Decoding random linear codes is a well studied problem with many applications in complexity theory and cryptography. The security of almost all coding and LPN/LWE-based schemes relies on the assumption that it is hard to decode random linear codes. Recently, there has been progress in improving the running time of the best decoding algorithms for binary random codes. The ball collision technique of Bernstein, Lange and Peters lowered the complexity of Stern's information set decoding algorithm to 2 0.0556n. Using representations this bound was improved to 2 0.0537n by May, Meurer and Thomae. We show how to further increase the number of representations and propose a new information set decoding algorithm with running time 2 0.0494n .

Decoding random linear codes is a fundamental problem in complexity theory and lies at the heart of almost all code-based cryptography. The best attacks on the most prominent code-based cryptosystems such as McEliece directly use decoding algorithms for linear codes. The asymptotically best decoding algorithm for random linear codes of length n was for a long time Stern's variant of information-set decoding running in timeÕ 2 0.05563n. Recently, Bernstein, Lange and Peters proposed a new technique called Ball-collision decoding which offers a speed-up over Stern's algorithm by improving the running time toÕ 2 0.05558n. In this paper, we present a new algorithm for decoding linear codes that is inspired by a representation technique due to Howgrave-Graham and Joux in the context of subset sum algorithms. Our decoding algorithm offers a rigorous complexity analysis for random linear codes and brings the time complexity down toÕ 2 0.05363n .

Abstract. Let pk = (N , e) be an RSA public key with corresponding secret key sk = (p, q, d , dp , dq , q −1 p ). Assume that we obtain partial error-free information of sk, e.g., assume that we obtain half of the most significant bits of p. Then there are well-known algorithms to recover the full secret key. As opposed to these algorithms that allow for correcting erasures of the key sk, we present for the first time a heuristic probabilistic algorithm that is capable of correcting errors in sk provided that e is small. That is, on input of a full but error-prone secret key sk we reconstruct the original sk by correcting the faults.More precisely, consider an error rate of δ ∈ [0,), where we flip each bit in sk with probability δ resulting in an erroneous key sk. Our Las-Vegas type algorithm allows to recover sk from sk in expected time polynomial in log N with success probability close to 1, provided that δ < 0.237. We also obtain a polynomial time Las-Vegas factorization algorithm for recovering the factorization (p, q) from an erroneous version with error rate δ < 0.084.

Abstract. This paper introduces a subset-sum algorithm with heuristic asymptotic cost exponent below 0.25. The new algorithm combines the 2010 Howgrave-Graham-Joux subset-sum algorithm with a new streamlined data structure for quantum walks on Johnson graphs.

Although crocodilians have attracted enormous attention in other research fields, from the cytogenetic point of view, this group remains understudied. Here, we analyzed the karyotypes of eight species formally described from the Alligatoridae family using differential staining, fluorescence in situ hybridization with rDNA and repetitive motifs as a probe, whole chromosome painting (WCP), and comparative genome hybridization. All Caimaninae species have a diploid chromosome number (2n) 42 and karyotypes dominated by acrocentric chromosomes, in contrast to both species of Alligatorinae, which have 2n = 32 and karyotypes that are predominantly metacentric, suggesting fusion/fission rearrangements. Our WCP results supported this scenario by revealing the homeology of the largest metacentric pair present in both Alligator spp. with two smaller pairs of acrocentrics in Caimaninae species. The clusters of 18S rDNA were found on one chromosome pair in all species, except for Paleosuchus spp., which possessed three chromosome pairs bearing these sites. Similarly, comparative genomic hybridization demonstrated an advanced stage of sequence divergence among the caiman genomes, with Paleosuchus standing out as the most divergent. Thus, although Alligatoridae exhibited rather low species diversity and some level of karyotype stasis, their genomic content indicates that they are not as conserved as previously thought. These new data deepen the discussion of cytotaxonomy in this family.