An Upper Bound on the Complexity of Multiplication of Polynomials Modulo a Power of an Irreducible Polynomial

Let μ q 2 (n,k) denote the minimum number of multiplications required to compute the coefficients of the product of two degree n k - 1 polynomials modulo the kth power of an irreducible polynomial of degree n over the q 2 element field \BBF q 2 . It is shown that for all odd q and all n = 1,2,..., l...

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Vydáno v:	IEEE transactions on information theory Ročník 59; číslo 10; s. 6845 - 6850
Hlavní autoři:	Kaminski, Michael, Chaoping Xing
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.10.2013 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Algebra Algebraic function fields Algorithms Applied sciences bilinear algorithms Coding theory Complexity theory Exact sciences and technology finite fields Indexes Information theory Information, signal and communications theory Poles and towers Poles and zeros polynomial multiplication Polynomials Telecommunications and information theory Terrorism the Garcia-Stichtenoth tower Upper bound Finite field finite fields Upper bound Multiplication Algebraic function fields bilinear algorithms polynomial multiplication Algorithm the Garcia―Stichtenoth tower
ISSN:	0018-9448, 1557-9654
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Popis
Shrnutí:	Let μ q 2 (n,k) denote the minimum number of multiplications required to compute the coefficients of the product of two degree n k - 1 polynomials modulo the kth power of an irreducible polynomial of degree n over the q 2 element field \BBF q 2 . It is shown that for all odd q and all n = 1,2,..., liminfk → ∞[( μ q 2 (n,k))/ k n] ≤ 2 (1 + [ 1/( q - 2)] ). For the proof of this upper bound, we show that for an odd prime power q, all algebraic function fields in the Garcia-Stichtenoth tower over \BBF q 2 have places of all degrees and apply a Chudnovsky like algorithm for multiplication of polynomials modulo a power of an irreducible polynomial.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14
ISSN:	0018-9448 1557-9654
DOI:	10.1109/TIT.2013.2272072