Introduction A fundamental objective of cryptography is to enable two persons to communicate over an insecure channel (a public channel such as the internet) in such a way that any other person is unable to recover their message (called the plaintext ) from what is sent in its place over the channel (the ciphertext ). The transformation of the plaintext into the ciphertext is called encryption , or enciphering. Encryption-decryption is the most ancient cryptographic activity (ciphers already existed four centuries b.c.), but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power. The encryption algorithm takes as input the plaintext and an encryption key K E , and it outputs the ciphertext. If the encryption key is secret, then we speak of conventional cryptography , of private key cryptography , or of symmetric cryptography . In practice, the principle of conventional cryptography relies on the sharing of a private key between the sender of a message (often called Alice in cryptography) and its receiver (often called Bob). If, on the contrary, the encryption key is public, then we speak of public key cryptography . Public key cryptography appeared in the literature in the late 1970s.

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Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes

Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.

A first course in coding theory, by Raymond Hill. Pp 251. £12·50. 1986. ISBN 0-19-853803-0 (Oxford University Press)

We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs), where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed–Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al. . Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates.

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Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

The recently developed algebraic attacks apply to all keystream generators whose internal state is updated by a linear transition function, including LFSR-based generators. Here, we describe this type of attacks and we present some open problems related to their complexity. We also investigate the design criteria which may guarantee a high resistance to algebraic attacks for keystream generators based on a linear transition function.

Open problems related to algebraic attacks on stream ciphers

Modern communications systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some applications in telecommunications.

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On some applications of Hadamard matrices

Information sets and partial permutation decoding for codes from finite geometries

We determine to what extent permutation decoding can be used for the codes from desarguesian projective and affine planes. We define the notion of s-PD-sets to correct s errors, and construct some specific small sets for s = 2 and 3 for desarguesian planes of prime order.

Partial permutation decoding for codes from finite planes

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q3, q2, q2−1/q−1) designs without symmetric (q, q)-subnets.

On symmetric nets and generalized Hadamard matrices from affine designs

An important characteristic of an affine space is that each of its hyperplanes is itself an affine space of one dimension lower. This property plays an essential part in the analysis of the structure of affine spaces. The purpose of this paper is to develop an analogous structure theory for general affine designs. This enables the introduction of a concept of dimension for affine designs and also yields techniques for constructing affine designs from smaller ones. The initial sections introduce the required basic results and terminology. In w is proved the main decomposition theorem which demonstrates how an affine design may be constructed with a given decomposition into sets of smaller affine designs. The results of this paper form part of my doctoral thesis at the University of London. To my supervisor Professor D.R. Hughes, I am indebted for invaluable assistance and guidance.

http://link.springer.com/content/pdf/10.1007/BF01110993.pdf

On the Structure of Affine Designs

We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529-538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p^m where p is a prime, and m>=1. This gives words of weight 2p^m+1-p^m-1p-1 in the dual of the p-ary code of the desarguesian plane of order p^m, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of Andre planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmaros and Mazzocca [Gabor Korchmaros, Francesco Mazzocca, On (q+t)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Philos. Soc. 108 (1990) 445-459].

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V. C. Mavron

Papers

Information sets and partial permutation decoding for codes from finite geometries

Partial permutation decoding for codes from finite planes

On symmetric nets and generalized Hadamard matrices from affine designs

On the Structure of Affine Designs

An upper bound for the minimum weight of the dual codes of desarguesian planes