**Some Applications of Lattice Based Root Finding Techniques**

*Santanu Sarkar and Subhamoy Maitra*

**Abstract: **In this paper we present some problems and their solutions exploiting
lattice based root finding techniques.

In CaLC 2001, Howgrave-Graham proposed a method to find the Greatest Common Divisor (GCD) of two large integers when one of the integers is exactly known and the other one is known approximately. In this paper, we present three applications of the technique. The first one is to show deterministic polynomial time equivalence between factoring $N$ ($N = pq$, where $p > q$ or $p, q$ are of same bit size) and knowledge of $q^{-1} \bmod p$. Next, we consider the problem of finding smooth integers in a short interval. The third one is to factorize $N$ given a multiple of the decryption exponent in RSA.

In Asiacrypt 2006, Jochemsz and May presented a general strategy for finding roots of a polynomial. We apply that technique for solving the following two problems. The first one is to factorize $N$ given an approximation of a multiple of the decryption exponent in RSA. The second one is to solve the implicit factorization problem given three RSA moduli considering certain portions of LSBs as well as MSBs of one set of three secret primes are same.

**Category / Keywords: **public-key cryptography / CRT-RSA, Greatest Common Divisor, Factorization, Integer Approximations, Lattice, LLL, RSA, Smooth Integers.

**Date: **received 17 Mar 2010, last revised 7 Apr 2010

**Contact author: **subho at isical ac in

**Available format(s): **PDF | BibTeX Citation

**Note: **Substantial extension to earlier version.

**Version: **20100407:130110 (All versions of this report)

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