Computing method of polynomial interpolation in cantor-type affine space over binary field

Abstract

The application discloses a calculation method of polynomial interpolation in a Cantor type affine space on a binary field. l‑1 The method first reduces the problem to calculating the remainder of the polynomial modulo a Cantor type three polynomial set and calculating the interpolation of the polynomials in a 2 l‑1 m-2 dimensional Cantor type affine space. Here, m is the dimension of the affine space, and

Description

Technical Field

[0001] computer Science. Technical Background

[0002] Fast computation of polynomial interpolation is an important topic in computer algorithm theory, playing a crucial role in computer science, coding theory, and cryptography. In computer science, it can be applied to fast polynomial root finding, fast polynomial multiplication, fast polynomial division, and the fast Euclidean algorithm. In coding theory, it can be applied to fast encoding of Reed-Solomon codes and fast decoding of BCH codes. In cryptography, it can be applied to the fast reconstruction of linear recursive sequences.

[0003] In 1989, Cantor, DG, published an article entitled "On arithmetic algorithms over finite fields" in the Journal of Combination Theory, ser. A, Vol. 50, No. 2. This article presented a method for computing polynomial interpolation in Cantor-type affine spaces over binary fields. The method requires O(n log₂ n) multiplications and O(n log₂ n) additions. The Cantor method has been applied to fast polynomial root finding, fast polynomial multiplication, fast polynomial division, fast Euclidean algorithm, Reed-Solomon fast encoding, and fast BCH code decoding.

[0004] In 2010, Gao, S. and Mater, T. published an article entitled "Additive Fast Fourier Transforms Over Finite Fields, T." in IEEE Trans. Inform., Volume 56, Issue 12. This article presents a method for computing 2T Fast Fourier Transforms over Finite Fields. l A method for polynomial interpolation in 1 / 2 Cantor-type affine spaces. This method requires O(n log₂ n) multiplications and O(n(log₂ n) log₂ log₂ n) additions. The Gao-Mateer method has applications in fast polynomial root finding and Reed-Solomon fast coding.

[0005] The core idea of ​​Cantor's method for computing polynomial interpolation in a Cantor-type affine space over a binary field is to decompose the Cantor-type affine space into the union of two lower-dimensional Cantor-type affine subspaces, and then reduce the problem to computing the interpolation of the two remainders of the polynomial in the two Cantor-type affine subspaces.

[0006] Gao-Mateer's method for computing 2 in a binary field l The method for polynomial interpolation in a Cantor-type affine space reduces the problem to computing the polynomial. The Gao-Mateer polynomials in 2 l-1 Interpolation in a Cantor-type affine space and computation of the polynomial The remainder in 2 l-1 Interpolation in a Cantor-type affine space.

[0007] Gao-Mateer also provided an algorithm for computing the Gao-Mateer polynomial of a polynomial, which we call the Gao-Mateer algorithm.

[0008] This invention discloses a method for calculating polynomial interpolation in a Cantor-type affine space over a binary field. The method first reduces the problem to calculating the remainder of the polynomial modulo a Cantor-type trinomial system and then calculating these remainders in a 22... l-1 Interpolation in a Cantor-type affine space, and then reducing the calculation of the remainder of the polynomial modulo a Cantor-type trinomial system to the calculation of the polynomial itself. The Gao-Mateer polynomials in m-2 l-1 Interpolation in an affine space of the dimensional Cantor type. Here, m is the dimension of the affine space, and...

[0009] The method presented in this invention requires O(n log₂ n) multiplications but O(n(log₂n) log₂ log₂ n) additions. This invention has applications in fast polynomial multiplication, fast polynomial division, fast Euclidean algorithm, Reed-Solomon fast encoding, and fast BCH code decoding.

[0010] Name Explanation

[0011] An affine space over a binary field is called a Cantor-type affine space if it is contained in a space of the form In a finite field, and its vector space has a basis β1,β2,…,β m Satisfying β1=1,β2 2 -β2=β1,…,β m 2 -β m =β m-1 .

[0012] A set of trinomials is called a Cantor-type trinomial set if there exists a positive integer l and a Cantor-type affine space B over a binary field such that this set of trinomials equals

[0013] A polynomial f(x) in The Gao-Mateer polynomial is a system of polynomials. It satisfies

[0014] Summary of the Invention

[0015] This invention discloses a method for calculating polynomial interpolation in a Cantor-type affine space over a binary field. This method has the following characteristics:

[0016] a) A method was called to calculate the remainder of a polynomial modulo a Cantor trinomial group;

[0017] b) The calculation of the interpolation of this polynomial in the Cantor-type affine space over the binary field is reduced to the calculation of the interpolation of these remainders in the subspaces of the affine space. When the dimension of the affine space is m, the dimension of these subspaces is... This invention discloses a method for calculating the remainder of a polynomial modulo a Cantor-type trinomial system. This method comprises the following steps:

[0018] a) Use the Gao-Mateer algorithm to compute the Gao-Mateer polynomial at a linearized binomial, which is the common linearized part of the trinomial.

[0019] b) Use the method disclosed in this invention to calculate the interpolation of these Gao-Mateer polynomials in the affine space composed of the constant term coefficients of the trinomial group;

[0020] c) Arrange the interpolation values ​​of each Gao-Mateer polynomial in the affine space formed by the constant terms of the trinomial group into a row, and then arrange these rows into a matrix.

[0021] d) Construct a polynomial using each column of this matrix as coefficients, and use all these newly constructed polynomials as the remainder. The method for calculating polynomial interpolation in a Cantor-type affine space over a binary field disclosed in this invention comprises the following steps:

[0022] a) Let m denote the dimension of the Cantor-type affine space over the binary field;

[0023] b) Let a be the orthogonal projection point of the Cantor-type affine space onto the binary domain;

[0024] c) When m = 1, use (f(a), f(a+1)) as the interpolation of the polynomial in the affine space and end the process;

[0025] d) When m>1, take And continue with the following steps;

[0026] e) Let

[0027] f) Take m-2 containing b l-1 A quintic space of type Cantor is denoted as B;

[0028] g) Calculate the modular trinomial of the polynomial using the method disclosed in this invention.

[0029] The remainder;

[0030] h) For each remainder, recursively call this method to calculate the interpolation of the remainder in the affine subspace formed by the zeros of the trinomial corresponding to the remainder.

[0031] i) Arrange the interpolation values ​​of each remainder in a row, and then arrange these rows in order into a matrix;

[0032] j) Sort the elements in this matrix column by column, and use the resulting sequence as the interpolation of the polynomial in the affine space. Attached Figure Description

[0033] Figure 1 is The present invention discloses pseudocode for a method of calculating polynomial interpolation in a Cantor-type affine space over a binary field.

[0034] Figure 2 is The present invention discloses a method for calculating the remainder of a polynomial modulo a Cantor trinomial group. Detailed Implementation

[0035] This invention can be programmed for fast encoding of Reed-Solomon codes. Combined with interpolation polynomial calculation methods, this invention can be applied to fast polynomial multiplication, fast polynomial division, fast Euclidean algorithm, and fast decoding of BCH codes.

Claims

1. A method for calculating polynomial interpolation in a Cantor-type affine space over a binary field, the method having the following characteristics: a) The method described above calls a method for calculating the remainder of a polynomial modulo a Cantor-type trinomial group; (b) The method reduces the calculation of the polynomial interpolation in the Cantor-type affine space over the binary field to the calculation of the interpolation of the remainder in the subspace of the affine space. When the dimension of the Cantor-type affine space over the binary field is m, the dimension of the subspace of the affine space is...

2. The method for calculating the remainder of a polynomial modulo a Cantor-type trinomial group according to claim 1 comprises the following steps: a) Calculate the Gao-Mateer polynomial of the polynomial at a linearized binomial, where the binomial is the linearized part common to the set of trinomials; b) Using the method of claim 1, calculate the interpolation of the Gao-Mateer polynomial in the affine space composed of the constant term coefficients of the trinomial; c) Arrange the interpolation values ​​of each of the Gao-Mateer polynomials in the affine space in order into a row, and arrange all the rows in order into a matrix; d) Construct a polynomial using each column of the matrix as coefficients, and use all the newly constructed polynomials as the remainder.

3. The method for calculating polynomial interpolation in a Cantor-type affine space over a binary field as described in claim 1, the method comprising the following steps: a) Let m denote the dimension of the Cantor-type affine space over the binary field; b) Let a be the orthogonal projection point of the Cantor-type affine space onto the binary domain; c) When m = 1, use (f(a), f(a+1)) as the interpolation of the polynomial in the affine space and end the process; d) When m>1, take And continue with the following steps; e) Let f) Take m-2 containing b l-1 A quintic space of type Cantor is denoted as B; g) Calculate the polynomial modular trinomial group by invoking the method of claim 2. The remainder of b∈B; h) For the remainder of each trinomial group, recursively call this method to calculate the interpolation of the remainder in the affine subspace composed of the zeros of the trinomial corresponding to the remainder. i) Arrange the interpolation values ​​of each remainder in a row, and arrange all the rows in order into a matrix; j) Sort the elements in the matrix column by column, and use the resulting sequence as the interpolation of the polynomial in the affine space.

Images