Paper 2022/1241

Continued Fractions Applied to a Family of RSA-like Cryptosystems

George Teseleanu, IMAR
Paul Cotan, IMAR

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 \emph{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.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. ISPEC 2022
continued fractionsMurru-Saettone cryptosystemsmall private key attack
Contact author(s)
george teseleanu @ yahoo com
paulcotan @ gmail com
2023-07-03: last of 2 revisions
2022-09-19: received
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      author = {George Teseleanu and Paul Cotan},
      title = {Continued Fractions Applied to a Family of RSA-like Cryptosystems},
      howpublished = {Cryptology ePrint Archive, Paper 2022/1241},
      year = {2022},
      note = {\url{}},
      url = {}
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