2023/04/15 by Gilda Rech Bansimba, Bansimba, Gilda Rech, Regis Freguin Babindamana +3 · 2 citations
Computer Science · Mathematics · #11T71 #11Y05 #11Y40 #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2304.07474
openalex publication_date 2023/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From the results in the literature, the algebraic set of the hyperbola with parameter n defined by Bn(X, Y, Z)_|x≥ 4n= \lbrace (X: Y: Z)∈ ℙ2(ℚ) \vert Y2=X2-4nXZ \rbrace where n is a semiprime is proved to be in relation with prime factors of n. In the affine space over ℤ\geqslant 4n× ℤ\geqslant 0, this set has exactly 5 points \lbrace P0, P1, P2, P3, P4 \rbrace with P2+P3=P1+2P2=P4 for which knowledge of P2 or P3 yields the factorization of n. However, The non cyclicity of this group structure over rationals and integers and moreover its non good reduction over finite fields constitute the main difficulty in finding its solutions. In this paper we describe an approach to finding P2 and P3. We introduce the concept of Hyperbola X-root and Y-root that the solution's greatest common divisors with n reveal prime factors of n. We prove that P2 and P3 can be found on a singular Weierstrass curve isomorphic to a Jacobi quartic using the Hyperbola X-root and Y-root. We present the mathematical framework for this approach.