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Elliptic Curve Factorization Using a "Partially Oblivious" Function.
"... . Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be nontrivial and that f(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Intr ..."
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. Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be nontrivial and that f(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Introduction It is not known how to eciently factor a large integer N . Currently, the algorithm with best asymptotic complexity is the Number Field Sieve (see [6] ). For numbers below a certain size (currently believed to be about 100 decimal digits), either the Quadratic Sieve [12] or Lenstra's Elliptic Curve Method (ECM) [7] are faster. Which of these algorithms to use depends on the size of N and of the smallest prime factor of N . When the size of the smallest factor is suciently smaller than p N , ECM is the fastest of the three. This note describes a speedup of ECM under special conditions. Suppose N = P R, where P is a prime not dividing R. We assume the size, in bits, of P is know...