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For a prime p and a matrix A ∈ Z n×n , write A as A = p(A quo p)+ (A rem p) where the remainder and quotient operations are applied element-wise. Write the p-adic expansion of A as A = A[0] + pA[1] + p 2A[2] + · · · where each A[i] ∈ Z n×n has entries between [0, p − 1]. Upper bounds are proven for the Z-ranks of A rem p, and A quo p. Also, upper bounds are proven for the Z/pZ-rank of A[i] for all i ≥ 0 when p = 2, and a conjecture is presented for odd primes.
We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the black-box model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an n × n matrix A over the ring F[z]/(f e ), where f e is a power of an irreducible polynomial f ∈ F[z] of degree d, our algorithm requires O(ηde2n) operations in F, where our black-box is assumed to require O(η) operations in F to compute a matrix-vector product by a vector over F[z]/(f e ) (and η is assumed greater than nde). The algorithm only requires additional storage for O(nde) elements of F. In particular, if η = O˜(nde), then our algorithm requires only O˜(n 2d 2 e 3 ) operations in F, which is an improvement on known dense methods for small d and e. For the ring Z/peZ, where p is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in µnre log p, where µ is the number of operations in Z/pZ to evaluate the black-box (assumed greater than n) and r is the total number of non-zero invariant factors. To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.
Proceedings of the 2005 …
Hybrid symbolic-numeric integration in multiple dimensions via tensor-product series2005 •
We devise an algorithm, L1, with the following specifications: It takes as input an arbitrary basis of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(d^(5+ε)β +d^(ω+1+ε)β^(1+ε)) where β = log max bits of a basis vector (for any ε > 0 and ω is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in β and polynomial in d. The backbone structure of L1 is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of cutting-edge ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.
2005 •
We show that for integers k > 1 and n > 2, the diameter of the Cayley graph of SL_n(Z/kZ) associated to a standard two-element generating set, is at most a constant times n^2 ln k. This answers a question of A. Lubotzky concerning SL_n(F_p) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL_n(Z/kZ).
2008 •
We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G and find that, if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in Q (Ö| G|) Θ (| G|) steps. This is the first proof of the correct bound which does not assume that every step of the algorithm produces an iid sample from G.
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