Get Computational and Algorithmic Problems in Finite Fields PDF

By Igor Shparlinski

ISBN-10: 9401047960

ISBN-13: 9789401047968

ISBN-10: 940111806X

ISBN-13: 9789401118064

This quantity offers an exhaustive remedy of computation and algorithms for finite fields.
issues lined comprise polynomial factorization, discovering irreducible and primitive polynomials, distribution of those primitive polynomials and of primitive issues on elliptic curves, developing bases of varied kinds, and new functions of finite fields to different araes of arithmetic. For completeness, additionally incorporated are distinct chapters on a few fresh advances and functions of the speculation of congruences (optimal coefficients, congruential pseudo-random quantity turbines, modular mathematics etc.), and computational quantity idea (primality checking out, factoring integers, computing in algebraic quantity conception, etc.) the issues thought of the following have many functions in computing device technological know-how, coding concept, cryptography, quantity idea and discrete arithmetic.
the extent of debate presuppose just a wisdom of the elemental proof on finite fields, and the e-book may be urged as supplementary graduate textual content.
For researchers and scholars attracted to computational and algorithmic difficulties in finite fields.

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Extra resources for Computational and Algorithmic Problems in Finite Fields

Example text

Here we show that very simple considerations enable us to obtain an algorithm with computing time (p log N)O(I) which for any N E N computes an irreducible polynomial of degree n = N + o(N) over IFp. 2. 1 ) n= N + O(N exp[-(loglogN)I/2-']). Proof. Let (4,3,5), { (d,Pl,P2)= (4,2,5), ifp=2, (2,2,3), if p > 3. ifp=3, CHAPTER 2 24 Then, in time (plogn)O(1), we choose tP E Gd(p) and the integer t of the form t = ptp~ that is nearest to N/d where k and m are non-negative integers. 35] that the polynomial of degree n = dt is irreducible.

Under the ERH one has hn{p) = O(log2n p). An analogous statement about Hn(P) was conjectured in [1146, 1147]. We show that the method of [10] with some considerations which were used in proving Artin's conjecture "on the average" (see [1151]) imply a more exact bound for "almost all" p E lID without any unproved hypotheses. 13. For n fixed and an arbitrary increasing function lIt(p) > 0 for alJ except possibJy o(1T(N» prime numbers p ::; N, the bound hn(p) = O(IIt(p» holds. Proof For x > 0 and k, r E ;Z we write IID( x, k, 1') = {I E lID 1 1 ~ x, 1 == l' (mod k)}; 1T(X, k, 1') = IIID(x, k, 1')1.

Without the ERH, finding an I-th nonresidue is the "bottleneck" of the algorithm. To avoid it, one algorithm of [1071, 1072] uses reduction to finding roots of binomials xl - b in finite fields. The versions of [1055] and [215] have computing time (np)O(l) (in [1055] only the case of fixed p was considered but, apparently, this is not essential). For the computing time of the algorithms of [332] and [10] the bound (nlogp)O(l) was proved under the ERH. The currently best bounds are given in [1071, 1072] (the result on factoring from [1071, 1073] were used in the proof).

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Computational and Algorithmic Problems in Finite Fields by Igor Shparlinski

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