In cryptography and the theory of computation, Yao's test is a test defined by Andrew Chi-Chih Yao in 1982
^{[1]}, against pseudo-random sequences. A sequence of words passes Yao's test if an attacker with reasonable computational power cannot distinguish it from a sequence generated uniformly at random.

## Formal statement[]

### Boolean circuits[]

Let be a polynomial, and be a collection of sets of -bit long sequences, and for each , let be a probability distribution on , and be a polynomial. A predicting collection is a collection of boolean circuits of size less than . Let be the probability that on input , a string randomly selected in with probability , , i.e.

Moreover, let be the probability that on input a -bit long sequence selected uniformly at random in . We say that passes Yao's test if for all predicting collection , for all but finitely many , for all polynomial :

### Probabilistic formulation[]

As in the case of the next-bit test, the predicting collection used in the above definition can be replaced by a probabilistic Turing machine, working in polynomial time. This also yields a strictly stronger definition of Yao's test (see Adleman's theorem). Indeed, One could decide undecidable properties of the pseudo-random sequence with the non-uniform circuits described above, whereas BPP machines can always be simulated by exponential-time deterministic Turing machines.

## References[]

- ↑ Andrew Chi-Chih Yao. Theory and applications of trapdoor functions. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982.