Orientable sequences, periodic sequences in which any n-tuple appears at most once in either direction, were introduced in the early 1990s for use in certain position location applications; constructions and upper bounds on the period for the binary case were published by Dai et al. More recent work has focussed on k-ary sequences for arbitrary k>2; one method of construction involves negative orientable sequences, in which an n-tuple appears at most once in either the sequence or the negative of its reverse. In this paper we show how additional n-tuples can be added to one previously described approach for generating negative orientable sequences, resulting in new sequences with asymptotically optimal period. These sequences can in turn be used to generate orientable sequences, again with asymptotically optimal period.

Negative avoiding sequences of span n are periodic sequences of elements from ℤ_k for some k with the property that no n-tuple occurs more than once in a period and if an n-tuple does occur then its negative does not. They are a special type of cut-down de Bruijn sequence with potential position-location applications. We establish a simple upper bound on the period of such a sequence, and refer to sequences meeting this bound as maximal negative avoiding sequences. We then go on to demonstrate the existence of maximal negative avoiding sequences for every k≥3 and every n≥2.

Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any tuple of n elements or its reverse occurs at most once as a contiguous subsequence (i.e. a substring or factor) in a period. They were introduced in the early 1990s in the context of possible applications in position sensing. Bounds on the period of such sequences and a range of methods of construction have been devised, although apart from very small cases a significant gap remains between the largest known period for such a sequence and the best known upper bound. In this paper we first give improved upper bounds on the period of such sequences. We then give a new general method of construction for orientable sequences involving subgraphs of the de Bruijn graph with special properties, and describe two different approaches for generating such subgraphs. This enables us to construct orientable sequences with periods meeting the improved upper bounds when n is 2 or 3, as well as n=4 and k odd. For 4≤ n≤ 8 , in some cases the sequences produced by the methods described have periods larger than for any previously known sequences.

In 2012, Ruskey, Sawada and Williams showed that a binary cut-down de Bruijn sequence can be constructed containing all n-tuples of Hamming weight either r or r+1 for every possible r, and no others. In this paper we examine extensions of this result that require choosing a weight-like function applying to n-tuples with symbols taken from an alphabet of arbitrary (finite) size. We consider three possible such weight functions, and in each case establish an analogous result to that of Ruskey et al.

Analogously to de Bruijn sequences, Orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., recursive methods of construction were described for orientable sequences over arbitrary finite alphabets, requiring 'starter sequences' with special properties. Some of these methods required as input special orientable sequences, i.e. orientable sequences which were simultaneously negative orientable. We exhibit methods for constructing special orientable sequences with properties appropriate for use in two of the recursive methods of Alhakim et al. As a result we are able to show how to construct special orientable sequences for arbitrary sizes of alphabet (larger than a small lower bound) and for all window sizes. These sequences have periods asymptotic to the optimal as the alphabet size increases.

Negative orientable sequences, i.e. periodic sequences with elements from a finite alphabet of size at least three in which an n-tuple or the negative of its reverse appears at most once in a period of the sequence, were introduced by Alhakim et al. in 2024. The main goal in defining them was as a means of generating orientable sequences, which have automatic position location applications, although they are potentially of interest in their own right. In this paper we develop new upper bounds on the period of negative orientable sequences, which are significantly sharper then the previous known bound for n>2. The approach used to develop the new bounds involves examining the nodes in the subgraph of the de Bruijn graph corresponding to a negative orientable sequence, and to consider the implications of the fact that the in-degree of every vertex in this subgraph must equal the out-degree. However, despite improving the bounds, a gap remains between the largest known period for a negative orientable sequence and the corresponding bounds for every n>2.

Analogously to de Bruijn sequences, orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., a range of methods of construction were described for orientable sequences over arbitrary finite alphabets; some of these methods involve using negative orientable sequences as a building block. In this paper we describe three techniques for generating such negative orientable sequences, as well as upper bounds on their period. We then go on to show how these negative orientable sequences can be used to generate orientable sequences with period close to the maximum possible for every non-binary alphabet size and for every tuple length. In doing so we use two closely related approaches described by Alhakim et al.

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either direction. In particular we establish how two variants of a generalised Lempel homomorphism can be used to recursively construct such sequences, generalising previous work on the binary case. We also derive an upper bound on the period of an orientable sequence.

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any n-tuple occurs at most once in a period in either direction. As has been previously described, such sequences have potential applications in automatic position-location systems, where the sequence is encoded onto a surface and a reader needs only examine n consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai et al.) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard ‘infinite periodic’ case, and also the aperiodic, finite sequence, case. Both the new methods build on the Lempel homomorphism, first introduced as a means of recursively generating de Bruijn sequences.

We present a novel approach that generalizes the well- known quantum SWAP gate to higher dimensions and construct a regular quantum gate composed entirely in terms of the generalized CNOT gate that cyclically permutes the states of d qudits for d prime. We also investigate the case for d other than prime. A key feature of the construction design relates to the periodicity evaluation for a family of linear recurrences which we achieve by exploiting generating functions and their factorization over the complex reals.

The SWAP gate plays a central role in network designs for qubit quantum computation. However, there has been a view to generalize qubit quantum computing to higher dimensional quantum systems. In this paper we construct a generalized SWAP gate using only instances of the generalized controlled-NOT gate to cyclically permute the states of d qudits for d prime.

In this paper, we consider authentication codes where the adversary has access to a verification oracle. We formally study two attack games: offline attack and online attack. In an offline impersonation attack with verification query of order i, the adversary launches its attack through two stages. In the first stage — the query stage — the adversary can adaptively choose i distinct messages to query the verification oracle. The verification oracle will answer whether these queried messages are valid or invalid under the secret encoding rule agreed by the transmitter and the receiver. In the later stage — the spoofing stage — the adversary creates a fraudulent message which is different from all its queried messages and sends this message to the receiver. The adversary wins if the receiver accepts the fraudulent message as a valid message. In an online impersonation attack with verification query of order i, the adversary has i + 1 chances to query the verification oracle and wins as soon as one of the queries is a valid message. We make use of strategy trees, which allow optimal strategies in both attack games to be identified, to establish a number of relationships between the value of the two games. This allows us to formally prove a relationship between the value of the game when the adversary has i queries, and the one in which he does not have any. The relationship, though widely believed to be true, was only recently proved for computationally secure systems. Our result complements this latter work for the information theoretic setting.

When an organisation chooses a system to make regular broadcasts to a changing user base, there is an inevitable trade off between the number of keys a user must store and the number of keys used in the broadcast. The Complete Subtree and Subset Difference Revocation Schemes were proposed as efficient solutions to this problem. However, all measurements of the broadcast size have been in terms of upper bounds on the worst-case. Also, the bound on the latter scheme is only relevant for small numbers of revoked users, despite the fact that both schemes allow any number of such users. Since the broadcast size can be critical for limited memory devices, we aid comparative analysis of these important techniques by establishing the worst-case broadcast size for both revocation schemes.

We give a quantum gate construction - composed entirely from incidents of the CNOT gate - that generalises the qubit SWAP gate to higher dimensions. This new construction is more regular than and is an improvement on the WilNOT quantum gate construction.

The qubit SWAP gate has been shown to be an integral component of quantum circuitry design. It permutes the states of two qubits and allows for the storage quantum information, teleportation of atomic or ionic states, and is a fundamental element in the circuit implementation of Shor's algorithm. We consider the problem of generalising the SWAP gate beyond the qubit setting. We show that quantum circuit architectures completely described by instances of the CNOT gate can not implement a transposition of a pair of qudits for dimensions $d \equiv 3 (mod 4)$. The task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation.

We derive a basis for the vector space of bounded operators acting on a $d$-dimensional system Hilbert space $C^d$. In the context of quantum computation the basis elements are identified as the generalised Pauli matrices - the error generators. As an application, we show how such matrices are used in the teleportation a single qudit.

A general method for deriving an identity-based public key cryptosystem from a one-way function is described. We construct both ID-based signature schemes and ID-based encryption schemes. We use a general technique which is applied to multi-signature versions of the one-time signature scheme of Lamport and to a public key encryption scheme based on a symmetric block cipher which we present. We make use of one-way functions and block designs with properties related to cover-free families to optimise the efficiency of our schemes.