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Many cryptographic solutions based on pseudorandom functions (for common problems like encryption, message-authentication or challenge-response protocols) have the following feature: There is a stateful (counter based) version of the scheme that has high security, but if, to avoid the use of state, we substitute a random value for the counter, the security of the scheme drops below the birthday bound. In some situations the use of counters or other forms of state is impractical or unsafe. Can we get security beyond the birthday bound without using counters? This paper presents a paradigm for strengthening pseudorandom function usages to this end, the idea of which is roughly to use the XOR of the values of a pseudorandom function on a small number of distinct random points in place of its value on a single point. We establish two general security properties of our construction, "pseudorandomness" and "integrity", with security beyond the birthday bound. These can be applied to derive encryption schemes, and MAC schemes (based on universal hash functions), that have security well beyond the birthday bound, without the use of state and at moderate computational cost.

Stateless Evaluation of Pseudorandom Functions: Security Beyond the Birthday Barrier