On oracle factoring of integers

Fast Factoring of Integers

An algorithm is given to factor an integer with N digits in lnN steps, with m approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a further exponential speedup. Factorization of large integers is important to many areas of pure mathematics and has practical applications in applied math including c...

Methods of Factoring Large Integers

On Factoring Arbitrary Integers with Known Bits

We study the factoring with known bits problem, where we are given a composite integer N = p1p2 . . . pr and oracle access to the bits of the prime factors pi, i = 1, . . . , r. Our goal is to find the full factorization of N in polynomial time with a minimal number of calls to the oracle. We present a rigorous algorithm that efficiently factors N given (1− 1 r Hr) log N bits, where Hr denotes ...

On Factoring Integers and Evaluating Discrete Logarithms

We present a survey of the state of two problems in mathematics and computer science: factoring integers and solving discrete logarithms. Included are applications in cryptography, a discussion of algorithms which solve the problems and the connections between these algorithms, and an analysis of the theoretical relationship between these problems and their cousins among hard problems of number...

Factoring Integers Using SIMD Sieves

We describe our single-instruction multiple data (SIMD) implementation of the multiple polynomial quadratic sieve integer factoring algorithm. On a 16K MasPar massively parallel computer, our implementation can factor 100 digit integers in a few days. Its most notable success was the factorization of the 110-digit RSA-challenge number, which took about a month.