Showing papers in "Information & Computation in 1999"

A calculus for cryptographic protocols

Abstract: We introduce the spi calculus, an extension of the pi calculus designed for describing and analyzing cryptographic protocols. We show how to use the spi calculus, particularly for studying authentication protocols. The pi calculus (without extension) suffices for some abstract protocols; the spi calculus enables us to consider cryptographic issues in more detail. We represent protocols as processes in the spi calculus and state their security properties in terms of coarse-grained notions of protocol equivalence.

Almost perfect nonlinear power functions on GF (2 n ): the Niho case

Abstract: Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography We prove for odd n and the exponent d=22r+2r?1, where 4r+1?0modn, that the power functions xd on GF(2n) is APN. The given proof is based on a new class of permutation polynomials which might be of independent interest. Our result supports a conjecture of Niho stating that the power function xd is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values ?1, ?1±2(n+1)/2.

Improved methods for approximating node weighted Steiner trees and connected dominating sets

Abstract: In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k, where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1 − o(1))ln k, assuming that \(NP ot\subseteq DTIME[n^{O(\log \log n)}]\), by a reduction from set cover.

Comparing object encodings

Abstract: Recent years have seen the development of several foundational models for statically typed object-oriented programming. But despite their intuitive similarity, differences in the technical machinery used to formulate the various proposals have made them difficult to compare. Using the typed lambda-calculus F < ω : as a common basis, we now offer a detailed comparison of four models: (1) a recursive-record encoding similar to the ones used by Cardelli [Car84], Reddy [Red88, KR94], Cook [Coo89, CHC90], and others; (2) Hofmann, Pierce, and Turner's existential encoding [PT94, HP95]; (3) Bruce's model based on existential and recursive types [Bru94]; and (4) Abadi, Cardelli, and Viswanathan's type-theoretic encoding [ACV96] of a calculus of primitive objects.

Macro Tree Transducers, Attribute Grammars, and MSO Definable Tree Translations

Abstract: A characterization is given of the class of tree translations definable in monadic second-order logic (MSO), in terms of macro tree transducers. The first main result is that the MSO definable tree translations are exactly those tree translations realized by macro tree transducers (MTTs) with regular look-ahead that are single use restricted. For this the single use restriction known from attribute grammars is generalized to MTTs. Since MTTs are closed under regular look-ahead, this implies that every MSO definable tree translation can be realized by an MTT. The second main result is that the class of MSO definable tree translations can also be obtained by restricting MTTs with regular look-ahead to be finite copying, i.e., to require that each input subtree is processed only a bounded number of times. The single use restriction is a rather strong, static restriction on the rules of an MTT, whereas the finite copying restriction is a more liberal, dynamic restriction on the derivations of an MTT.

Testing Preorders for Probabilistic Processes

Abstract: We present a testing preorder for probabilistic processes based on a quantification of the probability with which processes pass tests. The theory enjoys close connections with the classical testing theory of De Nicola and Hennessy in that whenever a process passes a test with probability 1 (respectively some nonzero probability) in our setting, then the process must (respectively may) pass the test in the classical theory. We also develop an alternative characterization of the probabilistic testing preorders that takes the form of a mapping from probabilistic traces to the interval 0, 1], where a probabilistic trace is an alternating sequence of actions and probability distributions over actions. Finally, we give proof techniques, derived from the alternative characterizations, for establishing preorder relationships between probabilistic processes. The utility of these techniques is demonstrated by means of some simple examples.

Piecemeal Graph Exploration by a Mobile Robot

Abstract: We study how a mobile robot can learn an unknown environment in a piecemeal manner. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position (for refueling, say). The environment is modeled as an arbitrary, undirected graph, which is initially unknown to the robot. We assume that the robot can distinguish vertices and edges that it has already explored. We present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph G=(V, E) in which the robot explores every vertex and edge in the graph by traversing at most O(E+V1+o(1)) edges. This nearly linear algorithm improves on the best previous algorithm, in which the robot traverses at most O(E+V2) edges. We also give an application of piecemeal learning to the problem of searching a graph for a “treasure.”

Function Field Sieve Method for Discrete Logarithms over Finite Fields

Abstract: We present a function field sieve method for discrete logarithms over finite fields. This method is an analog of the number field sieve method originally developed for factoring integers. It is asymptotically faster than the previously known algorithms when applied to finite fields Fpn, where p6?n.

A partial order approach to branching time logic model checking

Abstract: Partial order techniques enable reducing the size of the state space used for model checking, thus alleviating the “state space explosion” problem. These reductions are based on selecting a subset of the enabled operations from each program state. So far, these methods have been studied, implemented, and demonstrated for assertional languages that model the executions of a program as computation sequences, in particular the linear temporal logic. The present paper shows, for the first time, how this approach can be applied to languages that model the behavior of a program as a tree. We study here partial order reductions for branching temporal logics, e.g., the logics CTL and CTL* (with the next time operator removed) and process algebra logics such as Hennesy–Milner logic (withτactions). Conditions on the selection of subset of successors from each state during the state-space construction, which guarantee reduction that preserves CTL* properties, are given. The experimental results provided show that the reduction is substantial.

Incremental concept learning for bounded data mining

Abstract: Important refinements of concept learning in the limit from positive data considerably restricting the accessibility of input data are studied. Let c be any concept; every infinite sequence of elements exhausting c is called positive presentation of c. In all learning models considered the learning machine computes a sequence of hypotheses about the target concept from a positive presentation of it. With iterative learning, the learning machine, in making a conjecture, has access to its previous conjecture and the latest data items coming in. In k-bounded example-memory inference (k is a priori fixed) the learner is allowed to access, in making a conjecture, its previous hypothesis, its memory of up to k data items it has already seen, and the next element coming in. In the case of k-feedback identification, the learning machine, in making a conjecture, has access to its previous conjecture, the latest data item coming in, and, on the basis of this information, it can compute k items and query the database of previous data to find out, for each of the k items, whether or not it is in the database (k is again a priori fixed). In all cases, the sequence of conjectures has to converge to a hypothesis correctly describing the target concept. Our results are manyfold. An infinite hierarchy of more and more powerful feedback learners in dependence on the number k of queries allowed to be asked is established. However, the hierarchy collapses to 1-feedback inference if only indexed families of infinite concepts are considered, and moreover, its learning power is then equal to learning in the limit. But it remains infinite for concept classes of only infinite r.e. concepts. Both k-feedback inference and k-bounded example-memory identification are more powerful than iterative learning but incomparable to one another. Furthermore, there are cases where redundancy in the hypothesis space is shown to be a resource increasing the learning power of iterative learners. Finally, the union of at most k pattern languages is shown to be iteratively inferable.

An algorithm for the k -error linear complexity of sequences over GF ( p m ) with period p n , p a prime

Abstract: An algorithm is given for the k-error linear complexity of sequences over GF(pm) with period pn, p a prime. The algorithm is derived by the generalized Games?Chan algorithm for the linear complexity of sequences over GF(pm) with period pn and by using the modified cost different from that used in the Stamp?Martin algorithm for sequences over GF(2) with period 2n. A method is also given for computing an error vector which gives the k-error linear complexity.

Group Axioms for Iteration

Abstract: Iteration theories provide a sound and complete axiomatization of the equational properties of the iteration (or fixed point) operation in many models of theoretical computer science including ordered and metric structures, trees and synchronization trees. All known equational axiomatizations of iteration theories consist of a small set of equational axioms for Conway theories and a complicated equation scheme, the commutative identity. Here we associate an identity with each finite semigroup. We prove that the set consisting of the Conway identities and the group identities associated with the finite (simple) groups is complete. Moreover, we prove that the Conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite (simple) group divides one of the semigroups in the subclass. We also formulate a conjecture and study its consequences. The results are a generalization of Krob's axiomatization of the equational theory of the regular sets.

On the complexity of learning for spiking neurons with temporal coding

Abstract: Spiking neurons are models for the computational units in biological neural systems where information is considered to be encoded mainly in the temporal patterns of their activity. In a network of spiking neurons a new set of parameters becomes relevant which has no counterpart in traditional neural network models: the time that a pulse needs to travel through a connection between two neurons (also known as delay of a connection). It is known that these delays are tuned in biological neural systems through a variety of mechanisms. In this article we consider the arguably most simple model for a spiking neuron, which can also easily be implemented in pulsed VLSI. We investigate the Vapnik–Chervonenkis (VC) dimension of networks of spiking neurons, where the delays are viewed as programmable parameters and we prove tight bounds for this VC dimension. Thus, we get quantitative estimates for the diversity of functions that a network with fixed architecture can compute with different settings of its delays. In particular, it turns out that a network of spiking neurons with k adjustable delays is able to compute a much richer class of functions than a threshold circuit with k adjustable weights. The results also yield bounds for the number of training examples that an algorithm needs for tuning the delays of a network of spiking neurons. Results about the computational complexity of such algorithms are also given.

Perpetual Reductions in λ-Calculus

Abstract: This paper surveys a part of the theory ofs-reduction in?-calculus which might aptly be calledperpetual reductions. The theory is concerned withperpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from?-terms (when possible), and withperpetual redexes, i.e., redexes whose contraction in?-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in?-calculus and type theory.

Multireceiver authentication code4: models, bounds, constructions, and extensions

Abstract: Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message. In this paper, we give a formal definition of multireceiver authentication codes, derive information theoretic and combinatorial lower bounds on their performance, and give new efficient and flexible constructions for such codes. Finally, we extend the basic model to the case that multiple messages are sent and the case that the sender can be any member of the group.

Decidable integration graphs

Abstract: Integration graphsare a computational model developed in the attempt to identify simple hybrid systems with decidable analysis problems. We start with the class ofconstant slope hybrid systems(CSHS), in which the right-hand side of all differential equations is an integer constant. We refer to continuous variables whose right-hand side constants are always 1 astimers. All other continuous variables are calledintegrators. The first result shown in the paper is that simple questions such as reachability of a given state are undecidable for even this simple class of systems. To restrict the model even further, we impose the requirement that no test that refers to integrators may appear within a loop in the graph. This restricted class of CSHS is calledintegration graphs. The main results of the paper are that the reachability problem of integration graphs is decidable for two special cases: the case of a single timer and the case of a single test involving integrators. The expressive power of the integration-graphs formalism is demonstrated by showing that some typical problems studied within the context of the calculus of durations and timed statecharts can be formulated as reachability problems for restricted integration graphs, and a high fraction of these fall into the subclasses of a single timer or a single test involving integrators.

Byzantine-resistant total ordering algorithms

Abstract: Multicast group communication protocols are used extensively in fault-tolerant distributed systems. For many such protocols, the acknowledgments for individual messages define a causal order on messages. Maintaining the consistency of information, replicated on several processors to protect it against faults, is greatly simplified by a total order on messages. We present algorithms that incrementally convert a causal order on messages into a total order and that tolerate both crash and Byzantine process faults. Varying compromises between latency to message ordering and resilience to faults yield four distinct algorithms. All of these algorithms use a multistage voting strategy to achieve agreement on the total order and exploit the random structure of the causal order to ensure probabilistic termination.

Semi-explicit first-class polymorphism for ML

Abstract: We propose a modest conservative extension to ML that allows semi-explicit first-class polymorphism while preserving the essential properties of type inference. In our proposal, the introduction of polymorphic types is fully explicit, that is, both introduction points and exact polymorphic types are to be specified. However, the elimination of polymorphic types is semi-implicit: only elimination points are to be specified as polymorphic types themselves are inferred. This extension is particularly useful in objective ML where polymorphism replaces subtyping.

Relational interpretations of recursive types in an operational setting

Abstract: Relational interpretations of type systems are useful for establishing properties of programming languages. For languages with recursive types it is difficult to establish the existence of a relational interpretation. The usual approach is to pass to a domain-theoretic model of the language and, exploiting the structure of the model, to derive relational properties of it. We investigate the construction of relational interpretations of recursive types in a purely operational setting, drawing on recent ideas from domain theory and operational semantics as a guide. We prove syntactic minimal invariance for an extension of PCF with a recursive type, a syntactic analogue of the minimal invariance property used by Freyd and Pitts to characterize the domain interpretation of a recursive type. As Pitts has shown in the setting of domains, syntactic minimal invariance suffices to establish the existence of relational interpretations. We give two applications of this construction. First, we derive a notion of logical equivalence for expressions of the language that we show coincides with experimental equivalence and which, by virtue of its construction, validates useful induction and coinduction principles for reasoning about the recursive type. Second, we give a relational proof of correctness of the continuation-passing transformation, which is used in some compilers for functional languages.

Griebach normal form transformation revisited

Abstract: We develop a new method for placing a given context-free grammar into Greibach normal form with only polynomial increase of its size. Starting with an arbitrarye-free context-free grammarG, we transformGinto an equivalent context-free grammarHin extended Greibach normal form; i.e., in addition to rules, fulfilling the Greibach normal form properties, the grammar can have chain rules. The size ofHwill beO(|G|3), where |G| is the size ofG. Moreover, in the case thatGis chain rule free,Hwill be already in Greibach normal form. IfHis not chain rule free then we use the standard method for chain rule elimination for the transformation ofHinto Greibach normal form. The size of the constructed grammar isO(|G|4).

A Lattice-Based Public-Key Cryptosystem

Abstract: Ajtai recently found a random class of lattices of integer points for which he could prove the following worst-case/average-case equivalence result: If there is a probabilistic polynomial time algorithm which finds a short vector in a random lattice from the class, then there is also a probabilistic polynomial time algorithm which solves several problems related to the shortest lattice vector problem (SVP) in any n-dimensional lattice. Ajtai and Dwork then designed a public-key cryptosystem which is provably secure unless the worst case of a version of the SVP can be solved in probabilistic polynomial time. However, their cryptosystem suffers from massive data expansion because it encrypts data bit-by-bit. Here we present a public-key cryptosystem based on similar ideas, but with much less data expansion.

Automata, Boolean matrices, and ultimate periodicity

Abstract: This paper presents a Boolean-matrix-based method to automata theory, with an application to the study of regularity-preserving functions. A new characterization of such functions is derived in terms of the property of ultimate periodicity with respect to powers of Boolean matrices. This characterization reveals the intrinsic algebraic nature of regularity-preserving functions. It facilitates a concise proof of known, as well as previously unknown, properties of regularity-preserving functions, leading to the solution of the “subtraction problem,” left open by Kosaraju.

The synthesis of language learners

Abstract: An index for an r.e. class of languages (by definition) is a procedure which generates a sequence of grammars defining the class. An index for an indexed family of languages (by definition) is a procedure which generates a sequence of decision procedures defining the family. Studied is the metaproblem of synthesizing from indices for r.e. classes and for indexed families of languages various kinds of language learners for the corresponding classes or families indexed. Many positive results, as well as some negative results, are presented regarding the existence of such synthesizers. The negative results essentially provide lower bounds for the positive results. The proofs of some of the positive results yield, as pleasant corollaries, subset-principle or tell-tale style characterizations for the learnability of the corresponding classes or families indexed. For example, the indexed families of recursive languages that can be behaviorally correctly identified from positive data are surprisingly characterized by Angluin's condition 2 (the subset principle for circumventing overgeneralization).

On Cryptographic Propagation Criteria for Boolean Functions

Abstract: We determine the functions on GF(2)n which satisfy the propagation criterion of degree n?2, PC(n?2). We study subsequently the propagation criterion of degree ? and order k and its extended version EPC. We determine those Boolean functions on GF(2)n which satisfy PC(?) of order k?n???2. We show that none of them satisfies EPC(?) of the same order. We finally give a general construction of nonquadratic functions satisfying EPC(?) of order k. This construction uses the existence of nonlinear, systematic codes with good minimum distances and dual distances (e.g., Kerdock codes and Preparata codes).

Basic Observables for Processes

Abstract: A general approach for defining behavioral preorders over process terms as the maximal precongruences induced by basic observables is examined. Three different observables that provide information about the initial communication capabilities of processes and about the possibility that processes get engaged in divergent computations will be considered. We show that the precongruences induced by our basic observables coincide with intuitive and/or widely studied behavioral preorders. In particular, we retrieve in our setting the must preorder of De Nicola and Hennessy and the fair/should preorder introduced by Cleaveland and Natarajan and by Brinksma, Rensink, and Vogler. A new form of testing preorder, which we call safe-must , also emerges. The alternative characterizations we offer shed light on the differences between these pre- orders and on the role played in their definition by tests for divergence.

Recursive computational depth

Abstract: In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identified the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by defining the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep.

Further Results on Asymmetric Authentication Schemes

Abstract: This paper derives some further results on unconditionally secure asymmetric authentication schemes. It starts by giving a general framework for constructing A2-codes, identifying many known constructions as special cases. Then a full treatment of A3-codes (A2-codes protecting against arbiter's attacks) is given, including bounds on the parameters and optimal constructions. With these models as a basis, we proceed by giving constructions of general asymmetric authentication schemes, i.e., schemes protecting against specified arbitrary sets of participants collaborating in order to cheat someone else. As a consequence, we improve upon Chaum and Roijakkers interactive construction of unconditionally secure digital signatures and present a (noninteractive) construction in the form of a code. In addition, we also show a few bounds for this general model, proving the optimality of some constructions.

The Kleene-Schützenberger theorem for formal power series in partially commuting variables

Abstract: Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochmanski's theorems.

Set Constraints and Automata

Abstract: We define a new class of automata which is an acceptor model for mappings from the set of terms T? over a ranked alphabet ? into a set E of labels. When E is a set of tuples of binary values, an automaton can be viewed as an acceptor model for n-tuples of tree languages. We prove decidability of emptiness and closure properties for this class of automata. As a consequence of these results, we prove decidability of satisfiability of systems of positive and negative set constraints without projection symbols. We prove the decidability of the satisfiability problem for systems of positive and negative set constraints without projection symbols. Moreover we prove that a non-empty set of solutions always contain a regular solution (i.e., a n-tuple of regular tree languages). We also deduce decidability results for properties of sets of solutions of systems of set constraints.

Unfair Problems and Randomized Algorithms for Metrical Task Systems

Abstract: Borodin, Linial, and Saks introduced a general model for online systems calledmetrical task systems(1992,J. Assoc. Comput. Mach.39(4), 745?763). In this paper, the unfair two state problem, a natural generalization of the two state metrical task system problem, is studied. A randomized algorithm for this problem is presented, and it is shown that this algorithm is optimal. Using the analysis of the unfair two state problem, a proof of a decomposition theorem similar to that of Blum, Karloff, Rabani, and Saks (1992, “Proc. 33rd Symposium on Foundations of Computer Science,” pp. 197?207) is presented. This theorem allows one to design divide and conquer algorithms for specific metrical task systems. Our theorem gives the same bounds asymptotically, but it has less restrictive boundary conditions.