = 1\), instead of the classical RSA key equation \(ed - k (p-1)(q-1) = 1\). The authors claimed that their scheme is more secure than RSA. Unfortunately, the common attacks developed against RSA can be adapted for Elkamchouchi et al.’s scheme. In this paper, we introduce a family of RSA-like encryption schemes that uses the key equation \(ed - k (p^n-1)(q^n-1) = 1\), where \(n>0\) is an integer. Then, we show that regardless of the choice of n, there exists an attack based on continued fractions that recovers the secret exponent.

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Authors and Affiliations

  1. Advanced Technologies Institute, 10 Dinu Vintilă, Bucharest, Romania

    Paul Cotan & George Teşeleanu

  2. Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei, Bucharest, Romania

    Paul Cotan & George Teşeleanu

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  1. Paul Cotan
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  2. George Teşeleanu
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Correspondence to George Teşeleanu .

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Editors and Affiliations

  1. OsloMet – Oslo Metropolitan University, Oslo, Norway

    Lothar Fritsch

  2. OsloMet – Oslo Metropolitan University, Oslo, Norway

    Ismail Hassan

  3. OsloMet - Oslo Metropolitan University, Oslo, Norway

    Ebenezer Paintsil

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Cotan, P., Teşeleanu, G. (2024). Small Private Key Attack Against a Family of RSA-Like Cryptosystems. In: Fritsch, L., Hassan, I., Paintsil, E. (eds) Secure IT Systems. NordSec 2023. Lecture Notes in Computer Science, vol 14324. Springer, Cham. https://doi.org/10.1007/978-3-031-47748-5_4

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  • DOI: https://doi.org/10.1007/978-3-031-47748-5_4

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