3,748 research outputs found
A Discrete Logarithm-based Approach to Compute Low-Weight Multiples of Binary Polynomials
Being able to compute efficiently a low-weight multiple of a given binary
polynomial is often a key ingredient of correlation attacks to LFSR-based
stream ciphers. The best known general purpose algorithm is based on the
generalized birthday problem. We describe an alternative approach which is
based on discrete logarithms and has much lower memory complexity requirements
with a comparable time complexity.Comment: 12 page
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
A Multi-level Blocking Distinct Degree Factorization Algorithm
We give a new algorithm for performing the distinct-degree factorization of a
polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest
level of blocking replaces GCD computations by multiplications, as suggested by
Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our
approach is that a finer level of blocking replaces multiplications by
squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval
polynomials when P(x) is sparse. As an application we give a fast algorithm to
search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2),
while producing a certificate that can be checked in less time than the full
search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over
all trinomials of given degree r. Under a plausible assumption about the
distribution of factors of trinomials, the new algorithm has complexity O(r^2
(log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r.
Our implementation achieves a speedup of greater than a factor of 560 over the
naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our
program, we have found two new primitive trinomials of degree 24036583 over
GF(2) (the previous record degree was 6972593)
Efficiently Detecting Torsion Points and Subtori
Suppose X is the complex zero set of a finite collection of polynomials in
Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose
coordinates are d_th roots of unity can be done within NP^NP (relative to the
sparse encoding), under a plausible assumption on primes in arithmetic
progression. In particular, our hypothesis can still hold even under certain
failures of the Generalized Riemann Hypothesis, such as the presence of
Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional)
complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of
(C^*)^n we show that deciding whether X contains T is coNP-complete (relative
to an even more efficient encoding),unconditionally. We thus obtain new
non-trivial families of multivariate polynomial systems where deciding the
existence of complex roots can be done unconditionally in the polynomial
hierarchy -- a family of complexity classes lying between PSPACE and P,
intimately connected with the P=?NP Problem. We also discuss a connection to
Laurent's solution of Chabauty's Conjecture from arithmetic geometry.Comment: 21 pages, no figures. Final version, with additional commentary and
references. Also fixes a gap in Theorems 2 (now Theorem 1.3) regarding
translated subtor
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
- …