We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G, ·), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted.Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t < n/2, but it requires exponen- * This paper is an extended version of [13,28]. © International Association for Cryptologic Research 2011 558 Y. Desmedt et al. tial communication complexity O( 2t+1 t 2 · N g ) group elements and round complexity O( 2t+1 t · N g ), for a G-circuit of size N g . Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity Poly(n) · N g for any t ∈ O(n 1− ) where is any positive constant. For our third protocol, there is a probability δ (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t < n/2. It has communication complexity O(n 5.056 (n + log δ −1 ) 2 · N g ) and the number of rounds is O(n 2.528 · (n + log δ −1 ) · N g ).

We consider the stream authentication problem when an adversary has the ability to drop, reorder or inject data in the network. We propose a coding approach for multicast stream authentication using the list-decoding property of Reed-Solomon codes. We divide the data to be authenticated into a stream of packets and associate a single trapdoor hash collision for every λn packets where λ and n are predesignated parameters. Our scheme, which is also joinable at the boundary of any n-packet block, can be viewed as an extension of Lysyanskaya, Tamassia and Triandopoulos's technique in which λ = 1. We show that by choosing λ and n appropriately, our scheme outperforms theirs in time spent for processing data at the sender and receiver. Our approach relies on the dispersion process as SAIDA and eSAIDA. Assuming that we use RSA for signing and SHA-256 for hashing, we give an approximation of the proportion of extra packets per block which could be processed via our technique with respect to the previous scheme. As example when we process λ = 1000 blocks of 2650 64-byte-packets, the gain of our scheme with respect to Lysyanskaya et al.'s is about 30%.

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