- Next-bit test
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In cryptography and the theory of computation, the next-bit test[1] is a test against pseudo-random number generators. We say that a sequence of bits passes the next bit test for at any position i in the sequence, if an attacker knows the i first bits, he cannot predict the (i + 1)st with reasonable computational power.
Contents
Precise statement(s)
Let P be a polynomial, and S = {Sk} be a collection of sets such that Sk contains P(k)-bit long sequences. Moreover, let μk be the probability distribution of the strings in Sk.
We now define the next-bit test in two different ways.
Boolean circuit formulation
A predicting collection[2] is a collection of boolean circuits, such that each circuit has less than PC(k) gates and exactly i inputs. Let be the probability that, on input the i first bits of s, a string randomly selected in Sk with probability μk(s), the circuit correctly predicts si + 1, i.e. :
Now, we say that {Sk}k passes the next-bit test if for any predicting collection C, any polynomial Q :
Probabilistic Turing machines
We can also define the next-bit test in terms of probabilistic Turing machines, although this definition is somewhat stronger (see Adleman's theorem). Let be a probabilistic Turing machine, working in polynomial time. Let be the probability that predicts the (i + 1)st bit correctly, i.e.
We say that collection S = {Sk} passes the next-bit test if for all polynomial Q, for all but finitely many k, for all 0 < i < k:
Completeness for Yao's test
The next-bit test is a particular case of Yao's test for random sequences, and passing it is therefore a necessary condition for passing Yao's test. However, it has also been shown a sufficient condition by Yao.[1]
We prove it now in the case of probabilistic Turing machine, since Adleman has already done the work of replacing randomization with non-uniformity in his theorem. The case of boolean circuits cannot be derived from this case (since it involves deciding potentially undecidable problems), but the proof of Adleman's theorem can be easily adapted to the case of non-uniform boolean circuits families.
Let a distringuer for the probabilistic version of Yao's test, i.e. a probabilistic Turing machine, running in polynomial time, such that there is a polynomial Q such that for infinitely many k
Let . We have : Rk,0 = {0,1}P(k) and Rk,P(k) = Sk. Then, we notice that . Therefore, at least one of the should be no smaller than .
Next, we consider probability distributions μk,i and on Rk,i. Distribution μk,i is the probability distribution of choosing the i first bits in Sk with probability given by μk, and the P(k) − i remaining bits uniformly at random. We have thus :
We thus have (a simple calculus trick shows this), thus distributions μk,i + 1 and can be distinguished by . Without loss of generality, we can assume that , with R a polynomial.
This gives us a possible construction of a Turing machine solving the next-bit test : upon receiving the i first bits of a sequence, pads this input with a guess of bit l and then P(k) − i − 1 random bits, chosen with uniform probability. Then it runs , and outputs l if the result is 1, and 1 − l else.
References
- ^ a b Andrew Chi-Chih Yao. Theory and applications of trapdoor functions. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982.
- ^ Manuel Blum and Silvio Micali, How to generate cryptographically strong sequences of pseudo-random bits, in SIAM J. COMPUT., Vol. 13, No. 4, November 1984
Categories:- Pseudorandom number generators
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