By Henri Gilbert
This e-book constitutes the refereed lawsuits of the twenty ninth Annual overseas convention at the thought and functions of Cryptographic suggestions, EUROCRYPT 2010, hung on the French Riviera, in May/June 2010. The 33 revised complete papers provided including 1 invited lecture have been conscientiously reviewed and chosen from 188 submissions. The papers deal with all present foundational, theoretical and study facets of cryptology, cryptography, and cryptanalysis in addition to complicated functions. The papers are geared up in topical sections on cryptosystems; obfuscation and facet channel defense; 2-party protocols; cryptanalysis; automatic instruments and formal tools; types and proofs; multiparty protocols; hash and MAC; and foundational primitives.
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Extra info for Advances in Cryptology - EUROCRYPT 2010: 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, French Riviera, ... Computer Science / Security and Cryptology)
Given a sample (a, b) ← As,ψ , the transformation produces a sample (a , b ) = (a + v, b + vg/q) ∈ Rq × T, On Ideal Lattices and Learning with Errors over Rings 21 where v ∈ Rq is uniformly random modulo qi and is 0 modulo the other qj . First, notice that since a is uniformly distributed in Rq , so is a . Next, condition on any ﬁxed value of a . , it is uniformly random and independent modulo qj for all j < i, and is 0 modulo all the remaining qj . We consider two cases. First, assume that g = s ∈ Rq∨i .
4. For each zi with bi = 0, replace zi by zi ← (zi − parity(zi ))/2. ) Observe that when p rp (zi ), subtracting the parity bit does not change the quotient with respect to p, only the remainder. That is, qp (zi − parity(zi )) = qp (zi ). It follows that when we set zi ← (zi − parity(zi ))/2 in line 4 (where we know that qp (zi ) is even), we get qp (zi ) = qp (zi )/2 and rp (zi ) = rp (zi ) − parity(zi ) /2. We now show that the noise in z1 , z2 never grows too large in this process. Clearly, setting zi ← (zi − parity(zi ))/2 in line 4 we have |rp (zi )| ≤ (|rp (zi )| + 1)/2 ≤ |rp (zi )|.
Namely, we assume that given two smooth numbers p1 , p2 as above and given N such that one of the pi ’s divides φ(N ), it is hard to determine which of the two pi ’s divides φ(N ). In the full version we describe this optimization in more details, and provide a proof of security for it under this φ-hiding variant. 4 Security of the Somewhat Homomorphic Scheme We reduce the security of the scheme from Section 3 to the hardness of the approximate-gcd problem. Namely, given a set of integers x0 , x1 , .