The concept of computability is defined more exactly and illustrated as an example of Boolean functions and cryptanalysis. To define a Boolean function is not necessary to record its formula. To do that the reduced (c...The concept of computability is defined more exactly and illustrated as an example of Boolean functions and cryptanalysis. To define a Boolean function is not necessary to record its formula. To do that the reduced (compact) description of values is determined in the truth table or in the statement of the problem. We obtain estimates of computation time, the volume of a compact descriptions and the range of variables under which it takes the value 0 or 1, depending polynomially on the number of arguments.展开更多
We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve th...We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results: (i) we decrease from O(n 2 + n 1+o(1)logq) to O(n 1.9998 + n 1+o(1)logq) the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n × n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii) we decrease from O(m 1.575 n) to O(m 1.5356 n) the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.展开更多
文摘The concept of computability is defined more exactly and illustrated as an example of Boolean functions and cryptanalysis. To define a Boolean function is not necessary to record its formula. To do that the reduced (compact) description of values is determined in the truth table or in the statement of the problem. We obtain estimates of computation time, the volume of a compact descriptions and the range of variables under which it takes the value 0 or 1, depending polynomially on the number of arguments.
文摘We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results: (i) we decrease from O(n 2 + n 1+o(1)logq) to O(n 1.9998 + n 1+o(1)logq) the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n × n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii) we decrease from O(m 1.575 n) to O(m 1.5356 n) the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.
