## Cryptology ePrint Archive: Report 2009/062

On Deterministic Polynomial-Time Equivalence of Computing the CRT-RSA Secret Keys and Factoring

Subhamoy Maitra and Santanu Sarkar

Abstract: Let $N = pq$ be the product of two large primes. Consider CRT-RSA with the public encryption exponent $e$ and private decryption exponents $d_p, d_q$. It is well known that given any one of $d_p$ or $d_q$ (or both) one can factorize $N$ in probabilistic poly$(\log N)$ time with success probability almost equal to 1. Though this serves all the practical purposes, from theoretical point of view, this is not a deterministic polynomial time algorithm. In this paper, we present a lattice based deterministic poly$(\log N)$ time algorithm that uses both $d_p, d_q$ (in addition to the public information $e, N$) to factorize $N$ for certain ranges of $d_p, d_q$. We like to stress that proving the equivalence for all the values of $d_p, d_q$ may be a nontrivial task.

Category / Keywords: public-key cryptography / CRT-RSA, Cryptanalysis, Factorization, LLL Algorithm, RSA.

Publication Info: Presented in WCC 2009

Date: received 6 Feb 2009, last revised 11 Feb 2010

Contact author: subho at isical ac in

Available format(s): PDF | BibTeX Citation

Note: A revised and corrected version

Short URL: ia.cr/2009/062

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