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

All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [n,k] code. There is also a recent result on the complexity of computing the covering radius.

Covering radius---Survey and recent results

A general explicit construction of bent functions is described, which unifies well known constructions due to Maiorana-McFarland and Dillon as two opposite extremal cases. Within this framework we also find new ways to generate bent functions. Then it is shown how the constructed bent functions can be modified in order to obtain highly nonlinear balanced Boolean functions. Although their nonlinearity is the best known so far, it remains open whether this bound can still be improved.

Construction of bent functions and balanced Boolean functions with high nonlinearity

The covering radius R of a code is the maximal distance of any vector from the code This work gives a number of new results concerning t[n, k] , the minimal covering radius of any binary code of length n and dimension k  For example t[n, 4] and t[n, 5] are determined exactly, and reasonably tight bounds on t[n, k] are obtained for any k when n is large These results are found by using several new constructions for codes with small covering radius One construction, the amalgamated direct sum, involves a quantity called the norm of a code Codes with norm \leq 2 R + 1 are called normal, and may be combined efficiently The paper concludes with a table giving bounds on t[n, k] for n \leq 64 

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On the covering radius of codes

In this correspondence, two new methods for constructing sets of zero-correlation zone (ZCZ) sequences are proposed. These methods are based on perfect sequences and unitary matrices. Compared with previous methods, the proposed methods can generate sets of nonbinary ZCZ sequences having a longer zero-correlation zone. The sequences obtained by these methods are suitable for approximately synchronized code-division multiple-access (AS-CDMA) systems.

A new class of zero-correlation zone sequences

Two interesting cosets of the first-order Reed-Muller code of block length 2^{15} are described. They provide counterexamples to a conjecture on the covering radius.

N. Patterson

Papers

The covering radius of the (2^{15}, 16) Reed-Muller code is at least 16276