The RSA (Rivest–Shamir–Adleman) asymmetric-key cryptosystem is widely used for encryptions and digital signatures. Let $(n,e)$ be the RSA public key and d be the corresponding private key (or private exponent). One of the attacks on RSA is to find the private key d using continued fractions when d is small. In this paper, we present a new technique to improve a small private exponent attack on RSA using continued fractions and multicore systems. The idea of the proposed technique is to find an interval that contains $\varphi \left(n\right),$ and then propose a method to generate different points in the interval that can be used by continued fraction and multicore systems to recover the private key, where $\varphi$ is Euler’s totient function. The practical results of three small private exponent attacks on RSA show that we extended the previous bound of the private key that is discovered by continued fractions. When n is 1024 bits, we used 20 cores to extend the bound of d by $0.016$ for de Weger, Maitra-Sarkar, and Nassr et al. attacks in average times 7.67 h, 2.7 h, and 44 min, respectively.

We present a new version of the Secure Hash Algorithm-2 (SHA-2) formed on artificial sequences of deoxyribonucleic acid (DNA). This article is the first attempt to present the implementation of SHA-2 using DNA data processing. We called the new version DNSHA-2. We present new operations on an artificial DNA sequence, such as (1) $\bar {R}^{k}(\alpha)$R ̄ k ( α ) and $\bar {L}^{k}(\alpha)$L ̄ k ( α ) to mimic the right and left shift by k bits, respectively; (2) $\bar {S}^{k}(\alpha)$S ̄ k ( α ) to mimic the right rotation by k bits; and (3) DNA-nucleotide addition (mod 2⁶⁴) to mimic word-wise addition (mod 2⁶⁴). We also show, in particular, how to carry out the different steps of SHA-512 on an artificial DNA sequence. At the same time, the proposed nucleotide operations can be used to mimic any hash algorithm of its bitwise operations similar to bitwise operations specified in SHA-2. The proposed hash has the following features: (1) it can be applied to all data, such as text, video, and image; (2) it has the same security level of SHA-2; and (3) it can be performed in a biological environment or on DNA computers.

Generating prime numbers less than or equal to an integer number m plays an important role in many asymmetric key cryptosystems. Recently, a new sequential prime sieve algorithm was proposed based on set theory. The main drawback of this algorithm is that the running time and storage are high when the size of m is large. This paper introduces three new algorithms for a prime sieve based on two approaches. The first approach develops a fast sequential prime sieve algorithm based on set theory and some structural improvements to the recent prime sieve algorithm. The second approach introduces two new parallel algorithms in the shared memory parallel model based on static and dynamic strategies. The analysis of the experimental studies shows the following results. (1) The proposed sequential algorithm outperforms the recent prime sieve algorithm in terms of running time by 98% and memory consumption by 80%, on average. (2) The two proposed parallel algorithms outperform the proposed sequential algorithm by 72% and 67%, respectively, on average. (3) The maximum speedups achieved by the dynamic and static parallel algorithms using 16 threads are 7 and 4.5, respectively. As a result, the proposed algorithms are more effective than the recent algorithm in terms of running time, storage and scalability in generating primes.

Let h and (f, g) be the public and private keys of NTRU respectively. We present a new lattice to attack NTRU when the private polynomial g has one or more consecutive coefficients which are equal zeros, i.e., zero patterns. The target vector of the new lattice contains the private polynomial f, and the ratio between the size of the target vector and the expected size of a shortest non-zero vector in the new lattice is smaller than the ratio declared in the previous results. The zero patterns are not necessarily of the same or long lengths. Experimental results show that our attack succeeded to cryptanalyze NTRU faster than the previous attacks.