Using a well-known result of polynomial over the finite field p , we show that the Euler-Fermat theorem holds inN [x]. We present a multi-dimension RSA cryptosystem and point out that low exponent algorithm of attacki...Using a well-known result of polynomial over the finite field p , we show that the Euler-Fermat theorem holds inN [x]. We present a multi-dimension RSA cryptosystem and point out that low exponent algorithm of attacking RSA is not suitable for the multi-dimension RSA. Therefore, it is believed that the security of the new cryptosystem is mainly based on the factorization of large integers.展开更多
The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be b...The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over F<sub>q</sub>, it would belong to the template [n], and if f(x) split over F<sub>q</sub>, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of F<sub>q</sub>[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.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Using a well-known result of polynomial over the finite field p , we show that the Euler-Fermat theorem holds inN [x]. We present a multi-dimension RSA cryptosystem and point out that low exponent algorithm of attacking RSA is not suitable for the multi-dimension RSA. Therefore, it is believed that the security of the new cryptosystem is mainly based on the factorization of large integers.
文摘The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over F<sub>q</sub>, it would belong to the template [n], and if f(x) split over F<sub>q</sub>, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of F<sub>q</sub>[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.
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.
