Abstract. At Eurocrypt 2010 van Dijk et al. described a fully homomorphic encryption scheme over the integers. The main appeal of this scheme (compared to Gentry's) is its conceptual simplicity. This simplicity comes at the expense of a public key size inÕ(λ 10 ) which is too large for any practical system. In this paper we reduce the public key size toÕ(λ 7 ) by encrypting with a quadratic form in the public key elements, instead of a linear form. We prove that the scheme remains semantically secure, based on a stronger variant of the approximate-GCD problem, already considered by van Dijk et al.We also describe the first implementation of the resulting fully homomorphic scheme. Borrowing some optimizations from the recent GentryHalevi implementation of Gentry's scheme, we obtain roughly the same level of efficiency. This shows that fully homomorphic encryption can be implemented using simple arithmetic operations.

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Recently, NIST has selected 14 second round candidates of SHA3 competition. One of these candidates will win the competition and eventually become the new hash function standard. In TCC'04, Maurer et al introduced the notion of indifferentiability as a generalization of the concept of the indistinguishability of two systems. Indifferentiability is the appropriate notion of modeling a random oracle as well as a strong security criteria for a hash-design. In this paper we analyze the indifferentiability and preimage resistance of JH hash function which is one of the SHA3 second round candidates. JH uses a 2n bit fixed permutation based compression function and applies chopMD domain extension with specific padding.-We show under the assumption that the underlying permutations is a 2nbit random permutation, JH mode of operation with output length 2n − s bits, is indifferentiable from a random oracle with distinguisher's advantage bounded by O(q 2 σ 2 s + q 3 2 n) where σ is the total number of blocks queried by distinguisher.-We show that the padding rule used in JH is essential as there is a simple indifferentiablity distinguisher (with constant query complexity) against JH mode of operation without length padding outputting n bit digest.-We prove that a little modification (namely chopping different bits) of JH mode of operation enables us to construct a hash function based on random permutation (without any length padding) with similar bound of sponge constructions (with fixed output size) and with same efficiency.-On the other hand, we improve the preimage attack of query complexity 2 510.3 due to Mendel and Thompson. Using multicollisions in both forward and reverse direction, we show a preimage attack on JH with n = 512, s = 512 in 2 507 queries to the permutation.

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Abstract. We show that the Feistel construction with six rounds and random round functions is publicly indifferentiable from a random invertible permutation (a result that is not known to hold for full indifferentiability). Public indifferentiability (pub-indifferentiability for short) is a variant of indifferentiability introduced by Yoneyama et al. [29] and Dodis et al. [12] where the simulator knows all queries made by the distinguisher to the primitive it tries to simulate, and is useful to argue the security of cryptosystems where all the queries to the ideal primitive are public (as e.g. in many digital signature schemes). To prove the result, we introduce a new and simpler variant of indifferentiability, that we call sequential indifferentiability (seq-indifferentiability for short) and show that this notion is in fact equivalent to pub-indifferentiability for stateless ideal primitives. We then prove that the 6-round Feistel construction is seq-indifferentiable from a random invertible permutation. We also observe that sequential indifferentiability implies correlation intractability, so that the Feistel construction with six rounds and random round functions yields a correlation intractable invertible permutation, a notion we define analogously to correlation intractable functions introduced by Canetti et al. [4].