Discrete rings

Formal Groups over Discrete Rings

In this note we develop a theory of formal schemes and groups over arbitrary commutative rings which coincides with that of [5] if the base ring is a field, and generalizes that of [2]. We assume always our base ring is discrete and treat a formal scheme (resp. group) G, with two principal tools: A topology on the affine algebra (9(G) allows us to form its continuous linear dual B(G), the coalg...

Computing Discrete Logarithms with Quadratic Number Rings

At present, there are two competing index calculus variants for computing discrete logarithms in (Z/pZ)* in practice. The purpose of this paper is to summarize the recent practical experience with a generalized implementation covering both a variant of the Number Field Sieve and the Gaussian integer method. By this implementation we set a record with p consisting of 85 decimal digits. With rega...

On Ramified Complete Discrete Valuation Rings

Gersten’s conjecture for commutative discrete valuation rings

The purpose of this article is to prove that Gersten’s conjecture for a commutative discrete valuation ring is true. Combining with the result of [GL87], we learn that Gersten’s conjecture is true if the ring is a commutative regular local, smooth over a commutative discrete valuation ring.

Trapdooring Discrete Logarithms on Elliptic Curves over Rings

This paper introduces three new probabilistic encryption schemes using elliptic curves over rings. The cryptosystems are based on three specific trapdoor mechanisms allowing the recipient to recover discrete logarithms on different types of curves. The first scheme is an embodiment of Naccache and Stern’s cryptosystem and realizes a discrete log encryption as originally wanted in [23] by Vansto...