Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over *

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Abstract. We present an algorithm which speeds scalar multiplication on a general elliptic curve by an estimated 3.8% to 8.5% over the best known general methods when using affine coordinates. This is achieved by eliminating a field multiplication when we compute 2P +Q from given points P , Q on the curve. We give applications to simultaneous multiple scalar multiplication and to the Elliptic Curve Method of factorization. We show how this improvement together with another idea can speed the computation of the Weil and Tate pairings by up to 7.8%.

Let K be an algebraic function field of characteristic 2 with constant field C K . Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over C K and x algebraic over C(u) and such that K = C K (u, x). Then Hilbert's Tenth Problem over K is undecidable. Together with Shlapentokh's result for odd characteristic this implies that Hilbert's Tenth Problem for any such field K of finite characteristic is undecidable. In particular, Hilbert's Tenth Problem for any algebraic function field with finite constant field is undecidable.