K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is de ned using a generalization of A. Cobham's de nition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equivalent to the familiar poly time functions. We generalize this equivalence to type 2 functionals. This requires a de nition of the notion`poly time in the length of type 1 inputs'. The proof of this equivalence is not a simple generalization of the proof for type 1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation, and requires an analysis of the structure of oracle queries in time bounded computations. Key words. type 2 computability, polynomialtime, notational recursion, oracle Turing machine AMS subject classi cations. 68Q05,68Q15,03D65,03D20 2. A Computational Model for Functionals. Our model for type 2 computability is a generalization of the familiar multi-tape oracle Turing machine (OTM). However, we allow arbitrary type 1 functions as oracles, rather than subsets of N. Note A preliminary version of this paper appeared as 7].

The dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes which is undergoing a sequence of edge insertions and deletions, answer queries of the form q(a, b): "Is there a path between nodes a and b?" While data structures for this problem with polylogarithmic amortized time per operation have been known since the mid-1990's, these data structures have Θ(n) worst case time. In fact, no previously known solution has worst case time per operation which is o(√ n). We present a solution with worst case times O(log 4 n) per edge insertion, O(log 5 n) per edge deletion, and O(log n/ log log n) per query. The answer to each query is correct if the answer is "yes" and is correct with high probability if the answer is "no". The data structure is based on a simple novel idea which can be used to quickly identify an edge in a cutset. Our technique can be used to simplify and significantly speed up the preprocessing time for the emergency planning problem while matching previous bounds for an update, and to approximate the sizes of cutsets of dynamic graphs in timẽ O(min{|S|, |V \ S|}) for an oblivious adversary.

We resolve two long-standing open problems in distributed computation by showing that both Byzantine agreement and Leader Election can be solved in sub-exponential time in the asynchronous full information model. Surprisingly, our protocols for both problems run in only polylogarithmic time. We thus achieve a better than exponential speedup over previous results for asynchronous Byzantine agreement. In addition, to the best of our knowledge, ours is the first protocol for asynchronous full-information leader election. Our protocols work in the full information model with a non-adaptive adversary: the adversary is assumed to control up to a constant fraction of the processors, have unlimited computational power as well as access to all communications, but no access to processors' private random bits. The adversary is non-adaptive only in the sense that the corrupted processors must be chosen at the outset. Our protocols run in time that is polylogarithmic in the number of processors, n, and tolerate t < n 6+ faulty processors for any positive constant . Our protocols are Monte Carlo, succeeding with probability 1 − o(1) for Byzantine agreement, and constant probability for leader election.

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Computational Indistinguishability Logic (CIL) is a logic for reasoning about cryptographic primitives in computational models. It captures reasoning patterns that are common in provable security, such as simulations and reductions. CIL is sound for the standard model, but also supports reasoning in the random oracle and other idealized models. We illustrate the benefits of CIL by formally proving the security of the probabilistic signature scheme (PSS).