Nigel Boston

Some cases of the Fontaine-Mazur conjecture

Abstract We prove more special cases of the Fontaine–Mazur conjecture regarding p -adic Galois representations unramified at p , and we present evidence for and consequences of a generalization of it.

The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2^n-1 with two zeros @a^a,@a^b. Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros @a^3,@a^5 and @a,@a^1^1. These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n.

Genus Two Hyperelliptic Curve Coprocessor

Hyperelliptic curve cryptography with genus larger than one has not been seriously considered for cryptographic purposes because many existing implementations are significantly slower than elliptic curve versions with the same level of security. In this paper, the first ever complete hardware implementation of a hyperelliptic curve coprocessor is described. This coprocessor is designed for genus two curves over F2113. Additionally, a modification to the Extended Euclidean Algorithm is presented for the GCD calculation required by Cantors algorithm. On average, this new method computes the GCD in one-fourth the time required by the Extended Euclidean Algorithm.

The proportion of fixed-point-free elements of a transitive permutation group

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1 n is such a bound. Lenstra’s problem arose from his work on the number field sieve [2]. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion:

A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete d × N matrix is finitely rank-r completable if there are at most finitely many rank-r matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a characterization of finitely completable observation sets. We use this characterization to derive sufficient deterministic sampling conditions for unique completability. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if O(max{r,logd}) entries per column are observed.

A probabilistic generalization of the Riemann zeta function

There is an old problem that asks for the probability that two integers chosen at random are relatively prime. The informal solution goes as follows.

Near-Optimal Codebook Constructions for Limited Feedback Beamforming in Correlated MIMO Channels with Few Antennas

Transmit beamforming with receive combining is a low-complexity solution that achieves the full diversity afforded by a multi-antenna channel. Building on our recent result which shows that even channel statistics are sufficient to achieve perfect feedback performance (in the limit of antenna dimensions) with beamforming and combining, we propose near-optimal codebook designs for correlated channels with a focus on few antennas at the transmitter and the receiver. In the process, we refine the answer to the question: When are channel statistics sufficient to achieve near perfect feedback performance? We show that the condition number of the transmit and receive covariance matrices hold the key to this question. We partition the transmit and receive covariance spaces into 4 regions based on well and ill-conditioning of the covariance matrices and show that the number of bits required for near perfect feedback performance is dependent on the condition numbers of these matrices

A note on Beauville p-groups

We examine which p-groups of order ⩽p 6 are Beauville. We completely classify them for groups of order ⩽p 4. We also show that the proportion of 2-generated groups of order p 5 that are Beauville tends to 1 as p tends to infinity; this is not true, however, for groups of order p 6. For each prime p we determine the smallest nonabelian Beauville p-group.

Fusion of Summation Invariants in 3D Human Face Recognition

A novel family of 2D and 3D geometrically invariant features, called summation invariants is proposed for the recognition of the 3D surface of human faces. Focusing on a rectangular region surrounding the nose of a 3D facial depth map, a subset of the so called semi-local summation invariant features is extracted. Then the similarity between a pair of 3D facial depth maps is computed to determine whether they belong to the same person. Out of many possible combinations of these set of features, we select, through careful experimentation, a subset of features that yields best combined performance. Tested with the 3D facial data from the on-going Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit significant performance improvement over the baseline algorithm distributed with the datase

Optimal Linear Combination of Facial Regions for Improving Identification Performance

This paper presents a novel 3D multiregion face recognition algorithm that consists of new geometric summation invariant features and an optimal linear feature fusion method. A summation invariant, which captures local characteristics of a facial surface, is extracted from multiple subregions of a 3D range image as the discriminative features. Similarity scores between two range images are calculated from the selected subregions. A novel fusion method that is based on a linear discriminant analysis is developed to maximize the verification rate by a weighted combination of these similarity scores. Experiments on the Face Recognition Grand Challenge V2.0 dataset show that this new algorithm improves the recognition performance significantly in the presence of facial expressions.