By Henri Gilbert

ISBN-10: 3642131891

ISBN-13: 9783642131899

This booklet constitutes the refereed lawsuits of the twenty ninth Annual foreign convention at the conception and functions of Cryptographic recommendations, 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 handle all present foundational, theoretical and learn points of cryptology, cryptography, and cryptanalysis in addition to complicated purposes. The papers are equipped in topical sections on cryptosystems; obfuscation and facet channel safeguard; 2-party protocols; cryptanalysis; computerized instruments and formal equipment; types and proofs; multiparty protocols; hash and MAC; and foundational primitives.

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**Example text**

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.

In the full version we prove that for every secret key p and for all but a negligible fraction of the public keys (as generated by KeyGen for the secret key p), the procedure that B uses to generate ciphertexts in line 3 of the subroutine Learn-LSB produces a distribution which is statistically close to the ciphertext distribution of the scheme. This lets us analyze the success probability of B, as follows: Let P be the set of odd integers in [2η−1 , 2η ) for which A has more than ε/2 advantage def P = p ∈ [2η−1 , 2η ) : advantage(A) conditioned on sk = p is at least ε/2 A counting argument shows that the fraction of odd integers from [2η−1 , 2η ) that are in P is at least ε/2.

Xτ . Encrypt(pk, m ∈ {0, 1}). Choose a random subset S ⊆ {1, 2, . . , τ } and a random integer r in (−2ρ , 2ρ ), and output c ← m + 2r + 2 i∈S xi x0 . Evaluate(pk, C, c1 , . . , ct ). Given the (binary) circuit CE with t inputs, and t ciphertexts ci , apply the (integer) addition and multiplication gates of CE to the ciphertexts, performing all the operations over the integers, and return the resulting integer. Decrypt(sk, c). Output m ← (c mod p) mod 2. Remark 1. Recall that (c mod p) = c − p · c/p , and as p is odd we can instead decrypt using the formula m ← [c − c/p ]2 = (c mod 2) ⊕ ( c/p mod 2).

### Advances in Cryptology - EUROCRYPT 2010: 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, French Riviera, ... Computer Science Security and Cryptology) by Henri Gilbert

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