Several cryptographic primitives, especially succinct proofs of various forms, transform the satisfaction of high-level properties to the existence of a polynomial quotient between a polynomial that interpolates a set of values with a cleverly arranged divisor. Some examples are SNARKs, like Groth16, and polynomial commitments, such as KZG. Such a polynomial division naively takes O (n log n) time with Fast Fourier Transforms, and is usually the asymptotic bottleneck for these computations.

In the setting of searchable symmetric encryption (SSE), a data owner D outsources a database (or document/file collection) to a remote server E in encrypted form such that D can later search the collection at E while hiding information about the database and queries from E. Leakage to E is to be confined to well-defined forms of data-access and query patterns while preventing disclosure of explicit data and query plaintext values. Recently, Cash et al. presented a protocol, OXT, which can run arbitrary boolean queries in the SSE setting and which is remarkably efficient even for very large databases.

A new technique to lower-bound the minimum distance of certain types of quasi-cyclic codes with large dimension by reducing the problem to lower-bounding the minimum distance of a few significantly smaller codes has been developed. These codes have the property that they have extremely efficient software encoders. Using this technique, it is proved that a code which is similar to the SHA-1 (Secure Hash Algorithm, to be explained shortly) message expansion code has minimum distance 82, and that too in just the last 64 of the 80 expanded words. In fact, the proposed code has much greater distance than that of SHA-1 code, which makes our proposed hashing scheme robust against cryptographic attacks. The technique is further used to find the minimum weight of the SHA-1 code itself (25 in last 60 words), which was an open problem. Estimating minimum distance of a code given by its parity-check matrix is well known to be a hard problem. Our technique is expected to be helpful in estimating minimum distance of similar codes as well as in designing future practical cryptographic hash functions.

We present an efficient randomized algorithm to test if a given function f : F/sub p/ /sup n/ /spl rarr/ F/sub p/ (where p is a prime) is a low-degree polynomial. This gives a local test for generalized Reed-Muller codes over prime fields. For a given integer t and a given real /spl epsiv/ > 0, the algorithm queries f at 1//spl epsiv/ + t/spl middot/p/sup 2r/p-1+O(1)/ points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least e from every degree t polynomial, then our algorithm rejects f with probability at least 1/2. Our result is almost optimal since any such algorithm must query f on at least /spl Omega/(1//spl epsiv/ + p/sup r+1/p-1/) points.

The complexity of testing nonemptiness of finite state automata on infinite trees is investigated. It is shown that for tree automata with the pairs (or complemented pairs) acceptance condition having m states and n pairs, nonemptiness can be tested in deterministic time (mn) O (n) ; however, it is shown that the problem is in general NP-complete (or co-NP-complete, respectively). The new nonemptiness algorithm yields exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it follows that satisfiability for the full branching time logic CTL * can be tested in deterministic double exponential time. Another consequence is that satisfiability for propositional dynamic logic (PDL) with a repetition construct (PDL-delta) and for the propositional Mu-calculus ($L\mu$) can be tested in deterministic single exponential time.

Abstract We propose a methodology for designing sound and complete proof systems for proving progress properties of parallel programs under various fairness assumptions. Our methodology begins with a branching time temporal logic formula (CTL*) formula that expresses progress under a fairness assumption. The next step obtains an equivalent fixpoint characterization of this CTL*formula in theμ-calculus. The final step uses the fixpoint characterizations to extract proof systems for proving progress under the fairness constraint. The methodology guarantees that the proof rules so obtained are sound and relatively complete in the sense of Cook.

It is shown that the propositional mu-calculus is equivalent in expressive power to finite automata on infinite trees. Since complementation is trivial in the mu-calculus, the equivalence provides a radically simplified, alternative proof of M.O. Rabin's (1989) complementation lemma for tree automata, which is the heart of one of the deepest decidability results. It is also shown how mu-calculus can be used to establish determinacy of infinite games used in earlier proofs of complementation lemma, and certain games used in the theory of online algorithms.>

The authors give a construction to determine and complement simultaneously a Buchi automaton in infinite strings, with an exponential blowup in states and a linear blowup in the number of pairs. An exponential lower bound is already known. The previous best construction was double exponential. The present result permits exponentially improved essentially optimal decision procedures for various modal logics of programs. It also gives exponentially improved conversions between various kinds of omega automata.>

We present an extensional semantics for parallel programs based on predicate transformers.We identify the meaning of a parallel program with three concepts: a set of initial conditions, a predicate transformer, called wlt, characterizing progress properties of the program, and a predicate transformer wsafe concerned with safety aspects.For a given program, these predicate transformers are defined in terms of fixpoints of recursive equations.

The computational complexity of testing nonemptiness of finite-state automata on infinite trees is investigated. It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)/sup 3n/), even though the problem is in general NP-complete. The nonemptiness algorithm is used to obtain exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it is shown that satisfiability for the full branching time logic CTL* can be tested in deterministic double exponential time. It also follows that satisfiability for propositional dynamic logic with a repetition construct (PDL-delta) and for the propositional mu-calculus (L mu ) can be tested in deterministic single exponential time.>