## Cryptology ePrint Archive: Report 2005/208

On Finding Roots Without Factoring and A Special Purpose Factoring Algorithm

Daniel R. L. Brown

Abstract: For any integer $n$, some side information exists that allows roots of certain polynomials modulo $n$ to be found efficiently (in polynomial time). The quartics $q_{u,a,b}(x) = x^4 - 4ux^3 - 2ax^2 -(8b + 4ua)x + a^2 - 4ub$, where $a$ and $b$ are some fixed integers, can be solved with probability approximately $\frac{1}{4}$ over integers $u$ chosen randomly from in $\{0,1,\dots,n-1\}$. The side information depends on $a$ and $b$, and is derivable from the factorization of $n$. The side information does not necessarily seem to reveal the factorization of $n$. For certain other polynomials, such as $p_u(x) = x^3 - u$, it is an important unsolved problem of theoretical cryptology whether there exists an algorithm for finding roots that does not also reveal the factorization of $n$. Cheng's special-purpose factoring algorithm is also reviewed and some extensions suggested.

Category / Keywords: public-key cryptography / RSA, Factoring, Roots

Date: received 30 Jun 2005, last revised 13 Jul 2005, withdrawn 15 Jul 2005

Contact author: dbrown at certicom com

Available format(s): (-- withdrawn --)

Note: Steven Galbraith found a major flaw in this paper: the information used to find roots also reveals the factorization. My attempts to correct the flaw failed. A later revision of this paper may describe the flaw and attempted fixes.

Short URL: ia.cr/2005/208

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