Abstract. Secure computation consists of protocols for secure arithmetic: secret values are added and multiplied securely by networked processors. The striking feature of secure computation is that security is maintained even in the presence of an adversary who corrupts a quorum of the processors and who exercises full, malicious control over them. One of the fundamental primitives at the heart of secure computation is secret-sharing. Typically, the required secret-sharing techniques build on Shamir's scheme, which can be viewed as a cryptographic twist on the Reed-Solomon error correcting code. In this work we further the connections between secure computation and error correcting codes. We demonstrate that threshold secure computation in the secure channels model can be based on arbitrary codes. For a network of size n, we then show a reduction in communication for secure computation amounting to a multiplicative logarithmic factor (in n) compared to classical methods for small, e.g., constant size fields, while tolerating t < ( We also present a new method for constructing high information rate ramp schemes based on arbitrary codes, and in particular we give a new construction based on algebraic geometry codes.

Abstract. The problem of perfectly secure message transmission concerns two synchronized non-faulty processors sender (S) and receiver (R) that are connected by a synchronous network of n ≥ 2t + 1 noiseless 2-way communication channels. Their goal is to communicate privately and reliably, despite the presence of an adversary that may actively corrupt at most t of those channels. These properties should hold information theoretically and without error. We propose an asymptotically optimal solution for this problem. The proposed protocol consists of two communication rounds, and a total of O( n) bits are exchanged in order to transmit a message of bits. Earlier, at CRYPTO 2004, an equally optimal solution has been claimed. However, we give a counter-example showing that their result is not perfectly reliable. The flaw seems to be fundamental and non-trivial to repair. Our approach is overall entirely different, yet it also makes essential use of their neat communication efficient technique for reliably transmitting conflict graphs. What distinguishes our approach from previous ones is a technique that allows to identify all actively corrupted channels, initially trading it off against privacy. A perfectly secure and reliable secret key is then distilled by privacy amplification.

Abstract. In this work we introduce a novel paradigm for the construction of ramp schemes with strong multiplication that allows the secret to be chosen in an extension field, whereas the shares lie in a base field. When applied to the setting of Shamir's scheme, for example, this leads to a ramp scheme with strong multiplication from which protocols can be constructed for atomic secure multiplication with communication equal to a linear number of field elements in the size of the network. This is also achieved by the results from Cramer, Damgaard and de Haan from EUROCRYPT 2007. However, our new ramp scheme has an improved privacy bound that is essentially optimal and leads to a significant mathematical simplification of the earlier results on atomic secure multiplication.As a result, by considering high degree rational points on algebraic curves, this can now be generalized to algebraic geometric ramp schemes with strong multiplication over a constant size field, which in turn leads to low communication atomic secure multiplication where the base field can now be taken constant, as opposed to earlier work.

We provide a generic framework that, with the help of a preprocessing phase that is independent of the inputs of the users, allows an arbitrary number of users to securely outsource a computation to two non-colluding external servers. Our approach is shown to be provably secure in an adversarial model where one of the servers may arbitrarily deviate from the protocol specification, as well as employ an arbitrary number of dummy users.We use these techniques to implement a secure recommender system based on collaborative filtering that becomes more secure, and significantly more efficient than previously known implementations of such systems, when the preprocessing efforts are excluded. We suggest different alternatives for preprocessing, and discuss their merits and demerits.

Abstract. We consider the standard secure multi-party multiplication protocol due to M. Rabin. This protocol is based on Shamir's secret sharing scheme and it can be viewed as a practical variation on one of the central techniques in the foundational results of Ben-Or, Goldwasser, and Wigderson and Chaum, Crépeau, and Damgaard on secure multi-party computation. Rabin's idea is a key ingredient to virtually all practical protocols in threshold cryptography. Given a passive t-adversary in the secure channels model with synchronous communication, for example, secure multiplication of two secretshared elements from a finite field K based on this idea uses one communication round and has the network exchange O(n 2 ) field elements, if t = Θ(n) and t < n/2 and if n is the number of players. This is because each of O(n) players must perform Shamir secret sharing as part of the protocol. This paper demonstrates that under a few restrictions much more efficient protocols are possible; even at the level of a single multiplication. We demonstrate a twist on Rabin's idea that enables one-round secure multiplication with just O(n) bandwidth in certain settings, thus reducing it from quadratic to linear. The ideas involved can additionally be employed in the evaluation of arithmetic circuits, where under appropriate circumstances similar efficiency gains can be obtained.