Perfect Hash Families Research Articles

Detecting arrays for effects of multiple interacting factors

Detecting arrays provide test suites for complex engineered systems in which many factors interact. The determination of which interactions have a significant impact on system behaviour requires not only that each interaction appear in a test, but also that its effect can be distinguished from those of other significant interactions. In this paper, compact representations of detecting arrays using vectors over the finite field are developed. Covering strong separating hash families exploit linear independence over the field, while the weaker elongated covering perfect hash families permit some linear dependence. For both, probabilistic analyses are employed to establish effective upper bounds on the number of tests needed in a detecting array for a wide variety of parameters. The analyses underlie efficient algorithms for the explicit construction of detecting arrays.

In-Parameter-Order strategies for covering perfect hash families

Combinatorial testing makes it possible to test large systems effectively while maintaining certain coverage guarantees. At the same time, the construction of optimized covering arrays (CAs) with a large number of columns is a challenging task. Heuristic and Metaheuristic approaches often become inefficient when applied to large instances, as the computation of the quality for new moves or solutions during the search becomes too slow. Recently, the generation of covering perfect hash families (CPHFs) has led to vast improvements to the state of the art for many different instances of covering arrays. CPHFs can be considered a compact form of a specific family of covering arrays. Their compact representation makes it possible to apply heuristic methods for instances with a much larger number of columns. In this work, we adapt the ideas of the well-known In-Parameter-Order (IPO) strategy for covering array generation to efficiently construct CPHFs, and therefore implicitly covering arrays. We design a way to realize the concept of vertical extension steps in the context of CPHFs and discuss how a horizontal extension can be implemented in an efficient manner. Further, we develop a horizontal extension strategy for CPHFs with subspace restrictions that identifies candidate columns greedily based on conditional expectation. Then using a local optimization strategy, a candidate may be adjoined to the solution or may replace one of the existing columns. An extensive set of computational results yields many significant improvements on the sizes of the smallest known covering arrays.

Beating Fredman-Komlós for perfect k-hashing

We say a subset C⊆{1,2,…,k}n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (log2⁡|C|)/n, of a k-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family that maps a universe of N elements into {1,2,…,k}, and (ii) the zero-error capacity for decoding with lists of size less than k for a certain combinatorial channel.A general upper bound of k!/kk−1 on the rate of a k-hash code (in the limit of large n) was obtained by Fredman and Komlós in 1984 for any k≥4. While better bounds have been obtained for k=4, their original bound has remained the best known for each k≥5. In this work, we present a method to obtain the first improvement to the Fredman-Komlós bound for every k≥5.

Super-simple BIBDs with block size 4 and index 7

Super-simple designs are used in other constructions, such as coverings, superimposed codes and perfect hash families etc. The existence of super-simple (v,4,λ)-BIBDs has been determined for λ=2,3,4,5,6 and 9. When λ=7, the necessary conditions of such a design are that v≡1,4 (mod 12) and v≥16. In this paper, we show that these necessary conditions are sufficient.

Quantum Overlapping Tomography.

It is now experimentally possible to entangle thousands of qubits, and efficiently measure each qubit in parallel in a distinct basis. To fully characterize an unknown entangled state of n qubits, one requires an exponential number of measurements in n, which is experimentally unfeasible even for modest system sizes. By leveraging (i)that single-qubit measurements can be made in parallel, and (ii)the theory of perfect hash families, we show that all k-qubit reduced density matrices of an n qubit state can be determined with at most e^{O(k)}log^{2}(n) rounds of parallel measurements. We provide concrete measurement protocols which realize this bound. As an example, we argue that with near-term experiments, every two-point correlator in a system of 1024 qubits could be measured and completely characterized in a few days. This corresponds to determining nearly 4.5million correlators.

Improved covering arrays using covering perfect hash families with groups of restricted entries

Covering perfect hash families (CPHFs) are used to represent compactly some covering arrays. For CPHFs there is an efficient way to determine if the covering arrays derived from them are or not complete covering arrays; and this characteristic eases the construction of large covering arrays by greedy and metaheuristic methods working over the CPHF representation. CPHFs has been constructed in other works by backtracking, tabu search, simulated annealing, and greedy methods. The present work introduces a new simulated annealing algorithm that is able to construct CPHFs with groups of rows with restricted entries. In a CPHF of this kind the entries have some dependencies among them, that is, the values in some rows restrict the values that can appear in other rows. Restricted CPHFs produce covering arrays smaller than the ones derived from CPHFs without restricted entries. By using the new simulated annealing algorithm, we construct 135 new CPHFs whose derived covering arrays improve the upper bound of 19,669 covering array numbers.

Distributing hash families with few rows

Column replacement techniques for creating covering arrays rely on the construction of perfect and distributing hash families with few rows, having as many columns as possible for a specified number of symbols. To construct distributing hash families in which the number of rows is less than the strength, we examine a method due to Blackburn and extend it in three ways. First, the method is generalized from homogeneous hash families (in which every row has the same number of symbols) to heterogeneous ones. Second, the extension treats distributing hash families, in which only separation into a prescribed number of parts is required, rather than perfect hash families, in which columns must be completely separated. Third, the requirements on one of the main ingredients are relaxed to permit the use of a large class of distributing hash families, which we call fractal. Constructions for fractal perfect and distributing hash families are given, and applications to the construction of perfect hash families of large strength are developed.

Subspace restrictions and affine composition for covering perfect hash families

Covering perfect hash families provide a very compact representation of a useful family of covering arrays, leading to the best asymptotic upper bounds and fast, effective algorithms. Their compactness implies that an additional row in the hash family leads to many new rows in the covering array. In order to address this, subspace restrictions constrain covering perfect hash family so that a predictable set of many rows in the covering array can be removed without loss of coverage. Computing failure probabilities for random selections that must, or that need not, satisfy the restrictions, we identify a set of restrictions on which to focus. We use existing algorithms together with one novel method, affine composition, to accelerate the search. We report on a set of computational constructions for covering arrays to demonstrate that imposing restrictions often improves on previously known upper bounds.

A Simulated Annealing Algorithm to Construct Covering Perfect Hash Families

Covering perfect hash families (CPHFs) are combinatorial designs that represent certain covering arrays in a compact way. In previous works, CPHFs have been constructed using backtracking, tabu search, and greedy algorithms. Backtracking is convenient for small CPHFs, greedy algorithms are appropriate for large CPHFs, and metaheuristic algorithms provide a balance between execution time and quality of solution for small and medium-size CPHFs. This work explores the construction of CPHFs by means of a simulated annealing algorithm. The neighborhood function of this algorithm is composed of three perturbation operators which together provide exploration and exploitation capabilities to the algorithm. As main computational results we have the generation of 64 CPHFs whose derived covering arrays improve the best-known ones. In addition, we use the simulated annealing algorithm to construct quasi-CPHFs from which quasi covering arrays are derived that are then completed and postoptimized; in this case the number of new covering arrays is 183. Together, the 247 new covering arrays improved the upper bound of 683 covering array numbers.

A greedy-metaheuristic 3-stage approach to construct covering arrays

Covering arrays are combinatorial designs used as test-suites in software and hardware testing. Because of their practical applications, the construction of covering arrays with a smaller number of rows is desirable. In this work we develop a greedy-metaheuristic 3-stage approach to construct covering arrays that improve some of the best-known ones. In the first stage, a covering perfect hash family is created using a metaheuristic approach; this initial array may not be complete, and so the derived covering array may have missing tuples. In the second stage, the covering perfect hash family is converted to a covering array and, in case there are missing tuples, a greedy approach completes the covering array through the addition of some rows. The third stage is an iterative postoptimization stage that combines two greedy algorithms and a metaheuristic algorithm; the greedy algorithms detect and reduce redundancy in the covering array, and the metaheuristic algorithm covers the tuples that may become uncovered after the reduction of redundancy. The effectiveness of our greedy-metaheuristic 3-stage approach is assessed through the construction of covering arrays of order four and strengths 3–6; the main results are the improvement of 9473 covering arrays of strength three, 9303 of strength four, 2150 of strength five, and 291 of strength six. To see how to apply covering arrays to real testing scenarios, the final part of this work presents the use of covering arrays of order four for setting up a composting process.

Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials

We introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive polynomial-space O⁎(3.86k)-time (exponential-space O⁎(3.41k)-time) deterministic algorithm for k-Internal Out-Branching, improving upon the previously fastest exponential-space O⁎(5.14k)-time algorithm for this problem. (The notation O⁎ hides polynomial factors.) We also design polynomial-space O⁎((2e)k+o(k))-time (exponential-space O⁎(4.32k)-time) deterministic algorithms for k-ColorfulOut-Branching on arc-colored digraphs and k-Colorful Perfect Matching on planar edge-colored graphs. In k-Colorful Out-Branching, given an arc-colored digraph D, decide whether D has an out-branching with arcs of at least k colors. k-Colorful Perfect Matching is defined similarly. To obtain our polynomial-space algorithms, we show that (n,k,αk)-splitters (α⩾1) and in particular (n,k)-perfect hash families can be enumerated one by one with polynomial delay using polynomial space.

Fractal Perfect Hash Families (Extended Abstract)

Constructions of perfect hash families (PHFs) are developed in which the number of rows is less than the strength, using a method due to Blackburn. The constructions use coverings of strength d along with a restricted class of perfect hash families, called fractal PHFs. Constructions for fractal PHFs are given, and applications to the construction of PHFs of large strengths are developed.

Covering arrays of strength three from extended permutation vectors

A covering array $$\text{ CA }(N;t,k,v)$$ is an $$N\times k$$ array such that in every $$N\times t$$ subarray each possible t-tuple over a v-set appears as a row of the subarray at least once. The integers t and v are respectively the strength and the order of the covering array. Let v be a prime power and let $${\mathbb {F}}_v$$ denote the finite field with v elements. In this work the original concept of permutation vectors generated by a $$(t-1)$$ -tuple over $${\mathbb {F}}_v$$ is extended to vectors generated by a t-tuple over $${\mathbb {F}}_v$$ . We call these last vectors extended permutation vectors (EPVs). For every prime power v, a covering perfect hash family $$\text{ CPHF }(2;v^2-v+3,v^3,3)$$ is constructed from EPVs given by subintervals of a linear feedback shift register sequence over $${\mathbb {F}}_v$$ . When $$v\in \{7,9,11,13,16,17,19,23,25\}$$ the covering array $$\text{ CA }(2v^3-v;3,v^2-v+3,v)$$ generated by $$\text{ CPHF }(2;v^2-v+3,v^3,3)$$ has less rows than the best-known covering array with strength three, $$v^2-v+3$$ columns, and order v. CPHFs formed by EPVs are also constructed using simulated annealing; in this case the results improve the size of eighteen covering arrays of strength three.

Asymptotic and constructive methods for covering perfect hash families and covering arrays

Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when $$k \le 75$$ for strength seven, $$k \le 200$$ for strength six, $$k \le 600$$ for strength five, and $$k \le 2500$$ for strength four. When $$v > 3$$ , almost all known explicit constructions are improved upon. For strength $$t=3$$ , restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for $$t=3$$ and $$k \le 10{,}000$$ again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.

New Lower Bounds for Secure Codes and Related Hash Families: A Hypergraph Theoretical Approach

Various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials. The codewords in such a structure can be regarded as a subset of $Q^{N}$ , the set of all $q$ -ary vectors of given length $N$ , satisfying some constraints. We use a hypergraph model to characterize the combinatorial structure. By applying a result of Duke et al. on the lower bound of the independence number of hypergraphs, we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families. In particular, the general method is illustrated via the examples of existence results on some perfect hash families, frameproof codes, and separable codes.

Separating Hash Families: A Johnson-type bound and New Constructions

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography, and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before. It provides an affirmative answer to both Bazrafshan and Trung's open problem on separating hash families and Alon and Stav's conjecture on parent-identifying codes. Thirdly, let $p_t(N,q)$ denote the maximal cardinality of a $t$-perfect hash family of length $N$ over an alphabet of size $q$. Walker and Colbourn conjectured that $p_3(3,q)=o(q^2)$. We verify this conjecture by proving $q^{2-o(1)}<p_3(3,q)=o(q^2)$. Our proof can be viewed as an application of Ruzsa and Szemerédi's (6,3)-theorem. We also prove $q^{2-o(1)}<p_4(4,q)=o(q^2)$. Two new notions in graph theory and additive number theory, namely, rainbow cycles and $R$-sum-free sets, are introduced to prove this result. These two bounds support a question of Blackburn, Etzion, Stinson, and Zaverucha. Finally, we establish a bridge between perfect hash families and hypergraph Turán problems. This connection has not been noticed before. As a consequence, many new results and problems arise.

Perfect hash families of strength three with three rows from varieties on finite projective geometries

Let $${\mathcal {F}}$$F be a set of functions of $$f : X \rightarrow Y$$f:X?Y, where $$|X|=k, \,|Y|=v$$|X|=k,|Y|=v and $$|{\mathcal {F}}|=N$$|F|=N. If for any $$t$$t-subset $$C \subseteq X$$C⊆X there exists at least one function $$f\in \mathcal {F}$$f?F such that $$f|_{C}$$f|C is one-to-one, then $${\mathcal {F}}$$F is called a perfect hash family, denoted by PHF$$(N; k, v, t)$$(N?k,v,t). In this paper, we construct the simplest nontrivial PHFs of $$t=3$$t=3 and $$N=3$$N=3 using classic generalized quadrangles, quadrics in PG$$(4, q)$$(4,q) and Hermitian varieties in PG$$(4, q^2)$$(4,q2). We obtained PHF$$(3; q^2(q+1), q^2, 3)$$(3?q2(q+1),q2,3) and PHF$$(3; q^5, q^3, 3)$$(3?q5,q3,3) for $$q$$q a prime power. The curve $$k= v^{5/3}$$k=v5/3 is greater than known $$k$$k for $$v=q^3,\,q$$v=q3,q a prime power.

On the relationships between perfect nonlinear functions and universal hash families

In this paper, the relationships between perfect nonlinear (in brief, PN) functions and optimal universal hash families are discussed. We point out the equivalence of constructions between them, i.e., from PN functions, one can obtain optimal universal hash families and vice versa. As an application of our construction, a message authentication code is proposed, which provides better resistance to substitution attack than a known construction given by Carlet et al. in 2006. More generally, the connections between functions with given differential uniformity and some universal hash families are studied.

Super-simple balanced incomplete block designs with block size 5 and index 3

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (v,5,3) balanced incomplete block design and show that such a design exists if and only if v≡1,5(mod20) and v≥21 except possibly when v=45,65.

Improved bounds for separating hash families

An $${(N;n,m,\{w_1,\ldots, w_t\})}$$ -separating hash family is a set $${\mathcal{H}}$$ of N functions $${h: \; X \longrightarrow Y}$$ with $${|X|=n, |Y|=m, t \geq 2}$$ having the following property. For any pairwise disjoint subsets $${C_1, \ldots, C_t \subseteq X}$$ with $${|C_i|=w_i, i=1, \ldots, t}$$ , there exists at least one function $${h \in \mathcal{H}}$$ such that $${h(C_1), h(C_2), \ldots, h(C_t)}$$ are pairwise disjoint. Separating hash families generalize many known combinatorial structures such as perfect hash families, frameproof codes, secure frameproof codes, identifiable parent property codes. In this paper we present new upper bounds on n which improve many previously known bounds. Further we include constructions showing that some of these bounds are tight.