Straight-line Programs Research Articles

Communication-Efficient Multi-Party Computation for RMS Programs

Despite much progress, general-purpose secure multi-party computation (MPC) with active security may still be prohibitively expensive in settings with large input datasets. This particularly applies to the secure evaluation of graph algorithms, where each party holds a subset of a large graph. Recently, Araki et al. (ACM CCS '21) showed that dedicated solutions may provide significantly better efficiency if the input graph is sparse. In particular, they provide an efficient protocol for the secure evaluation of “message passing” algorithms, such as the PageRank algorithm. Their protocol's computation and communication complexity are both O ~ ( M · B ) instead of the O ( M 2 ) complexity achieved by general-purpose MPC protocols, where M denotes the number of nodes and B the (average) number of incoming edges per node. On the downside, their approach achieves only a relatively weak security notion; 1 -out-of- 3 malicious security with selective abort. In this work, we show that PageRank can instead be captured efficiently as a restricted multiplication straight-line (RMS) program, and present a new actively secure MPC protocol tailored to handle RMS programs. In particular, we show that the local knowledge of the participants can be leveraged towards the first maliciously-secure protocol with communication complexity linear in M , independently of the sparsity of the graph. We present two variants of our protocol. In our communication-optimized protocol, going from semi-honest to malicious security only introduces a small communication overhead, but results in quadratic computation complexity O ( M 2 ) . In our balanced protocol, we still achieve a linear communication complexity O ( M ) , although with worse constants, but a significantly better computational complexity scaling with O ( M · B ) . Additionally, our protocols achieve security with identifiable abort and can tolerate up to n − 1 corruptions.

There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups

Let [Formula: see text] be a finite set of finitary operation symbols and let [Formula: see text] be a nontrivial variety of [Formula: see text]-algebras. Assume that for some set [Formula: see text] of group operation symbols, all [Formula: see text]-algebras in [Formula: see text] are groups under the operations associated with the symbols in [Formula: see text]. In other words, [Formula: see text] is assumed to be a nontrivial variety of expanded groups. In particular, [Formula: see text] can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in [Formula: see text], even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families [Formula: see text] of computational and black-box [Formula: see text]-algebras (where [Formula: see text]) such that for every [Formula: see text], each element of [Formula: see text] is represented by a unique bit string of length polynomial in the length of d. In our main result, we use straight-line programs to represent nontrivial relations between elements of [Formula: see text]-algebras. Note that under certain conditions, this result depends on the classification of finite simple groups. Also, we define and study some types of post-quantum weak pseudo-freeness for families of computational and black-box [Formula: see text]-algebras.

The Power Word Problem in Graph Products

The power word problem for a group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} asks whether an expression u1x1⋯unxn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{u_1^{x_1} \\cdots u_n^{x_n}}$$\\end{document}, where the ui\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{u_i}$$\\end{document} are words over a finite set of generators of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} and the xi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x_i}$$\\end{document} binary encoded integers, is equal to the identity of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}. It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} is uNC1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{uNC}^{1}}$$\\end{document}-many-one reducible to the power word problem for a finite-index subgroup of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}. For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{AC} ^0}$$\\end{document}-Turing-reducible to the word problem for the free group F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F_2}$$\\end{document} and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mathcal {C}}$$\\end{document} without order two elements, the uniform power word problem in a graph product can be solved in AC0[C=LUPowWP(C)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{AC} ^0[\ extsf{C}_=\ extsf{L} ^{{{\\,\ extrm{UPowWP}\\,}}(\\mathcal {C})}]}$$\\end{document}, where UPowWP(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{{{\\,\ extrm{UPowWP}\\,}}(\\mathcal {C})}$$\\end{document} denotes the uniform power word problem for groups from the class C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mathcal {C}}$$\\end{document}. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{NP}}$$\\end{document}-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.

Optimizing implementations of linear layers using two and higher input XOR gates.

Maximum distance separable (MDS) matrices are often used in the linear layer of a block cipher due to their good diffusion property. A well-designed lightweight MDS matrix, especially an involutory one, can provide both security and performance benefits to the cipher. Finding the corresponding effective linear straight-line program (SLP) of the circuit of a linear layer is still a challenging problem. In this article, first, we propose a new heuristic algorithm called Superior Boyar-Peralta (SBP) in the computation of the minimum number of two-input Exclusive-OR (XOR) gates with the minimum circuit depth for the SLPs. Contrary to the existing global optimization methods supporting only two-input XOR gates, SBP heuristic algorithm provides the best global optimization solutions, especially for extracting low-latency circuits. Moreover, we give a new 4 × 4 involutory MDS matrix over F24, which requires only 41 XOR gates and depth 3 after applying SBP heuristic, whereas the previously best-known cost is 45 XOR gates with the same depth. In the second part of the article, for further optimization of the circuit area of linear layers with multiple-input XOR gates, we enhance the recently proposed BDKCI heuristic algorithm by incorporating circuit depth awareness, which limits the depth of the circuits created. By using the proposed circuit depth-bounded version of BDKCI, we present better circuit implementations of linear layers of block ciphers than those given in the literature. For instance, the given circuit for the AES MixColumn matrix only requires 44 XOR gates/depth 3/240.95 GE in the STM 130 nm (simply called ASIC4) library, while the previous best-known result is 55 XOR gates/depth 5/243.00 GE. Much better, our new 4 × 4 involutory MDS matrix requires only 19 XOR gates/depth3/79.75 GE in the STM 90 nm (simply called ASIC1) library, which is the lightest and superior to the state-of-the-art results.

Self-Supervised Learning to Prove Equivalence Between Straight-Line Programs via Rewrite Rules

Self-Supervised Learning to Prove Equivalence Between Straight-Line Programs via Rewrite Rules

A general construction for abstract interpretation of higher-order automatic differentiation

We present a novel, general construction to abstractly interpret higher-order automatic differentiation (AD). Our construction allows one to instantiate an abstract interpreter for computing derivatives up to a chosen order. Furthermore, since our construction reduces the problem of abstractly reasoning about derivatives to abstractly reasoning about real-valued straight-line programs, it can be instantiated with almost any numerical abstract domain, both relational and non-relational. We formally establish the soundness of this construction. We implement our technique by instantiating our construction with both the non-relational interval domain and the relational zonotope domain to compute both first and higher-order derivatives. In the latter case, we are the first to apply a relational domain to automatic differentiation for abstracting higher-order derivatives, and hence we are also the first abstract interpretation work to track correlations across not only different variables, but different orders of derivatives. We evaluate these instantiations on multiple case studies, namely robustly explaining a neural network and more precisely computing a neural network’s Lipschitz constant. For robust interpretation, first and second derivatives computed via zonotope AD are up to 4.76× and 6.98× more precise, respectively, compared to interval AD. For Lipschitz certification, we obtain bounds that are up to 11,850× more precise with zonotopes, compared to the state-of-the-art interval-based tool.

Sparse polynomial interpolation based on derivatives

We propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program (SLP). Both of our algorithms work over any finite fields Fq with large characteristic.The first algorithm is randomized of the Monte Carlo type. Its bit complexity is linear in the number of non-zero terms T and the number of variables n of f. Let D be the partial degree bound of f. If q is in O((nTD)(1)), our algorithm has better complexity than other existing algorithms.The second algorithm is deterministic; it has better complexity than any existing deterministic algorithms in the field with large characteristic. Its bit complexity is quadratic in n,T,log⁡D.

Logspace and compressed-word computations in nilpotent groups

For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

A similarity measure for Straight Line Programs and its application to control diversity in Genetic Programming

Finding a balance between diversity and convergence plays an important role in evolutionary algorithms to avoid premature convergence and to perform a better exploration of the search space. In the case of Genetic Programming, and more specifically for symbolic regression problems, different mechanisms have been devised to control diversity, ranging from novel crossover and/or mutation procedures to the design of distance measures that help genetic operators to increase diversity in the population. In this paper, we start from previous works where Straight Line Programs are used as an alternative representation to expression trees for symbolic regression, and develop a similarity measure based on edit distance in order to determine how different the Straight Line Programs in the population are. This measure is used in combination with the CHC algorithm strategy to control diversity in the population, and therefore to avoid local optima to solve symbolic regression problems. The proposal is first validated in a controlled scenario of benchmark datasets and it is compared with previous approaches to promote diversity in Genetic Programming. After that, the approach is also evaluated in a real world dataset of energy consumption data from a set of buildings of the University of Granada.

Асимптотические оценки средней сложности булевых функций

Some results obtained in recent years concerning the average-case complexity of computing Boolean functions by straight line programs with conditional stop are considered. These programs generalize the concept of a Boolean circuit and are a natural model of straight line computations, i.e., computations that do not involve conditional branching or indirect addressing, but allow an early termination under a certain condition. Such computations can be informally described as follows. Computations are executed by a processor with memory consisting of cells. The processor is capable of computing some number of elementary functions, which form the computational basis. At any moment of time, each memory cell is accessible to the processor for both reading and writing data. The processor is controlled by a program that is a sequence of elementary instructions of two types. Every instruction of the first type computes the value of a basis function whose arguments are contained in certain memory cells. The computed result is also placed in a memory cell. An instruction of the second type can terminate the execution of a program. Every such instruction has a single argument, which is contained in a memory cell. If the value of the argument is equal to a certain fixed number, say, unity, then the processor terminates the program execution. If the argument has a different value, then the next instruction of the program is executed. There are special memory cells whose values are declared to be the result of executing the program after its termination. A natural measure of the complexity of such a program is the running time averaged over all possible arguments.

Entropy Bounds for Grammar-Based Tree Compressors

The definition of $\text {k}^{th}$ -order empirical entropy of strings is extended to node-labelled binary trees: A notion of $\text {k}^{th}$ -order empirical entropy for node-labelled binary trees is proposed that is able to capture regularities in both labels and structure of a tree. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the $\text {k}^{th}$ -order empirical entropy plus some lower order terms. This result is then extended from node-labelled binary trees to node-labelled unranked trees. This generalizes recent results for grammar-based string compression to grammar-based tree compression. Additionally, experimental results with real XML document trees are presented, in which the proposed notion of $\text {k}^{th}$ -order empirical tree entropy is computed and compared to the performance of grammar-based tree compressors for those XML document trees.

Adaptive learning of compressible strings

Suppose an oracle knows a string S that is unknown to us and that we want to determine. The oracle can answer queries of the form “Is s a substring of S?”. In 1995, Skiena and Sundaram showed that, in the worst case, any algorithm needs to ask the oracle σn/4−O(n) queries in order to be able to reconstruct the hidden string, where σ is the size of the alphabet of S and n its length, and gave an algorithm that spends (σ−1)n+O(σn) queries to reconstruct S. The main contribution of our paper is to improve the above upper-bound in the context where the string is compressible. We first present a universal algorithm that, given a (computable) compressor that compresses the string to τ bits, performs q=O(τ) substring queries; this algorithm, however, runs in exponential time. For this reason, the second part of the paper focuses on more time-efficient algorithms whose number of queries is bounded by specific compressibility measures. We first show that any string of length n over an integer alphabet of size σ with rle runs can be reconstructed with q=O(rle(σ+log⁡nrle)) substring queries in linear time and space. We then present an algorithm that spends q∈O(σglog⁡n) substring queries and runs in O(n(log⁡n+log⁡σ)+q) time using linear space, where g is the size of a smallest straight-line program generating the string.

On the average-case complexity of Boolean functions under binomial distribution on their domains

Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.

The descriptional power of queue automata of constant length

We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

Balancing Straight-line Programs

We show that a context-free grammar of size that produces a single string of length (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for of size , whose unique derivation tree has depth . This solves an open problem in the area of grammar-based compression, improves many results in this area, and greatly simplifies many existing constructions. Similar results are shown for two formalisms for grammar-based tree compression: top dags and forest straight-line programs. These balancing results can be all deduced from a single meta-theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to with the cost of a constant multiplicative size increase. Here, refers to the size of the unfolding (or unravelling) of the circuit. In particular, this results applies to standard arithmetic circuits over (noncommutative) semirings.

Syntactic view of sigma-tau generation of permutations

We give a syntactic view of the Sawada-Williams (σ,τ)-generation of permutations. The corresponding sequence Seqn of στ-operations of length n!−1 is shown to have a compact description of size Θ(n2) in terms of straight-line programs. Using the compact description, we design almost linear time ranking and unranking algorithms.

Largest common prefix of a regular tree language

A family of tree automata of size n is presented such that the size of the largest common prefix (lcp) tree of all accepted trees is exponential in n. Moreover, it is shown that this prefix tree is not compressible via DAGs (directed acyclic graphs) or tree straight-line programs. We also show that determining whether or not the lcp trees of two given tree automata are equal is coNP-complete; the result holds even for deterministic bottom-up tree automata accepting finite tree languages. These results are in sharp contrast to the case of context-free string grammars.

An Ant Colony Optimization approach for symbolic regression using Straight Line Programs. Application to energy consumption modelling

The increase of energy consumption and their direct effects on pollution and global warming have motivated governments to develop new strategies to promote a better usage of energy. One of the most important aspects related to energy efficiency is the need for a suitable model of energy consumption that can be used to make predictions or to aid experts in high level decision making processes. Symbolic regression techniques can be used to discover an energy consumption model that fits these purposes. Traditionally, the problem of symbolic regression has been solved by using genetic programming approaches to find the algebraic expression that best fits the regression problem data, where each expression is encoded as a tree structure. In previous works, we found that a different approach using Straight Line Programs as a representation technique could provide promising results for symbolic regression, although the size of the resulting algebraic expression might be increased when compared to the traditional approach. This work proposes an Ant Colony Optimization algorithm for Straight Line Programs to solve the problem, and makes a study to compare the approach with traditional genetic programming in a real energy consumption modelling problem.

Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let f be an n-variate polynomial given by a straight-line program, which has a total degree bound D and a term bound T. Our deterministic algorithm is quadratic in n,T and cubic in log⁡D in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in nT and polynomial in log⁡D in the Soft-Oh sense.

Grammar-Based Compression of Unranked Trees

We introduce forest straight-line programs (FSLPs) as a compressed representation of unranked ordered node-labelled trees. FSLPs are based on the operations of forest algebra and generalize tree straight-line programs. We compare the succinctness of FSLPs with two other compression schemes for unranked trees: top dags and tree straight-line programs of first-child/next sibling encodings. Efficient translations between these formalisms are provided. Finally, we show that equality of unranked trees in the setting where certain symbols are associative or commutative can be tested in polynomial time. This generalizes previous results for testing isomorphism of compressed unordered ranked trees.