41,627 research outputs found
Fast Polynomial Multiplication over F_(2^60)
Can post-Schönhage–Strassen multiplication algorithms be competitive in practice for large input sizes? So far, the GMP library still outperforms all implementations of the recent, asymptotically more efficient algorithms for integer multiplication by Fürer, De–Kurur–Saha–Saptharishi, and ourselves. In this paper, we show how central ideas of our recent asymptotically fast algorithms turn out to be of practical interest for multiplication of polynomials over finite fields of characteristic two. Our Mathemagix implementation is based on the automatic generation of assembly codelets. It outperforms existing implementations in large degree, especially for polynomial matrix multiplication over finite fields
Decoding Generalized Reed-Solomon Codes and Its Application to RLCE Encryption Schemes
This paper compares the efficiency of various algorithms for implementing
quantum resistant public key encryption scheme RLCE on 64-bit CPUs. By
optimizing various algorithms for polynomial and matrix operations over finite
fields, we obtained several interesting (or even surprising) results. For
example, it is well known (e.g., Moenck 1976 \cite{moenck1976practical}) that
Karatsuba's algorithm outperforms classical polynomial multiplication algorithm
from the degree 15 and above (practically, Karatsuba's algorithm only
outperforms classical polynomial multiplication algorithm from the degree 35
and above ). Our experiments show that 64-bit optimized Karatsuba's algorithm
will only outperform 64-bit optimized classical polynomial multiplication
algorithm for polynomials of degree 115 and above over finite field
. The second interesting (surprising) result shows that 64-bit
optimized Chien's search algorithm ourperforms all other 64-bit optimized
polynomial root finding algorithms such as BTA and FFT for polynomials of all
degrees over finite field . The third interesting (surprising)
result shows that 64-bit optimized Strassen matrix multiplication algorithm
only outperforms 64-bit optimized classical matrix multiplication algorithm for
matrices of dimension 750 and above over finite field . It should
be noted that existing literatures and practices recommend Strassen matrix
multiplication algorithm for matrices of dimension 40 and above. All our
experiments are done on a 64-bit MacBook Pro with i7 CPU and single thread C
codes. It should be noted that the reported results should be appliable to 64
or larger bits CPU architectures. For 32 or smaller bits CPUs, these results
may not be applicable. The source code and library for the algorithms covered
in this paper are available at http://quantumca.org/
PUBLIC KEY CRYPTOGRAPHY USING PERMUTATION P-POLYNOMIALS OVER FINITE FIELDS
In this paper we propose an efficient multivariate
public key cryptosystem based on permutation p-polynomials over
finite fields. We first characterize a class of permutation
p-polynomials over finite fields and then construct a
trapdoor function using this class of permutation p-polynomials.
The complexity of encryption in our public key cryptosystem is
multiplication which is equivalent to other
multivariate public key cryptosystems. However the decryption is
much faster than other multivariate public key cryptosystems. In
decryption we need left cyclic shifts and
xor operations
Deterministic root finding over finite fields using Graeffe transforms
We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field F q . Our algorithms were designed to be particularly efficient in the case when the cardinality q − 1 of the multiplicative group of F q is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders
The parity of the number of irreducible factors for some pentanomials
AbstractIt is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely xm+xn+2+xn+1+xn+1∈F2[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F2 and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
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