arXiv subjects
Publications and source records attributed to Douglas R. Stinson.
A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size $\frac{1}{3}\binom{n}{2}-a$, for all $n \equiv 1 \pmod{6}$ except for $n=7$.
Given an STS(v), we ask if there is a permutation of the points of the design such that no $\ell$ consecutive points in this permutation contain a block of the design. Results are obtained in the cases $\ell = 3,4$.
In this expository paper, we mainly study orthogonal arrays (OAs) of strength two having a row that is repeated $m$ times. It turns out that the Plackett-Burman bound (\cite{PB}) can be strengthened by a factor of $m$ for orthogonal arrays of strength two that contain a row that is repeated $m$ times. This is a consequence of a more general result due to Mukerjee, Qian and Wu \cite{Muk} that applies to orthogonal arrays of arbitrary strength $t$. We examine several proofs of the Plackett-Burman bound and discuss which of these proofs can be strengthened to yield the aforementioned bound for OAs of strength two with repeated rows. We also briefly discuss related bounds for $t$-designs, and OAs of strength $t$, when $t > 2$.
We construct orthogonal arrays OA$_λ (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / λ$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n+1$, albeit with large $λ$. We also study basic OAs; these are optimal OAs in which $\gcd(m,λ) = 1$. We construct a basic OA with $n=2$ and $k =4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs wth $n=2$, modulo the Hadamard matrix conjecture.
A repairable threshold scheme (which we abbreviate to RTS) is a $(τ,n)$-threshold scheme in which a subset of players can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme. Combinatorial repairable threshold schemes (or combinatorial RTS) were recently introduced by Stinson and Wei. In these schemes, "multiple shares" are distributed to each player, as defined by a suitable combinatorial design called the distribution design. In this paper, we study the reliability of these combinatorial repairable threshold schemes in a setting where players may not be available to take part in a repair of a given player's share. Using techniques from network reliability theory, we consider the probability of existence of an available repair set, as well as the expected number of available repair sets, for various types of distribution designs.
We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size $v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such that every $t$-element subset of $X$ is contained in exactly $λ$ "blocks" $B\in \mathcal{B}$ for some fixed $λ$. We represent each block $B$ by a zero-one vector $\mathbf{c}_B$ of length $v$ and explore the ideal $\mathcal{I}(\mathcal{B})$ of polynomials in $v$ variables with complex coefficients which vanish on the set $\{ \mathbf{c}_B \mid B \in \mathcal{B}\}$. After setting up the basic theory, we investigate two parameters related to this ideal: $γ_1(\mathcal{B})$ is the smallest degree of a non-trivial polynomial in the ideal $\mathcal{I}(\mathcal{B})$ and $γ_2(\mathcal{B})$ is the smallest integer $s$ such that $\mathcal{I}(\mathcal{B})$ is generated by a set of polynomials of degree at most $s$. We first prove the general bounds $t/2 < γ_1(\mathcal{B}) \le γ_2(\mathcal{B}) \le k$. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have $γ_2(\mathcal{B}) \le t$. But we expect $γ_2(\mathcal{B})$ to be closer to $k$ for less structured designs and we indicate this by constructing infinitely many triple systems satisfying $γ_2(\mathcal{B})=k$.
The purpose of this paper is to describe and analyze the Cayley-Purser algorithm, which is a public-key cryptosystem proposed by Flannery in 1999. I will present two attacks on it, one of which is apparently new. I will also examine a variant of the Cayley-Purser algorithm that was patented by Slavin in 2008, and show that it is also insecure.
In 1996, Jackson and Martin proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist.
In this paper, we study the existence of $(v,m,k,λ)$-strong external difference families (SEDFs). We use character-theoretic techniques to show that no SEDF exists when $v$ is prime, $k > 1$ and $m > 2$. In the case where $v$ is the product of two distinct odd primes, some necessary conditions are derived, which can be used to rule out certain parameter sets. Further, we show that, when $m=3$ or $4$ and $v > m$, a $(v,m,k,λ)$-SEDF does not exist.
A $(t, s, v)$-all-or-nothing transform is a bijective mapping defined on $s$-tuples over an alphabet of size $v$, which satisfies the condition that the values of any $t$ input co-ordinates are completely undetermined, given only the values of any $s-t$ output co-ordinates. The main question we address in this paper is: for which choices of parameters does a $(t, s, v)$-all-or-nothing transform (AONT) exist? More specifically, if we fix $t$ and $v$, we want to determine the maximum integer $s$ such that a $(t, s, v)$-AONT exists. We mainly concentrate on the case $t=2$ for arbitrary values of $v$, where we obtain various necessary as well as sufficient conditions for existence of these objects. We consider both linear and general (linear or nonlinear) AONT. We also show some connections between AONT, orthogonal arrays and resilient functions.
In this paper, we consider methods whereby a subset of players in a $(k,n)$-threshold scheme can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme, and it should be efficient, where efficiency is measured in terms of the amount of information exchanged during the repairing process. We study two approaches to repairing. The first method is based on the "enrollment protocol" from \cite{NSG} which was originally developed to add a new player to a threshold scheme (without the participation of the dealer) after the scheme was set up. The second method distributes "multiple shares" to each player, as defined by a suitable combinatorial design. This method results in larger shares, but lower communication complexity, as compared to the first method.
There has been considerable recent interest in "cloud storage" wherein a user asks a server to store a large file. One issue is whether the user can verify that the server is actually storing the file, and typically a challenge-response protocol is employed to convince the user that the file is indeed being stored correctly. The security of these schemes is phrased in terms of an extractor which will recover the file given any "proving algorithm" that has a sufficiently high success probability. This forms the basis of \emph{proof-of-retrievability} ($\mathsf{PoR}$) systems. In this paper, we study multiple server $\mathsf{PoR}$ systems. We formalize security definitions for two possible scenarios: (i) when a threshold of servers succeed with high enough probability (worst-case) and (ii) when the average of the success probability of all the servers is above a threshold (average-case). We also motivate the study of confidentiality of the outsourced message. We give $\mathsf{M}\mbox{-}\mathsf{PoR}$ schemes which are secure under both these security definitions and provide reasonable confidentiality guarantees even when there is no restriction on the computational power of the servers. We also show how classical statistical techniques used by Paterson, Stinson and Upadhyay (Journal of Mathematical Cryptology: 7(3)) can be extended to evaluate whether the responses of the provers are accurate enough to permit successful extraction. We also look at one specific instantiation of our construction when instantiated with the unconditionally secure version of the Shacham-Waters scheme (Asiacrypt, 2008). This scheme gives reasonable security and privacy guarantee. We show that, in the multi-server setting with computationally unbounded provers, one can overcome the limitation that the verifier needs to store as much secret information as the provers.
We continue a study of unconditionally secure all-or-nothing transforms (AONT) begun in \cite{St}. An AONT is a bijective mapping that constructs s outputs from s inputs. We consider the security of t inputs, when s-t outputs are known. Previous work concerned the case t=1; here we consider the problem for general t, focussing on the case t=2. We investigate constructions of binary matrices for which the desired properties hold with the maximum probability. Upper bounds on these probabilities are obtained via a quadratic programming approach, while lower bounds can be obtained from combinatorial constructions based on symmetric BIBDs and cyclotomy. We also report some results on exhaustive searches and random constructions for small values of s.
In this paper, we present a new lower bound on the size of separating hash families of type {w_1^{q-1},w_2} where w_1 < w_2. Our result extends the paper by Guo et al. on binary frameproof codes. This bound compares well against known general bounds, and is especially useful when trying to bound the size of strong separating hash families. We also show that our new bound is tight by constructing hash families that meet the new bound with equality.
This paper provides a mathematical analysis of optimal algebraic manipulation detection (AMD) codes. We prove several lower bounds on the success probability of an adversary and we then give some combinatorial characterizations of AMD codes that meet the bounds with equality. These characterizations involve various types of generalized difference families. Constructing these difference families is an interesting problem in its own right.
In the generalized Russian cards problem, we have a card deck $X$ of $n$ cards and three participants, Alice, Bob, and Cathy, dealt $a$, $b$, and $c$ cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (weak $1$-security) to not gaining any probabilistic advantage in guessing the fate of some set of $δ$ cards (perfect $δ$-security). As we demonstrate, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly $δ$-secure strategies and $(c+δ)$-designs on $n$ points with block size $a$, when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect $1$-security against Cathy when $c=2$. We leverage a known combinatorial design to construct a strategy with $a=8$, $b=13$, and $c=3$ that is perfectly $2$-secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak $δ$-security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of $c = \lfloor \frac{a-δ}{2} \rfloor$ cards. Alternatively, the construction yields solutions for arbitrary $δ$, $c$ and any $a \geq δ+ 2c$.
In this paper, we study $w$-frameproof codes, which are equivalent to $\{1,w\}$-separating hash families. Our main results concern binary codes, which are defined over an alphabet of two symbols. For all $w \geq 3$, and for $w+1 \leq N \leq 3w$, we show that an $SHF(N; n,2, \{1,w \})$ exists only if $n \leq N$, and an $SHF(N; N,2, \{1,w \})$ must be a permutation matrix of degree $N$.
We study a method for key predistribution in a network of $n$ users where pairwise keys are computed by hashing users' IDs along with secret information that has been (pre)distributed to the network users by a trusted entity. A communication graph $G$ can be specified to indicate which pairs of users should be able to compute keys. We determine necessary and sufficient conditions for schemes of this type to be secure. We also consider the problem of minimizing the storage requirements of such a scheme; we are interested in the total storage as well as the maximum storage required by any user. Minimizing the total storage is NP-hard, whereas minimizing the maximum storage required by a user can be computed in polynomial time.