Buresh-Oppenheim proved that the NP search problem to find nontrivial factors of integers of a special form belongs to Papadimitriou’s class PPA, and is probabilistically reducible to a problem in PPP. In this paper, we use ideas from bounded arithmetic to extend these results to arbitrary integers. We show that general integer factoring is reducible in randomized polynomial time to a PPA probl...

Abstract Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner IEEE symposium information theory, ISIT 2020, 2020) defined ring integers modulo a prime power, inspired by paper Kamche Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric principal ideal rings. In t...

For a finite commutative ring R, let a,b,c∈R be fixed elements. Consider the equation ax+by=cz where x, y, and z are idempotents, units, any element in respectively. We say that R satisfies r-monochromatic clean condition if, for r-colouring χ of elements there exist x,y,z∈R with χ(x)=χ(y)=χ(z) such holds. define m(a,b,c)(R) to least positive integer r does not satisfy condition. This means exi...

In the field of mathematics, there are many branches study, especially in graph theory, mathematically a is pair sets, which consists non-empty set whose members called vertices and distinct unordered pairs edges. One example from group coprime graph, where defined as two with different x y neighbors if only (|x|,|y|)=1. this author discusses Harmonic Index Gutman Coprime Graph Integer Group Mo...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

The Manin ring is a family of quadratic algebras describing pointed stable curves of genus zero whose homology gives the solution of the Commutativity Equations. This solution was first observed by the physicist Losev. We show the Manin ring is the Stanley–Reisner ring of the standard triangulation of the n-cube modulo a system of parameters. Thus, the Hilbert series of the Manin ring is given ...

The ElGamal encryption scheme is described in the setting of any finite cyclic group G. Among the groups of most interest in cryptography are the multiplicative group Z of the ring of integers modulo a prime p, and the multiplicative groups F of finite fields of characteristic two. The later requires finding irreducible polynomials h(x) and constructing the quotient ring . El-Kassar et al. modi...

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P,Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the ca...