Approximate Integer Common Divisor Problem Relates to Implicit Factorization

In this paper, we analyze how to calculate the GCD of k ( ≥ 2) many large integers, given their approximations. This problem is known as the approximate integer common divisor problem in literature. Two versions of the problem, presented by Howgrave-Graham in CaLC 2001, turn out to be special cases of our analysis when k = 2. We relate the approximate common divisor problem to the implicit factorization problem as well. The later was introduced by May and Ritzenhofen in PKC 2009 and studied under the assumption that some of Least Significant Bits (LSBs) of certain primes are the same. Our strategy can be applied to the implicit factorization problem in a general framework considering the equality of (i) most significant bits (MSBs), (ii) least significant bits (LSBs), and (iii) MSBs and LSBs together. We present new and improved theoretical as well as experimental results in comparison with the state of the art work in this area.

[1]  Nick Howgrave-Graham,et al.  Finding Small Roots of Univariate Modular Equations Revisited , 1997, IMACC.

[2]  Alexander May,et al.  Implicit Factoring: On Polynomial Time Factoring Given Only an Implicit Hint , 2009, Public Key Cryptography.

[3]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[4]  S. J. Abbott,et al.  A classical introduction to modern number theory (2nd edition) , by Kenneth Ireland and Michael Rosen. Pp 394. DM 98. 1990. ISBN 3-540-97329-X (Springer) , 1992, The Mathematical Gazette.

[5]  Jeffrey C. Lagarias,et al.  The computational complexity of simultaneous Diophantine approximation problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[6]  Don Coppersmith,et al.  Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities , 1997, Journal of Cryptology.

[7]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[8]  Alexander May,et al.  Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits , 2008, ASIACRYPT.

[9]  Jean-Sébastien Coron,et al.  Deterministic Polynomial-Time Equivalence of Computing the RSA Secret Key and Factoring , 2006, Journal of Cryptology.

[10]  Alexander May,et al.  New RSA vulnerabilities using lattice reduction methods , 2003 .

[11]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[12]  Nick Howgrave-Graham,et al.  Approximate Integer Common Divisors , 2001, CaLC.

[13]  Jean-Charles Faugère,et al.  Implicit Factoring with Shared Most Significant and Middle Bits , 2010, Public Key Cryptography.

[14]  Craig Gentry,et al.  Fully Homomorphic Encryption over the Integers , 2010, EUROCRYPT.

[15]  Santanu Sarkar,et al.  Further results on implicit factoring in polynomial time , 2009, Adv. Math. Commun..

[16]  Douglas R. Stinson,et al.  Cryptography: Theory and Practice , 1995 .