2,838 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
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields
The fastest known algorithm for factoring univariate polynomials over finite
fields is the Kedlaya-Umans (fast modular composition) implementation of the
Kaltofen-Shoup algorithm. It is randomized and takes time to factor polynomials of degree over the finite field
with elements. A significant open problem is if the
exponent can be improved. We study a collection of algebraic problems and
establish a web of reductions between them. A consequence is that an algorithm
for any one of these problems with exponent better than would yield an
algorithm for polynomial factorization with exponent better than
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
We present two algorithms to compute the endomorphism ring of an ordinary
elliptic curve E defined over a finite field F_q. Under suitable heuristic
assumptions, both have subexponential complexity. We bound the complexity of
the first algorithm in terms of log q, while our bound for the second algorithm
depends primarily on log |D_E|, where D_E is the discriminant of the order
isomorphic to End(E). As a byproduct, our method yields a short certificate
that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits
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