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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 call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/N finds a highly regular nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no short relation m~ of length less than /spl alpha//2 exists for points x~ within half the x'-distance from x. The integer relation m for x' is for random x up to an average factor 2/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x.
We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of theL3-algorithm of Lenstra, Lenstra, Lovász (1982). We present a variant of theL3-algorithm with "deep insertions" and a practical algorithm for block Korkin—Zolotarev reduction, a concept introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel one-more-decyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts.
We introduce algorithms for lattice basis reduction that are improvements of the famous L3-algorithm. If a random L3-reduced lattice basis b1,b2,...,bn is given such that the vector of reduced Gram-Schmidt coefficients ({µi,j} 1<= j< i<= n) is uniformly distributed in [0,1)n(n-1)/2, then the pruned enumeration finds with positive probability a shortest lattice vector. We demonstrate the power of these algorithms by solving random subset sum problems of arbitrary density with 74 and 82 many weights, by breaking the Chor-Rivest cryptoscheme in dimensions 103 and 151 and by breaking Damgard's hash function.
We call a distribution on n bit strings (", e) locally random, if for every choice of e · n positions the induced distribution on e bit strings is in the L1 norm at most " away from the uniform distribution on e bit strings. We establish local randomness in polynomial random number generators (RNG) that are candidate one way functions. Let N be a squarefree integer and let f1, . . . , f be polynomials with coe±- cients in ZZN = ZZ/NZZ. We study the RNG that stretches a random x 2 ZZN into the sequence of least significant bits of f1(x), . . . , f(x). We show that this RNG provides local randomness if for every prime divisor p of N the polynomials f1, . . . , f are linearly independent modulo the subspace of polynomials of degree · 1 in ZZp[x]. We also establish local randomness in polynomial random function generators. This yields candidates for cryptographic hash functions. The concept of local randomness in families of functions extends the concept of universal families of hash functions by Carter and Wegman (1979). The proofs of our results rely on upper bounds for exponential sums.
