A Lattice Solution to Approximate Common Divisors
نویسندگان
چکیده
The approximate common divisor problem(ACDP) is to find one or more divisors which is the greatest common divisor of the approximate numbers a and b of two given numbers a0 and b0. Howgrave-Graham[7] has considered the special case of b = b0 and gave a continued fraction approach and a lattice approach to find divisors. Furthermore he raised another lattice approach for ACDP based on Coppersmiths method[3]. In this paper, we first propose a strategy to efficiently generate two bivariate independent polynomials via LLL reduction method. Then we use bivariate Newton’s method to find the roots of these two polynomials and thereby we find the divisors.
منابع مشابه
Approximate Integer Common Divisors
We show that recent results of Coppersmith, Boneh, Durfee and Howgrave-Graham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully” approximate common divisors, i.e. where both integers are only known by approximations. We explain the lattice techniques in both the partial and general cases. As an application of ...
متن کاملA New Algorithm for Solving the General Approximate Common Divisors Problem and Cryptanalysis of the FHE Based on the GACD problem
In this paper, we propose a new algorithm for solving the general approximate common divisors (GACD) problems, which is based on lattice reduction algorithms on certain special lattices and linear equation solving algorithms over integers. Through both theoretical arguments and experimental data, we show that our new algorithm works in polynomial time but under roughly the following condition: ...
متن کاملApproximate common divisors via lattices
We analyze the multivariate generalization of Howgrave-Graham’s algorithm for the approximate common divisor problem. In the m-variable case with modulus N and approximate common divisor of size Nβ , this improves the size of the error tolerated from Nβ 2 to Nβ (m+1)/m , under a commonly used heuristic assumption. This gives a more detailed analysis of the hardness assumption underlying the rec...
متن کاملComputing Approximate Greatest Common Right Divisors of Differential Polynomials
Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest Common Right Divisor Problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two ...
متن کاملExistence of common best proximity points of generalized $S$-proximal contractions
In this article, we introduce a new notion of proximal contraction, named as generalized S-proximal contraction and derive a common best proximity point theorem for proximally commuting non-self mappings, thereby yielding the common optimal approximate solution of some fixed point equations when there is no common solution. We furnish illustrative examples to highlight our results. We extend so...
متن کامل