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 ϕ(n), 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 ϕ 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.

/pdf/small-private-exponent-attacks-on-rsa-using-continued-2guchreb.pdf

Small Private Exponent Attacks on RSA Using Continued Fractions and Multicore Systems

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

New Semi-Prime Factorization and Application in Large RSA Key Attacks

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.

/pdf/efficient-sequential-and-parallel-prime-sieve-algorithms-jsa2xlp4.pdf

Efficient Sequential and Parallel Prime Sieve Algorithms

The Improved Estimation of the Least Upper Bound to Search for RSA's Private key

The security of many public-key cryptosystems and protocols relies on the difficulty of factoring a large positive integer n into prime factors. The Fermat factoring method is a core of some modern and important factorization methods, such as the quadratic sieve and number field sieve methods. It factors a composite integer n=pq in polynomial time if the difference between the prime factors is equal to ∆=p-q≤n^(0.25) , where p>q. The execution time of the Fermat factoring method increases rapidly as ∆ increases. One of the improvements to the Fermat factoring method is based on studying the possible values of (n mod 20). In this paper, we introduce an efficient algorithm to factorize a large integer based on the possible values of (n mod 20) and a precomputation strategy. The experimental results, on different sizes of n and ∆, demonstrate that our proposed algorithm is faster than the previous improvements of the Fermat factoring method by at least 48%.

Speeding Up Fermat’s Factoring Method using Precomputation

The Rivest-Shamir-Adleman (RSA) cryptosystem is one of the strong encryption approaches currently being used for secure data transmission over an insecure channel. The difficulty encountered in breaking RSA derives from the difficulty in finding a polynomial time for integer factorization. In integer factorization for RSA, given an odd composite number n, the goal is to find two prime numbers p and q such that n = p q. In this paper, we study several integer factorization algorithms that are based on Fermat’s strategy, and do the following: First, we classify these algorithms into three groups: Fermat, Fermat with sieving, and Fermat without perfect square. Second, we conduct extensive experimental studies on nine different integer factorization algorithms and measure the performance of each algorithm based on two parameters: the number of bits for the odd composite number n, and the number of bits for the difference between two prime factors, p and q. The results obtained by the algorithms when applied to five different data sets for each factor reveal that the algorithm that showed the best performance is the algorithms based on (1) the sieving of odd and even numbers strategy, and (2) Euler’s theorem with percentage of improvement of 44% and 36%, respectively compared to the original Fermat factorization algorithm. Finally, the future directions of research and development are presented.

/pdf/performance-analysis-of-fermat-factorization-algorithms-1i336cmhxp.pdf

Performance Analysis of Fermat Factorization Algorithms

The encryption efficiency of the Rivest-Shamir-Adleman cryptosystem is based on decreasing the number of multiplications in the modular exponentiation (ME) operation. An addition chain (AC) is one of the strategies used to reduce the time consumed by ME through generating a shortest/short chain. Due to the non-polynomial time required for generating a shortest AC, several algorithms have been proposed to find a short AC in a faster time. In this paper, we use the evolutionary algorithm (EA) to find a short AC for a natural number. We discuss and present the role of every component of the EA, including the population, mutation operator, and survivor selection. Then we study, practically, the effectiveness of the proposed method in terms of the length of chain it generates by comparing it with three kinds of algorithms: (1) exact, (2) non-exact deterministic, and (3) non-exact non-deterministic. The experiment is conducted on all natural numbers that have 10, 11, 12, 13, and 14 bits. The results demonstrate that the proposed algorithm has good performance compared to the other three types of algorithms.

/pdf/an-evolutionary-algorithm-for-short-addition-chains-ycqk8y9qg2.pdf

An Evolutionary Algorithm for Short Addition Chains

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Performance Analysis of Fermat Factorization Algorithms

An Evolutionary Algorithm for Short Addition Chains