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We present an efficient randomized algorithm to test if a given function f : F n p → F p (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields. For a given integer t and a given real ǫ > 0, the algorithm queries f at O 1 ǫ + t · p 2t p−1 +1 points to determine whether f can be described… (More)

Computation Tree Logic (CTL) is one of the most syntactically elegant and computationally attractive temporal logics for branching time model checking. In this paper, we observe that while CTL can be verified in time polynomial in the size of the state space times the length of the formula, there is a large set of reachability properties which cannot be… (More)

Recently, Wang, Yin, and Yu ([WYY05b]) have used a low weight codeword in the SHA-1 message expansion to show a better than brute force method to find collisions in SHA-1. The smallest minimum weight codeword they report has a (bit) weight of 25 in the last 60 of the 80 expanded words. In this paper we show, using a computer assisted method, that this is… (More)

We define a new family of error-correcting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraic-geometric (AG) codes via a (non-obvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy [11]. Our work shows that the PV… (More)

We develop a new computer assisted technique for lower bounding the minimum distance of linear codes similar to those used in SHA-1 message expansion. Using this technique, we prove that a modified SHA-1 like code has minimum distance at least 82, and that too in just the last 64 of the 80 expanded words. Further the minimum weight in the last 60 words… (More)

We argue that if the message expansion code of SHA-1 is replaced by a linear code with a better minimum distance, then the resulting hash function is collision resistant. To support this argument, we characterize the disturbance vectors which are used to build local collision attacks as a linear code. This linear code is the xor-sum of two codes, the… (More)

We develop a new technique to lower bound the minimum distance of quasi-cyclic codes with large dimension by reducing the problem to lower bounding the minimum distance of a few significantly smaller dimensional codes. Using this technique, we prove that a code which is similar to the SHA-1 message expansion code has minimum distance at least 82, and that… (More)

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