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On the Coefficients of Primitive Normal Polynomials

Author: WangXiaoZhe
Tutor: FanShuQin
School: PLA Information Engineering University
Course: Cryptography
Keywords: Finite field Galois ring p-adic Index and Primitive polynomial Normal basis. Hansen - Mullen conjecture Cohen sieve method
CLC: TN918.1
Type: Master's thesis
Year: 2010
Downloads: 22
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Abstract

In this thesis, the p-adic methods and Galois ring on the index and the estimated finite field primitive the formal polynomial coefficients distribution problem, a primitive regular Hansen-Mullen conjecture, and proved that n ≥ 9 the conjecture, that for any n ≥ 9, q is a prime power, there is a finite field Fq n primitive formal polynomial makes it to any one factor can be given in advance. The the primitive formal element in the finite field and Primitive formal polynomial has very important applications in cryptography, public key cryptography system implementation, distribution coefficient by a lot of the attention of scholars, and in the finite field fast computing the fast Fourier transform, error-correcting codes theory has important theoretical significance and application value. In order to solve the primitive formal Hansen-Mullen conjecture, this article first use of p-adic analysis methods and Galois ring of formal primitive roots of a polynomial coefficient relationship over finite fields formal primitive polynomial coefficients given in advance the existence problem into The Galois ring a trace equations primitive formal solution of the existence problem. Any one half of the coefficients in create traces Equations, primitive polynomial regular reciprocal polynomial may not be primitive polynomial regular problem, the introduction of a variable x-1 trace equation given primitive polynomial regular coefficient expressions; on the other hand, by introducing the z-criterion given of the primitive regular Hansen-Mullen suppose the the new problem of approximately O (Q1 / 2), so that the estimation result. Then use the index and estimates and Cohen sieve method to establish a general judgment the primitive formal Hansen-Mullen conjecture is established to determine the conditions and the introduction of the new form in Cohen sieve method under standard complementary factor. This new sieve method integer qn-1 factor and the polynomial x ^{n - 1 factor while doing sieve method, thus greatly improving the efficiency of the sieve method. Last for q GT respectively; n ≥ 15, 9 ≤ n ≤ 14 and q gt; 100, 16 ≤ q ≤ n, q LT; 16 four cases investigated determination condition is satisfied, by the type of integers under certain conditions the the number of prime factors and different polynomial irreducible number of factors detailed analysis, the use of some computing skills with the possible exception value falls within the range of the computer can search. Final the following conclusions theorem: Let n ≥ 9, q is a prime power. For any a ∈ Fq, (m, a) = (1, 0) for all 1 ≤ m lt; exist finite field Fq n-th primitive the regular polynomial f (x) = x ^{n -σ1x ^{<< / sup> sup> n-1 (-1) ^{n-1 σ ^{n-1 x ( -1) nσn coefficient m σm = a.}}}}}

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