Smoothing of binary codes, uniform distributions, and applications

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by R{\'e}nyi divergence of order $\alpha \in [1,\infty]$. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel

Parameterized Intractability of Even Set and Shortest Vector Problem

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W[1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W[1]-hard to approximate (under randomized reductions) to some constant factor

Trellis Decoding And Applications For Quantum Error Correction

Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost. Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products. Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.Ph.D

On Boolean functions, symmetric cryptography and algebraic coding theory

In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums. We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given

Cryptography based on the Hardness of Decoding

This thesis provides progress in the fields of for lattice and coding based cryptography. The first contribution consists of constructions of IND-CCA2 secure public key cryptosystems from both the McEliece and the low noise learning parity with noise assumption. The second contribution is a novel instantiation of the lattice-based learning with errors problem which uses uniform errors