59 research outputs found

    On Taking Square Roots without Quadratic Nonresidues over Finite Fields

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    We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in O~(log2q)\tilde{O}(\log^2 q) bit operations over finite fields with qq elements. As an application, we construct a deterministic primality proving algorithm, which runs in O~(log3N)\tilde{O}(\log^3 N) for some integers NN.Comment: 14 page

    Nilpotency in automorphic loops of prime power order

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    A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the vector space of 2×22\times 2 matrices over the field of prime order pp, we construct a family of automorphic loops of order p3p^3 with trivial center.Comment: 13 pages, amsart; v2: minor changes suggested by referee; to appear in J. Algebr

    On polynomials of small range sum

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    In order to reprove an old result of R\'edei's on the number of directions determined by a set of cardinality pp in Fp2\mathbb{F}_p^2, Somlai proved that the non-constant polynomials over the field Fp\mathbb{F}_p whose range sums are equal to pp are of degree at least p12\frac{p-1}{2}. Here we characterise all of these polynomials having degree exactly p12\frac{p-1}{2}, if pp is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to Lov\'asz and Schrijver using discrete Fourier analysis
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