Abstract. In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller's algorithm can be up to g times shorter than for the Tate pairing, with g the genus of the curve, and the pairing is automatically reduced, i.e. no final exponentiation is needed.
We present a fast addition algorithm in the Jacobian of a genus 3 non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and char(k) > 5, the computational cost for addition is 148M + 15SQ + 2I and 165M + 20SQ + 2I for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd q, we also show that the set of rational points of a nonhyperelliptic curve of genus 3 can not be an arc.
Abstract. In this paper we present a fast addition algorithm in the Jacobian of a Picard curve over a finite field Fq of characteristic different from 3. This algorithm has a nice geometric interpretation, comparable to the classic "chord and tangent" law for the elliptic curves. Computational cost for addition is 144M +12SQ+2I and 158M +16SQ+2I for doubling.
This short note investigates the effects of using expansions to the base of −2. The main applications we have in mind are cryptographic protocols, where the crucial operation is computation of scalar multiples. For the recently proposed groups arising from Picard curves this leads to a saving of at least 7% for the computation of an m-fold. For more general non-hyperelliptic genus 3 curves we expect a larger speed-up.
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