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Donald A. Preece gave two talks in 2013, in which he introduced the notion of tredoku patterns. These are certain configurations of diamond-shaped tiles, inspired by the sukodu-like puzzle that appears in newspapers. Very sadly, Prof. Preece died in January 2014, before he was able to complete his work on this problem. This paper reconstructs his work, and settles (in the affirmative) the conjectures he proposed., Comment: 36 pages, 31 figures. Conclusion revised, some minor typos corrected, and small changes to the introduction

We define and investigate the property of being `exponent-critical' for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. We explore properties of exponent-critical groups and give a characterization of such groups. This characterization generalises a classical result of Miller and Moreno on minimal non-abelian groups; interesting families of $p$-groups appear., Comment: 18 pages. Minor revisions to introduction and abstract

Let $F_n$ be a free group of rank $n$, with free generating set $X$. A subset $D$ of $F_n$ is a \emph{Distinct Difference Configuration} if the differences $g^{-1}h$ are distinct, where $g$ and $h$ range over all (ordered) pairs of distinct elements of $D$. The subset $D$ has diameter at most $d$ if these differences all have length at most $d$. When $n$ is fixed and $d$ is large, the paper shows that the largest distinct difference configuration in $F_n$ of diameter at most $d$ has size approximately $(2n-1)^{d/3}$., Comment: 14 pages

Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So \[ ||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}. \] A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is \emph{$(s,k)$-clash-free} if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein in 1964., Comment: 17 pages. Theorems etc renumbered

The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and K\"otter in 2010 to model data transmission using random linear network coding. The input and output of the channel are $n\times m$ matrices over a finite field $\mathbb{F}_q$. On input the matrix $X$, the channel outputs $Y=A(X+W)$ where $A$ is a uniformly chosen $n\times n$ invertible matrix over $\mathbb{F}_q$ and where $W$ is a uniformly chosen $n\times m$ matrix over $\mathbb{F}_q$ of rank $t$. Silva \emph{et al} considered the case when $2n\leq m$. They determined the asymptotic capacity of the AMMC when $t$, $n$ and $m$ are fixed and $q\rightarrow\infty$. They also determined the leading term of the capacity when $q$ is fixed, and $t$, $n$ and $m$ grow linearly. We generalise these results, showing that the condition $2n\geq m$ can be removed. (Our formula for the capacity falls into two cases, one of which generalises the $2n\geq m$ case.) We also improve the error term in the case when $q$ is fixed., Comment: 32 pages, 1 figure. Typos corrected, minor changes to proofs for clarity, more discussion added

Let $n$ be a fixed integer with $n\geq 2$. For $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a cycle. So $||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}$. For positive integers $s$ and $k$, the permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb{Z}_n$ is \emph{$(s,k)$-clash-free} if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n

Let $p$ be a fixed prime. We show that the number of isomorphism classes of finite rings of order $p^n$ is $p^\alpha$, where $\alpha=\frac{4}{27}n^3+O(n^{5/2})$. This result was stated (with a weaker error term) by Kruse and Price in 1969; a problem with their proof was pointed out by Knopfmacher in 1973. We also show that the number of isomorphism classes of finite commutative rings of order $p^n$ is $p^\beta$, where $\beta=\frac{2}{27}n^3+O(n^{5/2})$. This result was stated (again with a weaker error term) by Poonen in 2008, with a proof that relies on the problematic step in Kruse and Price's argument., Comment: 31 pages. Change of title, revised appendix, and various other small changes since previous version

Let $n$ and $\ell$ be positive integers. Recent papers by Kreher, Stinson and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system, directed triple system or Mendelsohn triple system) on $n$ points so that no block occurs in a segment of $\ell$ consecutive entries (thus the ordering is locally block-avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that $n$ is sufficiently large when compared to $\ell$. This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, for example, orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct Steiner triple systems and quadruple systems where $\ell$ can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalise the Stinson--Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson and Veitch were originally inspired by results of Alspach, Kreher and Pastine, who (motivated by zero-sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach~\emph{et al.}\ show that, when the system contains at most $k$ pairwise disjoint blocks, an ordering exists when the number of points is more than $15k-5$. By making use of a greedy approach, the paper improves this bound to $9k+O(k^{2/3})$., Comment: 38 pages. Typo in the statement of Theorem 17 corrected, and other minor changes made

An inglenook puzzle is a classic shunting (switching) puzzle often found on model railway layouts. A collection of wagons sits in a fan of sidings with a limited length headshunt (lead track). The aim of the puzzle is to rearrange the wagons into a desired order (often a randomly chosen order). This article answers the question: When can you be sure this can always be done? The problem of finding a solution in a minimum number of moves is also addressed., Comment: 23 pages, 4 figures. Minor typos in previous version corrected

Let $\pi$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,\pi(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(\pi)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(\pi)$ when $n\rightarrow\infty$ and $\pi$ is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when $d$ is fixed and $n\rightarrow\infty$, the probability that $d(\pi)\geq d+2$ tends to $e^{-d^2 - d}$. The minimum jump $mj(\pi)$ of $\pi$, defined by $mj(\pi)=\min_{1\leq i\leq n-1} |\pi(i+1)-\pi(i)|$, is another natural measure in this context. The paper computes the asymptotic moments of $mj(\pi)$, and the asymptotic probability that $mj(\pi)\geq d+1$ for any constant $d$., Comment: 20 pages, 3 figures. Minor changes throughout

In the classical model for (information theoretically secure) Private Information Retrieval (PIR), a user wishes to retrieve one bit of a database that is stored on a set of $n$ servers, in such a way that no individual server gains information about which bit the user is interested in. The aim is to design schemes that minimise communication between the user and the servers. More recently, there have been moves to consider more realistic models where the total storage of the set of servers, or the per server storage, should be minimised (possibly using techniques from distributed storage), and where the database is divided into $R$-bit records with $R>1$, and the user wishes to retrieve one record rather than one bit. When $R$ is large, downloads from the servers to the user dominate the communication complexity and so the aim is to minimise the total number of downloaded bits. Shah, Rashmi and Ramchandran show that at least $R+1$ bits must be downloaded from servers in the worst case, and provide PIR schemes meeting this bound. Sun and Jafar determine the best asymptotic download cost of a PIR scheme (as $R\rightarrow\infty$), where this cost is defined as the ratio of the message length $R$ and the total number of bits downloaded. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al. to the case when the number $n$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes, including explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average., Comment: 30 pages. Minor updates and corrections throughout, with updated bibliography

There has been much recent interest in Private information Retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient protocols. In this paper we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/\omegaR)$ of the bits of the database; here $\omegaR$ is a fixed rational number with $\omegaR > 1$. We study the maximum PIR rate of a PIR array code with the $k$-PIR property (which enables a $k$-server PIR protocol to be emulated on the $m$ servers), where the PIR rate is defined to be $k/m$. We present upper bounds on the achievable rate, some constructions, and ideas how to obtain PIR array codes with the highest possible PIR rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest PIR rate is obtained when $1 < \omegaR \leq 2$., Comment: A conference version for arXiv:1607.00235

The Algebraic Eraser has been gaining prominence as SecureRF, the company commercializing the algorithm, increases its marketing reach. The scheme is claimed to be well-suited to IoT applications but a lack of detail in available documentation has hampered peer-review. Recently more details of the system have emerged after a tag authentication protocol built using the Algebraic Eraser was proposed for standardization in ISO/IEC SC31 and SecureRF provided an open public description of the protocol. In this paper we describe a range of attacks on this protocol that include very efficient and practical tag impersonation as well as partial, and total, tag secret key recovery. Most of these results have been practically verified, they contrast with the 80-bit security that is claimed for the protocol, and they emphasize the importance of independent public review for any cryptographic proposal., Comment: 21 pages. Minor changes. Final version accepted for ACNS 2016

In 2010, Silva, Kschischang and K\"otter studied certain classes of finite field matrix channels in order to model random linear network coding where exactly $t$ random errors are introduced. In this paper we consider a generalisation of these matrix channels where the number of errors is not required to be constant, indeed the number of errors may follow any distribution. We show that a capacity-achieving input distribution can always be taken to have a very restricted form (the distribution should be uniform given the rank of the input matrix). This result complements, and is inspired by, a paper of Nobrega, Silva and Uchoa-Filho, that establishes a similar result for a class of matrix channels that model network coding with link erasures. Our result shows that the capacity of our channels can be expressed as a maximisation over probability distributions on the set of possible ranks of input matrices: a set of linear rather than exponential size., Comment: 21 pages. A significant revision: the main counting arguments shortened; computational results added; other minor revisions throughout

Anshel, Anshel, Goldfeld and Lemieaux introduced the Colored Burau Key Agreement Protocol (CBKAP) as the concrete instantiation of their Algebraic Eraser scheme. This scheme, based on techniques from permutation groups, matrix groups and braid groups, is designed for lightweight environments such as RFID tags and other IoT applications. It is proposed as an underlying technology for ISO/IEC 29167-20. SecureRF, the company owning the trademark Algebraic Eraser, has presented the scheme to the IRTF with a view towards standardisation. We present a novel cryptanalysis of this scheme. For parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key using less than 8 CPU hours, and less than 64MB of memory., Comment: 15 pages. Updated references, with brief comments added. Minor typos corrected. Final version, accepted for CRYPTO 2016

K.A.S. Immink and J.H. Weber recently defined and studied a channel with both gain and offset mismatch, modelling the behaviour of charge-leakage in flash memory. They proposed a decoding measure for this channel based on minimising Pearson distance (a notion from cluster analysis). The paper derives a formula for maximum likelihood decoding for this channel, and also defines and justifies a notion of minimum distance of a code in this context., Comment: 17 pages, 7 figures

The Pearson distance has been advocated for improving the error performance of noisy channels with unknown gain and offset. The Pearson distance can only fruitfully be used for sets of $q$-ary codewords, called Pearson codes, that satisfy specific properties. We will analyze constructions and properties of optimal Pearson codes. We will compare the redundancy of optimal Pearson codes with the redundancy of prior art $T$-constrained codes, which consist of $q$-ary sequences in which $T$ pre-determined reference symbols appear at least once. In particular, it will be shown that for $q\le 3$ the $2$-constrained codes are optimal Pearson codes, while for $q\ge 4$ these codes are not optimal., Comment: 17 pages. Minor revisions and corrections since previous version. Author biographies added. To appear in IEEE Trans. Inform. Theory

Separable codes were defined by Cheng and Miao in 2011, motivated by applications to the identification of pirates in a multimedia setting. Combinatorially, $\overline{t}$-separable codes lie somewhere between $t$-frameproof and $(t-1)$-frameproof codes: all $t$-frameproof codes are $\overline{t}$-separable, and all $\overline{t}$-separable codes are $(t-1)$-frameproof. Results for frameproof codes show that (when $q$ is large) there are $q$-ary $\overline{t}$-separable codes of length $n$ with approximately $q^{\lceil n/t\rceil}$ codewords, and that no $q$-ary $\overline{t}$-separable codes of length $n$ can have more than approximately $q^{\lceil n/(t-1)\rceil}$ codewords. The paper provides improved probabilistic existence results for $\overline{t}$-separable codes when $t\geq 3$. More precisely, for all $t\geq 3$ and all $n\geq 3$, there exists a constant $\kappa$ (depending only on $t$ and $n$) such that there exists a $q$-ary $\overline{t}$-separable code of length $n$ with at least $\kappa q^{n/(t-1)}$ codewords for all sufficiently large integers $q$. This shows, in particular, that the upper bound (derived from the bound on $(t-1)$-frameproof codes) on the number of codewords in a $\overline{t}$-separable code is realistic. The results above are more surprising after examining the situation when $t=2$. Results due to Gao and Ge show that a $q$-ary $\overline{2}$-separable code of length $n$ can contain at most $\frac{3}{2}q^{2\lceil n/3\rceil}-\frac{1}{2}q^{\lceil n/3\rceil}$ codewords, and that codes with at least $\kappa q^{2n/3}$ codewords exist. So optimal $\overline{2}$-separable codes behave neither like $2$-frameproof nor $1$-frameproof codes. Also, the Gao--Ge bound is strengthened to show that a $q$-ary $\overline{2}$-separable code of length $n$ can have at most \[ q^{\lceil 2n/3\rceil}+\tfrac{1}{2}q^{\lfloor n/3\rfloor}(q^{\lfloor n/3\rfloor}-1) \] codewords., Comment: 16 pages. Typos corrected and minor changes since last version. Accepted by IEEE Transactions on Information Theory

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. S\'ark\"ozy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

We say that a $q$-ary length $n$ code is \emph{non-overlapping} if the set of non-trivial prefixes of codewords and the set of non-trivial suffices of codewords are disjoint. These codes were first studied by Levenshtein in 1964, motivated by applications in synchronisation. More recently these codes were independently invented (under the name \emph{cross-bifix-free} codes) by Baji\'c and Stojanovi\'c. We provide a simple construction for a class of non-overlapping codes which has optimal cardinality whenever $n$ divides $q$. Moreover, for all parameters $n$ and $q$ we show that a code from this class is close to optimal, in the sense that it has cardinality within a constant factor of an upper bound due to Levenshtein from 1970. Previous constructions have cardinality within a constant factor of the upper bound only when $q$ is fixed. Chee, Kiah, Purkayastha and Wang showed that a $q$-ary length $n$ non-overlapping code contains at most $q^n/(2n-1)$ codewords; this bound is weaker than the Levenshtein bound. Their proof appealed to the application in synchronisation: we provide a direct combinatorial argument to establish the bound of Chee \emph{et al}. We also consider codes of short length, finding the leading term of the maximal cardinality of a non-overlapping code when $n$ is fixed and $q\rightarrow \infty$. The largest cardinality of non-overlapping codes of lengths $3$ or less is determined exactly., Comment: 14 pages. Extra explanations added at some points, and an extra citation. To appear in IEEE Trans Information Theory

A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all $y\in X$, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all $x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\rack y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest., Comment: 11 pages

Let $G$ be a triangulation of the sphere with vertex set $V$, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $A_W$ to be the abelian group generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $A_B$ can be defined similarly, using black triangles. The paper shows that $A_W$ and $A_B$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of $A_W$ and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group $A_W$ to the understanding of the embeddings of a partial latin square in an abelian group is also explained., Comment: 22 pages; citations added, and minor changes. To appear in Combinatorica

The iterated Johnson bound is the best known upper bound on a size of an error-correcting code in the Grassmannian $\mathcal{G}_q(n,k)$. The iterated Sch\"{o}nheim bound is the best known lower bound on the size of a covering code in $\mathcal{G}_q(n,k)$. We use probabilistic methods to prove that both bounds are asymptotically attained for fixed $k$ and fixed radius, as $n$ approaches infinity. We also determine the asymptotics of the size of the best Grassmannian codes and covering codes when $n-k$ and the radius are fixed, as $n$ approaches infinity., Comment: 5 pages

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq 1/4$, and shows that $G$ is abelian whenever $\kappa(G)|G| < 7/4$. It is also shown that $\kappa(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $\kappa(S_n) \leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $\kappa(S_n) \sim A/n^2$ as $n\rightarrow \infty$ for some constant $A$. The same techniques provide analogous results for $\rho(S_n)$, the probability that two elements of the symmetric group have conjugates that commute., Comment: 34 pages, corrected version, to appear in Journal of the London Mathematical Society

We cryptanalyse a matrix-based key transport protocol due to Baumslag, Camps, Fine, Rosenberger and Xu from 2006. We also cryptanalyse two recently proposed matrix-based key agreement protocols, due to Habeeb, Kahrobaei and Shpilrain, and due to Romanczuk and Ustimenko., Comment: 9 pages

Given a prime $p$, an elliptic curve $\E/\F_p$ over the finite field $\F_p$ of $p$ elements and a binary \lrs\ $\(u(n)\)_{n =1}^\infty$ of order~$r$, we study the distribution of the sequence of points $$ \sum_{j=0}^{r-1} u(n+j)P_j, \qquad n =1,..., N, $$ on average over all possible choices of $\F_p$-rational points $P_1,..., P_r$ on~$\E$. For a sufficiently large $N$ we improve and generalise a previous result in this direction due to E.~El~Mahassni.

Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of size $n$. A sequence over $F$ of length $m$ is a \emph{$k$-radius sequence} if any two distinct elements of $F$ occur within distance $k$ of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in order to produce an efficient caching strategy when computing certain functions on large data sets such as medical images. Let $f_k(n)$ be the length of the shortest $n$-ary $k$-radius sequence. The paper shows, using a probabilistic argument, that whenever $k$ is fixed and $n\rightarrow\infty$ \[ f_k(n)\sim \frac{1}{k}\binom{n}{2}. \] The paper observes that the same argument generalises to the situation when we require the following stronger property for some integer $t$ such that $2\leq t\leq k+1$: any $t$ distinct elements of $F$ must simultaneously occur within a distance $k$ of each other somewhere in the sequence., Comment: 8 pages. More papers cited, and a minor reorganisation of the last section, since last version. Typo corrected in the statement of Theorem 4

An n-ary k-radius sequence is a finite sequence of elements taken from an alphabet of size n such that any two distinct elements of the alphabet occur within distance k of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let f_k(n) be the shortest length of any k-radius sequence. We improve on earlier estimates for f_k(n) by using tilings and logarithms. The main result is that f_k(n) ~ n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists. In particular this result holds for infinitely many k, including all k < 195 and all k such that k+1 or 2k+1 is prime. For certain k, in particular when 2k+1 is prime, we get a sharper error term using the theory of logarithms., Comment: 21 pages, 1 figure. Revision after reviewer comments. Minor changes of phrasing and a few lines of extra explanation added at some points; update of status of an open problem

A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn, Etzion, Martin and Paterson has shown that (in contrast to the situation for Costas arrays) there are only finitely many examples of honeycomb arrays, though their bound on the maximal size of a honeycomb array is too large to permit an exhaustive search over all possibilities. The present paper contains a theorem that significantly limits the number of possibilities for a honeycomb array (in particular, the theorem implies that the number of dots in a honeycomb array must be odd). Computer searches for honeycomb arrays are summarised, and two new examples of honeycomb arrays with 15 dots are given., Comment: 12 pages, 11 figures

Given a right-angled triangle of squares in a grid whose horizontal and vertical sides are $n$ squares long, let N(n) denote the maximum number of dots that can be placed into the cells of the triangle such that each row, each column, and each diagonal parallel to the long side of the triangle contains at most one dot. It has been proven that $N(n) = \lfloor \frac{2n+1}{3} \rfloor$. In this note, we give a new proof of this result using linear programming techniques., Comment: 10 pages Minor rephrasing: final version to submit to journal.

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area., Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, UK

A distinct difference configuration is a set of points in $\mathbb{Z}^2$ with the property that the vectors (\emph{difference vectors}) connecting any two of the points are all distinct. Many specific examples of these configurations have been previously studied: the class of distinct difference configurations includes both Costas arrays and sonar sequences, for example. Motivated by an application of these structures in key predistribution for wireless sensor networks, we define the $k$-hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of $k$ or fewer difference vectors. This is an important parameter when distinct difference configurations are used in the wireless sensor application, as this parameter describes the density of nodes that can be reached by a short secure path in the network. We provide upper and lower bounds for the $k$-hop coverage of a distinct difference configuration with $m$ points, and exploit a connection with $B_{h}$ sequences to construct configurations with maximal $k$-hop coverage. We also construct distinct difference configurations that enable all small vectors to be expressed as the sum of two of the difference vectors of the configuration, an important task for local secure connectivity in the application., Comment: 11 pages More application information added; biographies added

A two-dimensional grid with dots is called a \emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid., Comment: 14 pages Changed definition of honeycomb array; biographies added

A \emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks. Such graphs arise in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph $G(n,m,k)$ when $n\to \infty$ with $k$ a function of $n$ such that $k\geq 2$ and $m=\lfloor n^\alpha\rfloor$ for some fixed positive real number $\alpha$. In this situation, $G(n,m,k)$ is almost surely connected when \[ \liminf k^2n/m\log n>1, \] and $G(n,m,k)$ is almost surely disconnected when \[ \limsup k^2n/m\log n<1. \], Comment: 19 pages New version with rewritten intro, and a discussion section added. The results and proofs are unchanged from the previous version