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Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.25
We present a novel parallel one-more signature forgery against blind Okamoto-Schnorr and blind Schnorr signatures in which an attacker interacts some times with a legitimate signer and produces from these interactions signatures. Security against the new attack requires that the following ROS-problem is intractable: find an overdetermined, solvable system of linear equations modulo with random inhomogenities (right sides). There is an inherent weakness in the security result of POINTCHEVAL AND STERN. Theorem 26 [PS00] does not cover attacks with 4 parallel interactions for elliptic curves of order 2200. That would require the intractability of the ROS-problem, a plausible but novel complexity assumption. Conversely, assuming the intractability of the ROS-problem, we show that Schnorr signatures are secure in the random oracle and generic group model against the one-more signature forgery.
Let G be a group of prime order q with generator g. We study hardcore subsets H is include in G of the discrete logarithm (DL) log g in the model of generic algorithms. In this model we count group operations such as multiplication, division while computations with non-group data are for free. It is known from Nechaev (1994) and Shoup (1997) that generic DL-algorithms for the entire group G must perform p2q generic steps. We show that DL-algorithms for small subsets H is include in G require m/ 2 + o(m) generic steps for almost all H of size #H = m with m <= sqrt(q). Conversely, m/2 + 1 generic steps are su±cient for all H is include in G of even size m. Our main result justifies to generate secret DL-keys from seeds that are only 1/2 * log2 q bits long.
Public key signature schemes are necessary for the access control to communication networks and for proving the authenticity of sensitive messages such as electronic fund transfers. Since the invention of the RSA scheme by Rivest, Shamir and Adleman (1978) research has focused on improving the e±ciency of these schemes. In this paper we present an efficient algorithm for generating public key signatures which is particularly suited for interactions between smart cards and terminals.
The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short nonzero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463 . . . in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408 . . . if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.
Let b1, . . . , bm 2 IRn be an arbitrary basis of lattice L that is a block Korkin Zolotarev basis with block size ¯ and let ¸i(L) denote the successive minima of lattice L. We prove that for i = 1, . . . ,m 4 i + 3 ° 2 i 1 ¯ 1 ¯ · kbik2/¸i(L)2 · ° 2m i ¯ 1 ¯ i + 3 4 where °¯ is the Hermite constant. For ¯ = 3 we establish the optimal upper bound kb1k2/¸1(L)2 · µ3 2¶m 1 2 1 and we present block Korkin Zolotarev lattice bases for which this bound is tight. We improve the Nearest Plane Algorithm of Babai (1986) using block Korkin Zolotarev bases. Given a block Korkin Zolotarev basis b1, . . . , bm with block size ¯ and x 2 L(b1, . . . , bm) a lattice point v can be found in time ¯O(¯) satisfying kx vk2 · m° 2m ¯ 1 ¯ minu2L kx uk2.
Parallel FFT-hashing
(1994)
We propose two families of scalable hash functions for collision resistant hashing that are highly parallel and based on the generalized fast Fourier transform (FFT). FFT hashing is based on multipermutations. This is a basic cryptographic primitive for perfect generation of diffusion and confusion which generalizes the boxes of the classic FFT. The slower FFT hash functions iterate a compression function. For the faster FFT hash functions all rounds are alike with the same number of message words entering each round.
We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2^((n+2)/4) best possible. Given a real vector x=(x_1,...,x_{n-1},1) in R^n this CFA generates a sequence of vectors (p_1^(k),...,p_{n-1}^(k),q^(k)) in Z^n, k=1,2,... with increasing integers |q^{(k)}| satisfying for i=1,...,n-1 | x_i - p_i^(k)/q^(k) | <= 2^((n+2)/4) sqrt(1+x_i^2) |q^(k)|^(1+1/(n-1)) By a theorem of Dirichlet this bound is best possible in that the exponent 1+1/(n-1) can in general not be increased.