By Henri Gilbert

This booklet constitutes the refereed lawsuits of the twenty ninth Annual overseas convention at the conception and functions of Cryptographic thoughts, EUROCRYPT 2010, hung on the French Riviera, in May/June 2010. The 33 revised complete papers offered including 1 invited lecture have been conscientiously reviewed and chosen from 188 submissions. The papers handle all present foundational, theoretical and study facets of cryptology, cryptography, and cryptanalysis in addition to complicated functions. The papers are prepared in topical sections on cryptosystems; obfuscation and aspect channel safeguard; 2-party protocols; cryptanalysis; computerized instruments and formal tools; types and proofs; multiparty protocols; hash and MAC; and foundational primitives.

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Hence we use Gentry’s technique from [5] that takes advantage of the fact that all but θ of the ai ’s are zero. Denote the bit representation of each number ai by ai,0 • ai,−1 ai,−2 . . ai,−n . n That is, ai = j=0 2−j ai,−j . The heart of this procedure is a subroutine for computing integers W−j , j = 0, 1, . . , n, where W−j is the Hamming weight of the “column” of bits (a1,−j , a2,−j , . . , aΘ,−j ) (see an illustration in Figure 1). Since at most θ of the ai ’s are nonzero, then the W−j ’s are no larger than θ, and hence can be represented by log(θ + 1) < n bits.

This is shorter than our target solution when t + 1 < γ/η. 5 On the other hand, when t is large, v likely is the shortest vector in L, but known lattice reductions algorithms will not be able to ﬁnd it eﬃciently. Specifically, as a rule of thumb, they require time roughly 2t/k to output a 2k approximation of the shortest vector. Since clearly there are √ √ exponentially (in t) many vectors in L of length at most x0 t + 1 < 2γ t + 1, which is about 2η−ρ times longer than v, we need better than a 2η−ρ approximation.

Hence the “real challenge” in constructing fully homomorphic encryption comes from the compactness property, which essentially means that the size of the ciphertext that Evaluate generates does not depend on the size of the circuit C. Definition 3 (Compact Homomorphic Encryption). The scheme E = (KeyGen, Encrypt, Decrypt, Evaluate) is compact if there exists a ﬁxed polynomial 1 Formally, C is an ensemble, parametrized by the security parameter. Fully Homomorphic Encryption over the Integers 27 bound b(λ) so that for any key-pair (sk, pk) output by KeyGen(λ), any circuit C and any sequence of ciphertext c = c1 , .