4,397 research outputs found
Integer Factorization with a Neuromorphic Sieve
The bound to factor large integers is dominated by the computational effort
to discover numbers that are smooth, typically performed by sieving a
polynomial sequence. On a von Neumann architecture, sieving has log-log
amortized time complexity to check each value for smoothness. This work
presents a neuromorphic sieve that achieves a constant time check for
smoothness by exploiting two characteristic properties of neuromorphic
architectures: constant time synaptic integration and massively parallel
computation. The approach is validated by modifying msieve, one of the fastest
publicly available integer factorization implementations, to use the IBM
Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6;
Section III, par. 2) and phase calculation (Section IV, par 2
A construction of polynomials with squarefree discriminants
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we
give an unconditional construction of infinitely many monic irreducible
polynomials of degree n with integer coefficients having squarefree
discriminant and exactly r real roots. These give rise to number fields of
degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we
may also force the discriminant to be coprime to any given integer. The number
of fields produced with discriminant in the range [-N, N] is at least c
N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic
number fields admit everywhere unramified degree n extensions whose normal
closures have Galois group A_n. This generalizes results of Yamamura, who
treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not
control the real place.Comment: 10 pages; v2: refereed version, minor edits onl
Practical improvements to class group and regulator computation of real quadratic fields
We present improvements to the index-calculus algorithm for the computation
of the ideal class group and regulator of a real quadratic field. Our
improvements consist of applying the double large prime strategy, an improved
structured Gaussian elimination strategy, and the use of Bernstein's batch
smoothness algorithm. We achieve a significant speed-up and are able to compute
the ideal class group structure and the regulator corresponding to a number
field with a 110-decimal digit discriminant
Root optimization of polynomials in the number field sieve
The general number field sieve (GNFS) is the most efficient algorithm known
for factoring large integers. It consists of several stages, the first one
being polynomial selection. The quality of the chosen polynomials in polynomial
selection can be modelled in terms of size and root properties. In this paper,
we describe some algorithms for selecting polynomials with very good root
properties.Comment: 16 pages, 18 reference
Improvements in the computation of ideal class groups of imaginary quadratic number fields
We investigate improvements to the algorithm for the computation of ideal
class groups described by Jacobson in the imaginary quadratic case. These
improvements rely on the large prime strategy and a new method for performing
the linear algebra phase. We achieve a significant speed-up and are able to
compute ideal class groups with discriminants of 110 decimal digits in less
than a week.Comment: 14 pages, 5 figure
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