Directed Acyclic Graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of a given number of vertices has been solved in the 70’s by Robinson. In many applications one needs to construct connected DAGs and to control their number of edges, but the adaptation of Robinson’s enumeration to take this into account led to counting formulas based on the inclusion-exclusion principle, inducing a high computational cost for the uniform random sampling of DAGs based on this formula. In the present paper we propose two contributions. First we enumerate a new class of DAGs, enriched with an independent ordering of the children of each vertex, according to their numbers of vertices and edges. We obtain a constructive recursive counting formula for them (i.e. without using the inclusion-exclusion principle) using a new decomposition scheme. Then we show the applicability of our method by proposing a constructive enumeration of Robinson’s labelled DAGs, by vertices and edges, based on the same decomposition. As a consequence we are able to derive efficient uniform random samplers for both models.

In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process is a directed acyclic graph (DAG). Since the quantitative study of increasingly labeled DAG (directly related to processes) seems out of reach (this is a #P-complete problem), we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the classical sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of a uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.

We present counting methods for some special classes of multivariate polynomials over a finite field, namely, the reducible ones, the $s$-powerful ones (divisible by the $s$th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, and another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.

Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameterq, 0qqcannot be optimal for any other value ofq. This is in sharp contrast to the one-dimensional (1-D) case, where codes are optimal for positive-length intervals of the parameterq. Thus, in the 2-D case, it is infeasible to give a compact characterization of optimal codes for all values of the parameterq, as was done in the 1-D case. Instead, optimal codes are characterized for a discrete sequence of values ofqthat provides good coverage of the unit interval. Specifically, optimal prefix codes are described forq= 2-1/k(k≥ 1), covering the rangeq≥ [1/2], andq= 2-k(k> 1), covering the rangeq<; [1/2]. The described codes produce the expected reduction in redundancy with respect to the 1-D case, while maintaining low-complexity coding operations.

Boolean functions are very important cryptographic primitives in stream or block ciphers. In order to be useful for cryptographic applications, these functions should satisfy some properties like high algebraic degree, high non linearity or being correlation immune. Since for most of the cryptographic criteria presented in the literature there is no complete characterization of the set of functions that optimally satisfy any of them, the possibility of finding an enumerative encoding of any such class of functions is extremely hard. In a recent paper Le Bars and Viola have presented an innovative recursive decomposition of the first order correlation immune Boolean functions. It is not a trivial task, however, to derive from this characterization an enumerative encoding. This paper presents an enumerative encoding for first order correlation immune functions. It provides the first enumerative encoding of a class of Boolean functions with cryptographic applications. The encoding naturally leads to efficient random generation algorithms. For example, we may construct, with uniform probability, any 1-resilient function (balanced first order correlation immune function) with 8 variables in less than 30 seconds, from a universe of around 1068functions

Bi-directional context sets extend the classical context-tree modeling framework to situations in which the observations consist of two tracks or directions. In this paper, we study the problem of efficiently finding an optimal bi-directional context set for a given data sequence and loss function. This problem has applications in data compression, prediction, and denoising. The main tool in our construction is a new data structure, the compact bi-directional context graph, which generalizes compact suffix trees to two directions.

Quickselect with median-of-3 is largely used in practice and its behavior is fairly well understood. However, the following natural adaptive variant, which we call proportion-from-3 , had not been previously analyzed: “choose as pivot the smallest of the sample if the relative rank of the sought element is below 1/3, the largest if the relative rank is above 2/3, and the median if the relative rank is between 1/3 and 2/3.” We first analyze the average number of comparisons made when using proportion-from-2 and then for proportion-from-3. We also analyze ν-find, a generalization of proportion-from-3 with interval breakpoints at ν and 1-ν. We show that there exists an optimal value of ν and we also provide the range of values of ν where ν-find outperforms median-of-3. Then, we consider the average total cost of these strategies, which takes into account the cost of both comparisons and exchanges. Our results strongly suggest that a suitable implementation of ν-find could be the method of choice in a practical setting. We also study the behavior of proportion-from- s with s >3 and in particular we show that proportion-from- s -like strategies are optimal when s →∞.

In 2002, in two independent papers, Bellare, Kohno, and Namprempre and Joux, Martinet, and Valette introduced the notion of blockwise security for modes of operations. This notion stems from common practice, since in many applications, modes of operation for block ciphers do not process messages as atomic entities but in a incremental manner, block after block. Soon afterward, several papers showed that many modes of operation are already blockwise secure and that others can be made secure by simple modifications. In this paper, we revisit these results, by comparing possible attacks on modes of operation after the birthday bound is reached. Amusingly, in spite having essentially identical security proofs up to this bound, modes of operation in the blockwise model behave very differently than their counterparts in the regular model, once the birthday paradox bound is crossed.

Boolean functions are very important cryptographic primitives in stream or block ciphers. In this context, these functions need to satisfy good properties like high algebraic degree, nonlinearity and correlation immunity. We present here an original and efficient method to enumerate all the correlation-immune functions of a fixed Hamming weight, in particular the class of 1-resilient functions. The key idea consists in defining equivalent classes to split boolean functions along their distance from correlation-immune boolean functions. These classes, called first-order correlation classes, are built using a recursive decomposition of smaller classes. We derive from this method several algorithms to enumerate their elements and to count their cardinality. We first show that the exact number of 1-resilient boolean functions with 7 variables is 23478015754788854439497622689296 and we obtain a tight estimation of their number with 8 variables, between 4 1067and 5.6 1068. We then present a general lower bound for the number of 1-resilient boolean functions and improve Schneider's upper bound. We also propose a general lower bound for the number of k-resilient functions. Most of the bounds presented in this paper, substantially improve the best known bounds in the literature. We finally establish that the probability of a Boolean function being 1-resilient is asymptotically between (npi)n/2/2n2-3/2n-1en-1/2.

Lossless compression is studied for pairs of independent integer-valued symbols emitted by a source with a geometric probability distribution of parameter q /spl isin/ (0,1). Optimal prefix codes are described for q = 1/2/sup k/ (k > 1) and q = 1/k/spl radic/2 (k > 0). The codes described differ from previously characterized cases related to the geometric distribution in that their corresponding trees are of unbounded width, and in that an infinite set of distinct optimal codes is required to cover any interval (0, /spl epsi/), /spl epsi/ > 0, of values of q.

Lossless compression is studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter q, 0k(k > 1) and q = 1/knthroot2 (k > 0). These codes retain some of the low-complexity and low-latency advantage of symbol by symbol coding of geometric distributions, which is widely used in practice, while improving on the inherent redundancy of the approach. From a combinatorial standpoint, the codes described differ from previously characterized cases related to the geometric distribution in that their corresponding trees are of unbounded width, and in that an infinite set of distinct optimal codes is required to cover any interval (0,epsi), epsi > 0, of values of q

This paper studies the distribution of individual displacements for the standard and the Robin Hood linear probing hashing algorithms. When the a table of size m has n elements, the distribution of the search cost of a random element is studied for both algorithms. Specifically, exact distributions for fixed m and n are found as well as when the table is α-full, and α strictly smaller than 1. Moreover, for full tables, limit laws for both algorithms are derived.

We consider open addressing hashing and implement it by using the Robin Hood strategy; that is, in case of collision, the element that has traveled the farthest can stay in the slot. We hash $\sim \alpha n$ elements into a table of size n where each probe is independent and uniformly distributed over the table, and $\alpha < 1$ is a constant. Let $M_n$ be the maximum search time for any of the elements in the table. We show that with probability tending to one, $M_n \in [ \log_2 \log n + \sigma, \log_2 \log n + \tau ]$ for some constants $\sigma, \tau$ depending upon $\alpha$ only. This is an exponential improvement over the maximum search time in case of the standard FCFS (firstcome first served) collision strategy and virtually matches the performance of multiple-choice hash methods.