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

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.

Reducing Rational Polynomial: A Proposition of a Strategy to Deal With Floating Points Numbers Using SVD

In this article, we present a new strategy to reduce rational polynomial based on the kernel of a linear map defined by the matrix's Sylvester. The strategy does not hold the computation of the Greatest common divisors (GCD) of two polynomials, as other algorithms do, but produce the reduced fraction directly. This strategy was inspired when we consider elements of Padé approximant as a basis is the Proper Generalised Decomposition for solving Partial Differential equation. The algorithm can use the Singular value decomposition technique when dealing with a polynomial with floating-point arithmetic. We compare it with Brown's algorithm in two wedges: multiplication for finite fields and large integers. Results are shown in term of time computation and robustness. The proposed algorithm shows that the time accuracy of computing the reduced fractional is at the same order as the Brown algorithm for finite field and large integers when the GCD of both polynomials has a small degree and an improving when the GCD's degree increase with the degree of polynomials. Also, robustness is more dynamic when arithmetic with the floating-point operation.

The μ-Basis of Improper Rational Parametric Surface and Its Application

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.

The Smith normal form distribution of a random integer matrix

International audience We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.