A geometric view of cryptographic equation solving

This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algorithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both algorithms can be viewed as being equivalent to the problem of finding a matrix of low rank in the linear span of a collection of matrices, a problem sometimes known as the MinRank problem. Furthermore, we generalise the XL algorithm to a geometrically invariant algorithm, which we term the GeometricXL algorithm. The GeometricXL algorithm is a technique which can solve certain equation systems that are not easily soluble by the XL algorithm or by Groebner basis methods

Recursive n-gram hashing is pairwise independent, at best

Many applications use sequences of n consecutive symbols (n-grams). Hashing these n-grams can be a performance bottleneck. For more speed, recursive hash families compute hash values by updating previous values. We prove that recursive hash families cannot be more than pairwise independent. While hashing by irreducible polynomials is pairwise independent, our implementations either run in time O(n) or use an exponential amount of memory. As a more scalable alternative, we make hashing by cyclic polynomials pairwise independent by ignoring n-1 bits. Experimentally, we show that hashing by cyclic polynomials is is twice as fast as hashing by irreducible polynomials. We also show that randomized Karp-Rabin hash families are not pairwise independent.Comment: See software at https://github.com/lemire/rollinghashcp

XL-NBT: A Cross-lingual Neural Belief Tracking Framework

Task-oriented dialog systems are becoming pervasive, and many companies heavily rely on them to complement human agents for customer service in call centers. With globalization, the need for providing cross-lingual customer support becomes more urgent than ever. However, cross-lingual support poses great challenges---it requires a large amount of additional annotated data from native speakers. In order to bypass the expensive human annotation and achieve the first step towards the ultimate goal of building a universal dialog system, we set out to build a cross-lingual state tracking framework. Specifically, we assume that there exists a source language with dialog belief tracking annotations while the target languages have no annotated dialog data of any form. Then, we pre-train a state tracker for the source language as a teacher, which is able to exploit easy-to-access parallel data. We then distill and transfer its own knowledge to the student state tracker in target languages. We specifically discuss two types of common parallel resources: bilingual corpus and bilingual dictionary, and design different transfer learning strategies accordingly. Experimentally, we successfully use English state tracker as the teacher to transfer its knowledge to both Italian and German trackers and achieve promising results.Comment: 13 pages, 5 figures, 3 tables, accepted to EMNLP 2018 conferenc

Density Evolution for Asymmetric Memoryless Channels

Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor

Search Problems in Vector Spaces

We consider the following $q$-analog of the basic combinatorial search problem: let $q$ be a prime power and \GF(q) the finite field of $q$ elements. Let $V$ denote an $n$-dimensional vector space over \GF(q) and let $\mathbf{v}$ be an unknown 1-dimensional subspace of $V$. We will be interested in determining the minimum number of queries that is needed to find $\mathbf{v}$ provided all queries are subspaces of $V$ and the answer to a query $U$ is YES if $\mathbf{v} \leqslant U$ and NO if $\mathbf{v} \not\leqslant U$. This number will be denoted by $A(n,q)$ in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and $M(n,q)$ in the non-adaptive case (when all queries must be made in advance). In the case $n=3$ we prove $2q-1=A(3,q)<M(3,q)$ if $q$ is large enough. While for general values of $n$ and $q$ we establish the bounds $$n\log q \le A(n,q) \le (1+o(1))nq$$ and $$(1-o(1))nq \le M(n,q) \le 2nq,$$ provided $q$ tends to infinity