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Continued Fractions Applied to a Family of RSA-like Cryptosystems

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Information Security Practice and Experience (ISPEC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13620))

Abstract

Let \(N=pq\) be the product of two balanced prime numbers p and q. Murru and Saettone presented in 2017 an interesting RSA-like cryptosystem that uses the key equation \(ed \,-\, k (p^2\,+\,p\,+\,1)(q^2\,+\,q\,+\,1) = 1\), instead of the classical RSA key equation \(ed - k (p-1)(q-1) = 1\). The authors claimed that their scheme is immune to Wiener’s continued fraction attack. Unfortunately, Nitaj et. al. developed exactly such an attack. In this paper, we introduce a family of RSA-like encryption schemes that uses the key equation \(ed \,-\, k [(p^n\,-\,1)(q^n\,-\,1)]/[(p\,-\,1)(q\,-\,1)] = 1\), where \(n>1\) 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. University of Aizu, Fukushima, Japan

    Chunhua Su

  2. Athens University of Economics and Business, Athens, Greece

    Dimitris Gritzalis

  3. Università degli Studi di Milano, Milan, Italy

    Vincenzo Piuri

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Cotan, P., Teşeleanu, G. (2022). Continued Fractions Applied to a Family of RSA-like Cryptosystems. In: Su, C., Gritzalis, D., Piuri, V. (eds) Information Security Practice and Experience. ISPEC 2022. Lecture Notes in Computer Science, vol 13620. Springer, Cham. https://doi.org/10.1007/978-3-031-21280-2_33

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  • DOI: https://doi.org/10.1007/978-3-031-21280-2_33

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