Robust Multiplication-Based Tests for Reed-Muller Codes

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime. Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial. We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,...P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014). One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

Quantum Locally Testable Codes

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, together with some basic results using those definitions. The most crucial parameter of such codes is their soundness, $R(\delta)$, namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code ($\delta$, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This phenomenon, which can be attributed to monogamy of entanglement, stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on poor small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs.Comment: Some of the results presented here appeared in an initial form in our quant-ph submission arXiv:1301.3407. This is a much extended and improved version. 30 pages, no figure

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Mixed radix design flow for security applications

The purpose of secure devices, such as smartcards, is to protect sensitive information against software and hardware attacks. Implementation of the appropriate protection techniques often implies non-standard methods that are not supported by the conventional design tools. In the recent decade the designers of secure devices have been working hard on customising the workflow. The presented research aims at collecting the up-to-date experiences in this area and create a generic approach to the secure design flow that can be used as guidance by engineers. Well-known countermeasures to hardware attacks imply the use of specific signal encodings. Therefore, multi-valued logic has been considered as a primary aspect of the secure design. The choice of radix is crucial for multi-valued logic synthesis. Practical examples reveal that it is not always possible to find the optimal radix when taking into account actual physical parameters of multi-valued operations. In other words, each radix has its advantages and disadvantages. Our proposal is to synthesise logic in different radices, so it could benefit from their combination. With respect to the design opportunities of the existing tools and the possibilities of developing new tools that would fill the gaps in the flow, two distinct design approaches have been formed: conversion driven design and pre-synthesis. The conversion driven design approach takes the outputs of mature and time-proven electronic design automation (EDA) synthesis tools to generate mixed radix datapath circuits in an endeavour to investigate the added relative advantages or disadvantages. An algorithm underpinning the approach is presented and formally described together with secure gate-level implementations. The obtained results are reported showing an increase in power consumption, thus giving further motivation for the second approach. The pre-synthesis approach is aimed at improving the efficiency by using multivalued logic synthesis techniques to produce an abstract component-level circuit before mapping it into technology libary. Reed-Muller expansions over Galois field arithmetic have been chosen as a theoretical foundation for this approach. In order to enable the combination of radices at the mathematical level, the multi-valued Reed-Muller expansions have been developed into mixed radix Reed-Muller expansions. The goals of the work is to estimate the potential of the new approach and to analyse its impact on circuit parameters down to the level of physical gates. The benchmark results show the approach extends the search space for optimisation and provides information on how the implemented functions are related to different radices. The theory of two-level radix models and corresponding computation methods are the primary theoretical contribution. It has been implemented in RMMixed tool and interfaced to the standard EDA tools to form a complete security-aware design flow.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

D2.1 - Report on Selected TRNG and PUF Principles

This report represents the final version of Deliverable 2.1 of the HECTOR work package WP2. It is a result of discussions and work on Task 2.1 of all HECTOR partners involved in WP2. The aim of the Deliverable 2.1 is to select principles of random number generators (RNGs) and physical unclonable functions (PUFs) that fulfill strict technology, design and security criteria. For example, the selected RNGs must be suitable for implementation in logic devices according to the German AIS20/31 standard. Correspondingly, the selected PUFs must be suitable for applying similar security approach. A standard PUF evaluation approach does not exist, yet, but it should be proposed in the framework of the project. Selected RNGs and PUFs should be then thoroughly evaluated from the point of view of security and the most suitable principles should be implemented in logic devices, such as Field Programmable Logic Arrays (FPGAs) and Application Specific Integrated Circuits (ASICs) during the next phases of the project