Advances in Cryptology - EUROCRYPT 2010: 29th Annual by Henri Gilbert

By Henri Gilbert

This booklet constitutes the refereed complaints of the twenty ninth Annual foreign convention at the conception and purposes of Cryptographic strategies, EUROCRYPT 2010, hung on the French Riviera, in May/June 2010. The 33 revised complete papers awarded including 1 invited lecture have been conscientiously reviewed and chosen from 188 submissions. The papers deal with all present foundational, theoretical and study points of cryptology, cryptography, and cryptanalysis in addition to complex purposes. The papers are geared up in topical sections on cryptosystems; obfuscation and part channel safety; 2-party protocols; cryptanalysis; computerized instruments and formal equipment; versions and proofs; multiparty protocols; hash and MAC; and foundational primitives.

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From the Θ rational numbers {ai }Θ i=1 , generate other n + 1 rational numbers {wj }nj=0 , each with less than n bits of precision, such that j wj = i ai (mod 2). 3. Output c∗ − ( j wj ) mod 2. 40 M. van Dijk et al. The first step can be performed with a 1-level sub-circuit of multiplication gates. However, the second and third steps require more complicated sub-circuits. k The problem of using a shallow boolean circuit to compute the sum i=1 ri of k rational numbers in binary representation is well-studied.

Using this technique, we can compute the entire polynomial Pσ (z) with complexity t · polylog(t). Remark 6. Note that our first circuit implementation of the procedure from above is not “shallow”. , up to degree 2i ), then by Lemma 3 it is a permitted circuit. 3 Security of the Squashed Scheme Putting the hint y in the public key induces another computational assumption, related to the sparse subset sum problem (SSSP) used by Gentry [5], and studied previously (sometimes under the name “low-weight” knapsack) in the context of server-aided cryptography [16] and in connection to the Chor-Rivest cryptosystem [18].

Set yi = ui /2κ and y = i∈S ui = xp {y1 , . . , yΘ }. Hence each yi is a positive number smaller than two, with κ bits of precision after the binary point. Also, [ i∈S yi ]2 = (1/p) − Δp for some |Δp | < 2−κ . Output the secret key sk = s and public key pk = (pk∗ , y). Encrypt and Evaluate. , an integer). Then for i ∈ {1, . . , Θ}, set zi ← [c∗ · yi ]2 , keeping only n = log θ + 3 bits of precision after the binary point for each zi . Output both c∗ and z = z 1 , . . , zΘ . Decrypt. 2. We proved that our somewhat homomorphic scheme was correct for the set C(PE ) of circuit that compute permitted polynomials, and we now show that this is true also of the modified scheme.

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