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Abstract. Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm. 1

Abstract. This paper introduces GMP-EECM, a fast implementation of the elliptic-curve method of factoring integers. GMP-EECM is based on, but faster than, the well-known GMP-ECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted Edwards coordinates; (3) use signedsliding-window addition chains; (4) batch primes to increase the window size; (5) choose curves with small parameters a, d, X1, Y1, Z1; (6) choose curves with larger torsion.