In this article we investigate the Bernoulli numbers B̂n associated to the formal group laws whose canonical invariant differentials generate the Lucas sequences {Un} and {Vn}. We give explicit expressions for these numbers and prove analogues of Kummer congruences for them.

For the finite field IFq of q elements (q odd) and a quadratic nonresidue α ∈ IFq, we define the distance function δ ( u+ v √ α, x+ y √ α ) = (u− x)2 − α(v − y)2 vy on the upper half plane Hq = {x + y √ α | x ∈ IFq, y ∈ IFq} ⊆ IFq2 . For two sets E ,F ⊂ Hq with #E = E, #F = F and a non-trivial additive character ψ on IFq, we give the following estimate ∣∣∣∣∣ ∑ w∈E,z∈F ψ(δ(w, z)) ∣∣∣∣∣ ≤ min {√ ...

We investigate the possibility of disposing of interaction between Prover and Veriier in a zero-knowledge proof if they share beforehand a short random string. Without any assumption, we prove that non-interactive zero-knowledge proofs exist for some number theoretic languages for which no eecient algorithm is known. If deciding quadratic residuosity (modulo composite integers whose factorizati...

Steven Finch December 4, 2013 Fifty years separate two computations: the mean value of a certain function () over primes , mentioned in [1], and the mean value of () over all positive integers . We anticipate that the overlap between number theory and probability will only deepen with time. 0.1. Quadratic. Let () be the smallest positive quadratic nonresidue modulo   2. Erdős [2] prov...

We show how the theory of real quadratic congruence function fields can be used to produce a secure key distribution protocol. The technique is similar to that advocated by Diffie and Hellman in 1976, but instead of making use of a group for its underlying structure, makes use of a structure which is “almost” a group. The method is an extension of the recent ideas of Scheidler, Buchmann and Wil...

Generally, communication network systems provide data transfer services for customers. Further requirements such as performance, security, and reliability characterize the quality of the transfer service. Network and information security refer to confidence that information and services existing on a network cannot be accessed by unauthorized users (eavesdropper). However, these service require...

In this work, we show how to construct IBE schemes that are secure against a bounded number of collusions, starting with underlying PKE schemes which possess linear homomorphisms over their keys. In particular, this enables us to exhibit a new (bounded-collusion) IBE construction based on the quadratic residuosity assumption, without any need to assume the existence of random oracles. The new I...