Download PDF by Igor Shparlinski: Computational and Algorithmic Problems in Finite Fields

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 forms, and new purposes of finite fields to different araes of arithmetic. For completeness, additionally integrated are unique chapters on a few contemporary advances and purposes of the speculation of congruences (optimal coefficients, congruential pseudo-random quantity turbines, modular mathematics etc.), and computational quantity conception (primality checking out, factoring integers, computing in algebraic quantity idea, etc.) the issues thought of the following have many functions in machine technological know-how, coding idea, cryptography, quantity concept and discrete arithmetic.
the extent of dialogue presuppose just a wisdom of the elemental proof on finite fields, and the ebook could be prompt as supplementary graduate textual content.
For researchers and scholars attracted to computational and algorithmic difficulties in finite fields.

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If we write x = Xl Wl + ... + XnW n , where Wl, ... ,Wn is a basis lFpn over lFp, then we can replace the equation ft(x) = 0, x E lFpn, t E T, by a system of the kind i=l, ... ,n, over IFp with t E T. Several results on the distribution of rational points (and on exponential sums) over varieties over finite fields are given in [61, 357, 587, 589, 833, 1027, 1031, 1063, 1099, 1111], but these are not good enough for the applications mentioned. For the distribution of rational points on varieties over Q and other fields see the papers mentioned above and [70, 658, 737,1106,1107,1109,1110,1233] (for results obtained by algebraic methods) and [124, 235, 264, 287, 502, 515, 1025, 1028-1030, 1252] (for results obtained by analytic methods).

For arbitrary fields an analogous result was proved in [1Ol6]. It was shown in [1084] that there is a polynomial f E Gn (2) of weight W(f) ~ n/4 + o(n). 3 above). 3. Sparse Polynomials Here we present a result of [1093] on sparse polynomials which completely split over IFp into linear factors. Our main tool is an upper bound for the number of zeros of polynomials of a given weight which is based on the bound for the number of solutions of exponential equations of [1146]. 5) 1 ~ tl < ... < tn < p - 1, for which there exist ai, .

It was shown in [1084] that there is a polynomial f E Gn (2) of weight W(f) ~ n/4 + o(n). 3 above). 3. Sparse Polynomials Here we present a result of [1093] on sparse polynomials which completely split over IFp into linear factors. Our main tool is an upper bound for the number of zeros of polynomials of a given weight which is based on the bound for the number of solutions of exponential equations of [1146]. 5) 1 ~ tl < ... < tn < p - 1, for which there exist ai, . 6) is completely decomposed over IFp.

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Computational and Algorithmic Problems in Finite Fields by Igor Shparlinski


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