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Modular multiplication without trial division
TLDR
A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.
Speeding the Pollard and elliptic curve methods of factorization
Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1
Five, six, and seven-term Karatsuba-like formulae
  • P. L. Montgomery
  • Mathematics, Computer Science
    IEEE Transactions on Computers
  • 1 March 2005
TLDR
This work presents division-free formulae, which multiply two 5-term polynomials with 13 scalar multiplications, two 6- term polynmials with 17 scalarmultiplications, and two 7-termPolynomial with 22 scalar multiplier, and describes their application to elliptic curve arithmetic over binary fields.
Factorization of a 768-Bit RSA Modulus
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.
Trading Inversions for Multiplications in Elliptic Curve Cryptography
TLDR
A variant which is faster whenever a field inversion is more expensive than six field multiplications is proposed, an improvement when tripling a point, and a ternary/binary method to perform efficient scalar multiplication are presented.
A Block Lanczos Algorithm for Finding Dependencies Over GF(2)
TLDR
The Lanczos algorithm is modified to produce a sequence of orthogonal subspaces of GF(2)n, each having dimension almost N, by applying the given matrix and its transpose to N binary vectors at once.
Fast Elliptic Curve Arithmetic and Improved Weil Pairing Evaluation
TLDR
This paper converts the width-w NAF to an SPA-resistant addition chain and generates a scalar sequence with the fixed pattern, where x is positive odd points < 2w, and the size of the table is 2w-1, which is optimal in the construction of the Spa-resistant chain based on width-W NAF.
Division by invariant integers using multiplication
TLDR
This paper presents code sequences for division by arbitrary nonzero integer constants and run-time invariants using integer multiplication using a two's complement architecture, and treats unsigned division, signed division, and division where the result is known a priori.
An FFT extension of the elliptic curve method of factorization
TLDR
This thesis describes how to apply convolutions modulo N and last polynomial arithmetic algorithms in the search of factors of a large integer N, which effectively increases the range of ECM by a factor of 100 with about twice the combined Step 1/Step 2 execution time previously required.
Factorization of a 512-Bit RSA Modulus
This paper reports on the factorization of the 512-bit number RSA-155 by the Number Field Sieve factoring method (NFS) and discusses the implications for RSA.
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