Paper 2022/611

Further Cryptanalysis of a Type of RSA Variants

Gongyu Shi, Shanghai Jiao Tong University
Geng Wang, Shanghai Jiao Tong University
Dawu Gu, Shanghai Jiao Tong University

To enhance the security or the efficiency of the standard RSA cryptosystem, some variants have been proposed based on elliptic curves, Gaussian integers or Lucas sequences. A typical type of these variants which we called Type-A variants have the specified modified Euler's totient function $\psi(N)=(p^2-1)(q^2-1)$. But in 2018, based on cubic Pell equation, Murru and Saettone presented a new RSA-like cryptosystem, and it is another type of RSA variants which we called Type-B variants, since their scheme has $\psi(N)=(p^2+p+1)(q^2+q+1)$. For RSA-like cryptosystems, four key-related attacks have been widely analyzed, i.e., the small private key attack, the multiple private keys attack, the partial key exposure attack and the small prime difference attack. These attacks are well-studied on both standard RSA and Type-A variants. Recently, the small private key attack on Type-B variants has also been analyzed. In this paper, we make further cryptanalysis of Type-B variants, that is, we propose the first theoretical results of multiple private keys attack, partial key exposure attack as well as small prime difference attack on Type-B variants, and the validity of our attacks are verified by experiments. Our results show that for all three attacks, Type-B variants are less secure than standard RSA.

Available format(s)
Attacks and cryptanalysis
Publication info
Published elsewhere. ISC 2022
Cryptanalysis RSA variants Coppersmith's method Lattice reduction
Contact author(s)
gy_shi @ sjtu edu cn
2022-12-09: revised
2022-05-23: received
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Creative Commons Attribution


      author = {Gongyu Shi and Geng Wang and Dawu Gu},
      title = {Further Cryptanalysis of a Type of RSA Variants},
      howpublished = {Cryptology ePrint Archive, Paper 2022/611},
      year = {2022},
      doi = {10.1007/978-3-031-22390-7_9},
      note = {\url{}},
      url = {}
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