Abstract
Let N = pq be an RSA modulus where p and q are primes not necessarily of the same bit size. Previous cryptanalysis results on the difficulty of factoring the public modulus N = pq deployed on variants of RSA cryptosystem are revisited. Each of these variants share a common key relation utilizing the modified Euler quotient (p 2 - 1)(q 2 - 1), given by the key relation ed - k(p 2 - 1)(q 2 - 1) = 1 where e and d are the public and private keys respectively. By conducting continuous midpoint subdivision analysis upon an interval containing (p 2 - 1)(q 2 - 1) together with continued fractions on the key relation, we increase the security bound for d exponentially.
Highlights
With the realization of the quantum computer coming into reality in the near future, expected in 2030 [20], the demise of traditional asymmetric encryption schemes is imminent
PRELIMINARIES This section reviews the fundamental concept of the continued fractions and presents some existing results relevant to our algebraic cryptanalysis method
Remark 5: In the case when the primes p and q are of arbitrary sizes, one can observe that the continuous midpoint subdivision analysis increases the upper bound of private d≈
Summary
With the realization of the quantum computer coming into reality in the near future, expected in 2030 [20], the demise of traditional asymmetric encryption schemes is imminent. In RSA key generation algorithm, the positive integers e and d are associated by the modular relation ed ≡ 1 (mod φ(N )) where the Euler’s totient function or Euler quotient be represented by φ(N ) = (p − 1)(q − 1). A. Ruzai et al.: On the Improvement Attack Upon Some Variants of RSA Cryptosystem via the Continued Fractions Method. In 2017, Bunder et al [12] extended their previous work in [3] by considering the general key equation of the form ex − y(p2 − 1)(q2 − 1) = z where the unknown parameters x, y and z fulfill the conditions xy < 2N −.
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