In this paper, we introduce two new cryptographic primitives: functional digital signatures and functional pseudorandom functions.In a functional signature scheme, in addition to a master signing key that can be used to sign any message, there are signing keys for a function f , which allow one to sign any message in the range of f . As a special case, this implies the ability to generate keys for predicates P , which allow one to sign any message m, for which P (m) = 1.We show applications of functional signatures to constructing succinct non-interactive arguments and delegation schemes. We give several general constructions for this primitive based on different computational hardness assumptions, and describe the trade-offs between them in terms of the assumptions they require and the size of the signatures.In a functional pseudorandom function, in addition to a master secret key that can be used to evaluate the pseudorandom function F on any point in the domain, there are additional secret keys for a function f , which allow one to evaluate F on any y for which there exists an x such that f (x) = y. As a special case, this implies pseudorandom functions with selective access, where one can delegate the ability to evaluate the pseudorandom function on inputs y for which a predicate P (y) = 1 holds. We define and provide a sample construction of a functional pseudorandom function family for prefix-fixing functions.This work appeared in part as the Master Thesis of Ioana Ivan filed May 22 at MIT. We note that independently the notion of pseudorandom functions with selective access was studied by BonehWaters under the name of constrained pseudorandom functions

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Khanna and Sudan [2] studied a natural model of continuous time channels where signals are corrupted by the effects of both noise and delay, and showed that, surprisingly, in some cases both are not enough to prevent such channels from achieving unbounded capacity. Inspired by their work, we consider channels that model continuous time communication with adversarial delay errors. The sender is allowed to subdivide time into an arbitrarily large number M of micro-units in which binary symbols may be sent, but the symbols are subject to unpredictable delays and may interfere with each other. We model interference by having symbols that land in the same micro-unit of time be summed, and we study k-interference channels, which allow receivers to distinguish sums up to the value k. We consider both a channel adversary that has a limit on the maximum number of steps it can delay each symbol, and a more powerful adversary that only has a bound on the average delay.We give precise characterizations of the threshold between finite and infinite capacity depending on the interference behavior and on the type of channel adversary: for max-bounded delay, the threshold is at D max = Θ (M log (min{k, M })), and for average bounded delay the threshold is at Davg = Θ M min{k, M } .