Consensus problems occur in many contexts and have therefore been intensively studied in the past. In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem [1]. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Coming up with a self-stabilizing rule is easy without adversarial involvement, but we allow some T -bounded adversary to manipulate any T processes at any time. In this situation, a perfect consensus is impossible to reach, so we only require that there is a time point t and value v so that at any point after t, all but up to O(T ) processes agree on v, which we call an almost stable consensus. As we will demonstrate, there is a surprisingly simple rule for the standard message passing model that just needs O(log n log log n) time for any √ n-bounded adversary and just O(log n) time without adversarial involvement, with high probability, to reach an (almost) stable consensus from any initial state. A stable consensus is reached, with high probability, in the absence of adversarial involvement.Dagstuhl Seminar Proceedings 09371 Algorithmic Methods for Distributed Cooperative Systems

This document describes a Fully-Specified Forward Error Correction (FEC) scheme, corresponding to FEC Encoding ID 6, for the RaptorQ FEC code and its application to reliable delivery of data objects.RaptorQ codes are a new family of codes that provide superior flexibility, support for larger source block sizes, and better coding efficiency than Raptor codes in RFC 5053. RaptorQ is also a fountain code, i.e., as many encoding symbols as needed can be generated on the fly by the encoder from the source symbols of a source block of data. The decoder is able to recover the source block from almost any set of encoding symbols of sufficient cardinality --in most cases, a set of cardinality equal to the number of source symbols is sufficient; in rare cases, a set of cardinality slightly more than the number of source symbols is required.

Abstract. We present a structural attack against the Sidelnikov cryptosystem [8]. The attack creates a private key from a given public key. Its running time is subexponential and is effective if the parameters of the Reed-Muller code allow for efficient sampling of minimum weight codewords. For example, the length 2048, 3rd-order Reed-Muller code as proposed in [8] takes roughly an hour to break on a stock PC using the presented method.

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Consider the following problem: Given k = 2 q random lists of n-bit vectors, L 1 , . . . , L k , each of length m, find x 1 ∈ L 1 , . . . , x k ∈ L k such that x 1 + · · · + x k = 0, where + is the XOR operation. This problem has applications in a number of areas, including cryptanalysis, coding theory, finding shortest lattice vectors, and learning theory. The so-called k-tree algorithm, due to Wagner, solves this problem inÕ(2 q+n/(q+1) ) expected time provided the length m of the lists is large enough, specifically if m ≥ 2 n/(q+1) . In many applications, however, it is necessary to work with lists of smaller length, where the above algorithm breaks down. In this paper we generalize the algorithm to work for significantly smaller values of the list length m, all the way down to the threshold value for which a solution exists with reasonable probability. Our algorithm exhibits a tradeoff between the value of m and the running time. We also provide the first rigorous bounds on the failure probability of both our algorithm and that of Wagner.As a third contribution, we give an extension of this algorithm to the case where the vectors are not binary, but defined over an arbitrary finite field F r , and a solution to λ 1 x 1 +· · ·+λ k x k = 0 with λ i ∈ F * r and x i ∈ L i is sought.