By Dario Catalano, Ronald Cramer, Ivan Damgard, Giovanni Di Crescenzo, David Pointcheval, Tsuyoshi Takagi
The goal of this article is to regard chosen themes of the topic of up to date cryptology, established in 5 rather self sufficient yet comparable topics: effective dispensed computation modulo a shared mystery, multiparty computation, smooth cryptography, provable defense for public key schemes, and effective and safe public-key cryptosystems.
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Notice that the ≈ n least signiﬁcant bits of c do not inﬂuence the computation of the quotient. For this reason we could eliminate these bits from c using the truncation algorithm described in Figure 2. Denoting with c the “truncated” c, one can compute the required d as c − c p 2−k−t+ p, which has the advantage of requiring a public modulus q of smaller size. This solution however requires a slightly more complicate analysis (more parameters have to be considered). Thus, even though reducing the size of public modulus is of primary importance for practical applications, in our context it may be preferable to describe a slightly less eﬃcient but simpler solution.
Consider for example the case of a blind signature scheme, which is useful in electronic cash systems. We can think of this as a two-party secure computation where the signer enters his private signing key sk as input, the user enters a message m to be signed, and the function f (sk, m) = (y1 , y2 ), where y1 is for the signer and is empty, and where y2 is for the user and the signature on m. Again, security means exactly what we want: the user gets the signature and nothing else, while the signer learns nothing new.
To perform this check in a distributed way one may simply compute (for each bj ) the value a − bj mod p, multiply it with a jointly generated random element and check if the obtained result is zero or not. Unfortunately, however, this solution does not quite solve the problem in our setting. e. a value that is oﬀ by some small multiple i of p from the actual solution). Eﬃcient Distributed Computation Modulo a Shared Secret 25 However since i is less (in absolute value) than 3n we can distributely compute j≤3(n+1) A = j=−3(n+1) a − bj − jp mod p and then check if A is zero or not.