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Factoring the Modulus of Type N = p 2 q by Finding Small Solutions of the Equation e r − ( N s + t ) = α p 2 + β q 2

Author

Listed:
  • Muhammad Asyraf Asbullah

    (Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research, University Putra Malaysia, UPM, Serdang 43400, Malaysia
    Centre of Foundation Studies for Agricultural Science, University Putra Malaysia, UPM, Serdang 43400, Malaysia
    These authors contributed equally to this work.)

  • Normahirah Nek Abd Rahman

    (Pusat GENIUS@Pintar Negara, University Kebangsaan Malaysia, UKM, Bangi 43600, Malaysia
    These authors contributed equally to this work.)

  • Muhammad Rezal Kamel Ariffin

    (Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research, University Putra Malaysia, UPM, Serdang 43400, Malaysia
    Department of Mathematics & Statistics, Faculty of Science, University Putra Malaysia, UPM, Serdang 43400, Malaysia
    These authors contributed equally to this work.)

  • Nur Raidah Salim

    (Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research, University Putra Malaysia, UPM, Serdang 43400, Malaysia
    These authors contributed equally to this work.)

Abstract

The modulus of type N = p 2 q is often used in many variants of factoring-based cryptosystems due to its ability to fasten the decryption process. Faster decryption is suitable for securing small devices in the Internet of Things (IoT) environment or securing fast-forwarding encryption services used in mobile applications. Taking this into account, the security analysis of such modulus is indeed paramount. This paper presents two cryptanalyses that use new enabling conditions to factor the modulus N = p 2 q of the factoring-based cryptosystem. The first cryptanalysis considers a single user with a public key pair ( e , N ) related via an arbitrary relation to equation e r − ( N s + t ) = α p 2 + β q 2 , where r , s , t are unknown parameters. The second cryptanalysis considers two distinct cases in the situation of k -users (i.e., multiple users) for k ≥ 2 , given the instances of ( N i , e i ) where i = 1 , … , k . By using the lattice basis reduction algorithm for solving simultaneous Diophantine approximation, the k -instances of ( N i , e i ) can be successfully factored in polynomial time.

Suggested Citation

  • Muhammad Asyraf Asbullah & Normahirah Nek Abd Rahman & Muhammad Rezal Kamel Ariffin & Nur Raidah Salim, 2021. "Factoring the Modulus of Type N = p 2 q by Finding Small Solutions of the Equation e r − ( N s + t ) = α p 2 + β q 2," Mathematics, MDPI, vol. 9(22), pages 1-16, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2931-:d:681519
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    References listed on IDEAS

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    1. Nurul Nur Hanisah Adenan & Muhammad Rezal Kamel Ariffin & Faridah Yunos & Siti Hasana Sapar & Muhammad Asyraf Asbullah, 2021. "Analytical cryptanalysis upon N = p2q utilizing Jochemsz-May strategy," PLOS ONE, Public Library of Science, vol. 16(3), pages 1-11, March.
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