Abstract. Let N > 1. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one N. The addition and subtraction algorithms are unchanged. To validate REDC, observe mN = TN'N = -Tmod R, so t is an integer. Also, tR = Tmod N so t = TR'X mod N. Thirdly, 0 < T + mN < RN + RN, so 0 < t < 2N.If R and N are large, then T + mN may exceed the largest double-precision value. One can circumvent this by adjusting m so -R < m < 0.Given two numbers x and y between 0 and N -1 inclusive, let z = REDC(xy). Then z = (xy)R~x mod N, so (xR-l)(yR~x) = zRx mod N. Also, 0 < z < N, so z is the product of x and y in this representation.Other algorithms for operating on N-residues in this representation can be derived from the algorithms normally used. The addition algorithm is unchanged, since xR~x + yR~x = zR~x mod N if and only if x + y = z mod N. Also unchanged are

Abstract. Let N > 1. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one N. The addition and subtraction algorithms are unchanged. To validate REDC, observe mN = TN'N = -Tmod R, so t is an integer. Also, tR = Tmod N so t = TR'X mod N. Thirdly, 0 < T + mN < RN + RN, so 0 < t < 2N.If R and N are large, then T + mN may exceed the largest double-precision value. One can circumvent this by adjusting m so -R < m < 0.Given two numbers x and y between 0 and N -1 inclusive, let z = REDC(xy). Then z = (xy)R~x mod N, so (xR-l)(yR~x) = zRx mod N. Also, 0 < z < N, so z is the product of x and y in this representation.Other algorithms for operating on N-residues in this representation can be derived from the algorithms normally used. The addition algorithm is unchanged, since xR~x + yR~x = zR~x mod N if and only if x + y = z mod N. Also unchanged are

Abstract. Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo N and occasionally taking GCDs with N. These began with Pollard's p -1 and Monte Carlo methods. More recently, Williams published a p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ± 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 log2 n multiplications for arbitrary P.

Abstract. Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo N and occasionally taking GCDs with N. These began with Pollard's p -1 and Monte Carlo methods. More recently, Williams published a p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ± 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 log2 n multiplications for arbitrary P.

Abstract. This paper reports on the factorization of the 768-bit number RSA-768 by the number field sieve factoring method and discusses some implications for RSA.