Some Arithmetical Properties on Hyperbola

Bansimba, Gilda R.; Babindamana, Regis F.; Bossoto, Basile Guy Richard

doi:10.17654/nt050010045

“…Public key encryption uses Euler's totient function [56,57], which provides the basis for factoring algorithms that have a major impact on cybersecurity. Hence, past works on number theory, primality testing, and the implementation of algorithms [58][59][60][61][62][63] have been the backbone of this research work to arrive at a new method. Further, the speeding up of existing methods using more efficient methods and the adoption of more computing power, including parallelism [64][65][66][67], have fueled this ongoing research.…”

Section: Improvements and Future Researchmentioning

confidence: 99%

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Overmars,

Venkatraman

2024

JCP

0

View full text Add to dashboard Cite

The RSA (Rivest–Shamir–Adleman) cryptosystem is an asymmetric public key cryptosystem popular for its use in encryptions and digital signatures. However, the Wiener’s attack on the RSA cryptosystem utilizes continued fractions, which has generated much interest in developing competitive factoring algorithms. A general-purpose integer factorization method first proposed by Lehmer and Powers formed the basis of the well-known Continued Fraction Factorization (CFRAC) method. Recent work on the one line factoring algorithm by Hart and its connection with Lehman’s factoring method have motivated this paper. The emphasis of this paper is to explore the representations of PQ as continued fractions and the suitability of lower ordered convergences as representations of ab. These simpler convergences are then prescribed to Hart’s one line factoring algorithm. As an illustration, we demonstrate the working of our approach with two numbers: one smaller number and another larger number occupying 95 bits. Using our method, the fourth convergence finds the factors as the solution for the smaller number, while the eleventh convergence finds the factors for the larger number. The security of the RSA public key cryptosystem relies on the computational difficulty of factoring large integers. Among the challenges in breaking RSA semi-primes, RSA250, which is an 829-bit semi-prime, continues to hold a research record. In this paper, we apply our method to factorize RSA250 and present the practical implementation of our algorithm. Our approach’s theoretical and experimental findings demonstrate the reduction of the search space and a faster solution to the semi-prime factorization problem, resulting in key contributions and practical implications. We identify further research to extend our approach by exploring limitations and additional considerations such as the difference of squares method, paving the way for further research in this direction.

show abstract

“…Still, in the previous works, algebraic properties and distribution of lattice points and cardinalities on Fermat's equation are not presented. More recently, in [16], Gilda et al investigated algebraic and arithmetical properties on the group structure B N (x, y) � (x, y) ∈ Q × Q/y 2 � x 2 − 4Nx 􏼈 􏼉, mainly isomorphisms, integral solutions, and a description of a factorization method with no generalization to the Fermat factorization equation.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we use the hyperbola parametrization introduced in [16] to study algebraic properties of lattice points and their distribution for Fermat's factorization equation for which we find exact upper and lower bounds and we present the forms and cardinalities with a generalization of results for most of special cases of N, using results from the particular hyperbola parametrization.…”

Section: Introductionmentioning

confidence: 99%

Lattice Points on the Fermat Factorization Method

Babindamana

¹

,

Bansimba

²

,

Bossoto

³

2022

Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

In this paper, we study algebraic properties of lattice points of the arc on the conics x 2 − d y 2 = N especially for d = 1 , which is the Fermat factorization equation that is the main idea of many important factorization methods like the quadratic field sieve, using arithmetical results of a particular hyperbola parametrization. As a result, we present a generalization of the forms, the cardinal, and the distribution of its lattice points over the integers. In particular, we prove that if N − 6 ≡ 0 mod 4 , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, 2 α + 1 , 1 − δ 2 p i 2 n + 1 , and 2 ∏ i = 1 n α i + 1 lattice pointts if N is an odd prime, N = N a × N b with N a and N b being odd primes, N = N a α with N a being prime, N = ∏ i = 1 n p i with p i being distinct primes, and N = ∏ i = 1 n N i α i with N i being odd primes. These results are important since they provide further arithmetical understanding and information on the integer solutions revealing factors of N . These results could be particularly investigated for the purpose of improving the underlying integer factorization methods.

show abstract

“…Everyone who learned the analytic geometry knows that all the textbooks focus a lot on the so-called standard form in the Cartesian coordinate system while another form xy = N has been less exploited because the later is thought to be a transformed form from the standard form via coordinate transformations. Our recent study on integer factorization problem came across the literatures [8] and [9]. Accordingly we were aroused to have a study on the kind of the hyperbola.…”

Section: Introductionmentioning

confidence: 99%

New Fantastic Curves Discovered from Rectangular Hyperbola

Wang

¹

,

Han

²

2022

JAMCS

View full text Add to dashboard Cite

By means of constructing on the hyperbola xy = N a triangle whose shape is changed under the rule that one vertex is fixed at the vertex of the hyperbola, one vertex is moving on the hyperbola and the two laterals with respect to the fixed and the moving vertices respectively keep their directions unchanged, it is discovered that the loci of the triangle's centroid and orthocenter are respectively a hyperbola and a line, the locus of the circumcenter is a new cubic algebraic curve, and those of the incenter and ex-centers are planar curves that have not been reported before. All the loci of the centers form a fantastic graph like a ying insect. Meanwhile, the discovered hyperbola and curves are merely N-dependant and can be used to estimate the distribution of y divided by x with respect to xy = N.

show abstract

Some Arithmetical Properties on Hyperbola

Cited by 4 publications

References 0 publications

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Lattice Points on the Fermat Factorization Method

New Fantastic Curves Discovered from Rectangular Hyperbola

Contact Info

Product

Resources

About