New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method

Kamel Ariffin, Muhammad Rezal; Abubakar, Saidu Isah; Yunos, Faridah; Asbullah, Muhammad Asyraf

doi:10.3390/cryptography3010002

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New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method

Muhammad Rezal Kamel Ariffin

^1,2,†,

Saidu Isah Abubakar

^1,*,†

Faridah Yunos

^1,2,† and

Muhammad Asyraf Asbullah

^1,3,†

Al-Kindi Cryptography Research Laboratory, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, Malaysia

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Selangor 43400, Malaysia

Centre of Foundation Studies for Agriculture Science, Universiti Putra Malaysia, Selangor 43400, Malaysia

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Cryptography 2019, 3(1), 2; https://doi.org/10.3390/cryptography3010002

Received: 1 November 2018 / Revised: 14 December 2018 / Accepted: 15 December 2018 / Published: 20 December 2018

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Abstract

This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus

N = p q

in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form

| b^{2} p - a^{2} q | < N^{γ}

where the ratio of

\frac{q}{p}

is close to

\frac{b^{2}}{a^{2}}

, which yields a bound

d < \frac{\sqrt{3}}{\sqrt{2}} N^{\frac{3}{4} - γ}

from the convergents of the continued fraction expansion of

\frac{e}{N - ⌈ \frac{a^{2} + b^{2}}{a b} \sqrt{N} ⌉ + 1}

. The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli

N_{s} = p_{s} q_{s}

for

s = 1, 2, \dots, t

where we use

N - ⌈ \frac{a^{2} + b^{2}}{a b} \sqrt{N} ⌉ + 1

as an approximation of

ϕ (N)

satisfying generalized key equations of the shape

e_{s} d - k_{s} ϕ (N_{s}) = 1

e_{s} d_{s} - k ϕ (N_{s}) = 1

e_{s} d - k_{s} ϕ (N_{s}) = z_{s}

, and

e_{s} d_{s} - k ϕ (N_{s}) = z_{s}

for unknown positive integers

d, k_{s}, d_{s}, k_{s}

, and

z_{s}

, where we establish that t RSA moduli can be simultaneously factored in polynomial time using combinations of simultaneous Diophantine approximations and lattice basis reduction methods. In all the reported attacks, we have found an improved short secret exponent bound, which is considered to be better than some bounds as reported in the literature. View Full-Text

Keywords: RSA modulus; primes difference; cryptanalysis; short decryption exponent; attacks; continued fraction

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Kamel Ariffin, M.R.; Abubakar, S.I.; Yunos, F.; Asbullah, M.A. New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method. Cryptography 2019, 3, 2. https://doi.org/10.3390/cryptography3010002

AMA Style

Kamel Ariffin MR, Abubakar SI, Yunos F, Asbullah MA. New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method. Cryptography. 2019; 3(1):2. https://doi.org/10.3390/cryptography3010002

Chicago/Turabian Style

Kamel Ariffin, Muhammad Rezal, Saidu Isah Abubakar, Faridah Yunos, and Muhammad Asyraf Asbullah. 2019. "New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method" Cryptography 3, no. 1: 2. https://doi.org/10.3390/cryptography3010002

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