59 research outputs found
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
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 bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
Nilpotency in automorphic loops of prime power order
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 matrices over the field of prime
order , we construct a family of automorphic loops of order with
trivial center.Comment: 13 pages, amsart; v2: minor changes suggested by referee; to appear
in J. Algebr
On polynomials of small range sum
In order to reprove an old result of R\'edei's on the number of directions
determined by a set of cardinality in , Somlai proved that
the non-constant polynomials over the field whose range sums are
equal to are of degree at least . Here we characterise all
of these polynomials having degree exactly , if 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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