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Cited by 2 publications
References 14 publications
A security analysis of two classes of RSA-like cryptosystems
Let N = p q N=pq be the product of two balanced prime numbers p p and q q . In Elkamchouchi et al. (Extended RSA cryptosystem and digital signature schemes in the domain of Gaussian integers. In: ICCS 2002. vol. 1. IEEE Computer Society; 2002. p. 91–5.) introduced an Rivest-Shamir-Adleman (RSA)-like cryptosystem that uses the key equation e d − k ( p 2 − 1 ) ( q 2 − 1 ) = 1 ed-k\left({p}^{2}-1)\left({q}^{2}-1)=1 , instead of the classical RSA key equation e d − k ( p − 1 ) ( q − 1 ) = 1 ed-k\left(p-1)\left(q-1)=1 . Another variant of RSA, presented in Murru and Saettone (A novel RSA-like cryptosystem based on a generalization of the Rédei rational functions. In: NuTMiC 2017. vol. 10737 of Lecture Notes in Computer Science. Springer; 2017. p. 91–103), uses the key equation e d − k ( p 2 + p + 1 ) ( q 2 + q + 1 ) = 1 ed-k\left({p}^{2}+p+1)\left({q}^{2}+q+1)=1 . Despite the authors’ claims of enhanced security, both schemes remain vulnerable to adaptations of common RSA attacks. Let n n be an integer. This article proposes two families of RSA-like encryption schemes: one employs the key equation e d − k ( p n − 1 ) ( q n − 1 ) = 1 ed-k\left({p}^{n}-1)\left({q}^{n}-1)=1 for n > 0 n\gt 0 , while the other uses e d − k [ ( p n − 1 ) ( q n − 1 ) ] ⁄ [ ( p − 1 ) ( q − 1 ) ] = 1 ed-k\left[\left({p}^{n}-1)\left({q}^{n}-1)]/\left[\left(p-1)\left(q-1)]=1 for n > 1 n\gt 1 . Note that we remove the conventional assumption of primes having equal bit sizes. In this scenario, we show that regardless of the choice of n n , continued fraction-based attacks can still recover the secret exponent. Additionally, this work fills a gap in the literature by establishing an equivalent of Wiener’s attack when the primes do not have the same bit size.
A security analysis of two classes of RSA-like cryptosystems
Let N = p q N=pq be the product of two balanced prime numbers p p and q q . In Elkamchouchi et al. (Extended RSA cryptosystem and digital signature schemes in the domain of Gaussian integers. In: ICCS 2002. vol. 1. IEEE Computer Society; 2002. p. 91–5.) introduced an Rivest-Shamir-Adleman (RSA)-like cryptosystem that uses the key equation e d − k ( p 2 − 1 ) ( q 2 − 1 ) = 1 ed-k\left({p}^{2}-1)\left({q}^{2}-1)=1 , instead of the classical RSA key equation e d − k ( p − 1 ) ( q − 1 ) = 1 ed-k\left(p-1)\left(q-1)=1 . Another variant of RSA, presented in Murru and Saettone (A novel RSA-like cryptosystem based on a generalization of the Rédei rational functions. In: NuTMiC 2017. vol. 10737 of Lecture Notes in Computer Science. Springer; 2017. p. 91–103), uses the key equation e d − k ( p 2 + p + 1 ) ( q 2 + q + 1 ) = 1 ed-k\left({p}^{2}+p+1)\left({q}^{2}+q+1)=1 . Despite the authors’ claims of enhanced security, both schemes remain vulnerable to adaptations of common RSA attacks. Let n n be an integer. This article proposes two families of RSA-like encryption schemes: one employs the key equation e d − k ( p n − 1 ) ( q n − 1 ) = 1 ed-k\left({p}^{n}-1)\left({q}^{n}-1)=1 for n > 0 n\gt 0 , while the other uses e d − k [ ( p n − 1 ) ( q n − 1 ) ] ⁄ [ ( p − 1 ) ( q − 1 ) ] = 1 ed-k\left[\left({p}^{n}-1)\left({q}^{n}-1)]/\left[\left(p-1)\left(q-1)]=1 for n > 1 n\gt 1 . Note that we remove the conventional assumption of primes having equal bit sizes. In this scenario, we show that regardless of the choice of n n , continued fraction-based attacks can still recover the secret exponent. Additionally, this work fills a gap in the literature by establishing an equivalent of Wiener’s attack when the primes do not have the same bit size.
The Diophantine equation is a strong research domain in number theory with extensive cryptography applications. The goal of this paper is to describe certain geometric properties of positive integral solutions of the quadratic Diophantine equation x12+x22=y12+y22(x1,x2,y1,y2>0), as well as their use in communication protocols. Given one pair (x1,y1), finding another pair (x2,y2) satisfying x12+x22=y12+y22 is a challenge. A novel secure authentication mechanism based on the positive integral solutions of the quadratic Diophantine which can be employed in the generation of one-time passwords or e-tokens for cryptography applications is presented. Further, the constructive cost models are applied to predict the initial effort and cost of the proposed authentication schemes.
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