Cryptographically Secure Pseudorandom Number Generator - Requirements

Requirements

The requirements of an ordinary PRNG are also satisfied by a cryptographically secure PRNG, but the reverse is not true. CSPRNG requirements fall into two groups: first, that they pass statistical randomness tests; and secondly, that they hold up well under serious attack, even when part of their initial or running state becomes available to an attacker.

• Every CSPRNG should satisfy the next-bit test. That is, given the first k bits of a random sequence, there is no polynomial-time algorithm that can predict the (k+1)th bit with probability of success better than 50%. Andrew Yao proved in 1982 that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness.

• Every CSPRNG should withstand "state compromise extensions". In the event that part or all of its state has been revealed (or guessed correctly), it should be impossible to reconstruct the stream of random numbers prior to the revelation. Additionally, if there is an entropy input while running, it should be infeasible to use knowledge of the input's state to predict future conditions of the CSPRNG state.

Example: If the CSPRNG under consideration produces output by computing bits of π in sequence, starting from some unknown point in the binary expansion, it may well satisfy the next-bit test and thus be statistically random, as π appears to be a random sequence. (This would be guaranteed if π is a normal number, for example.) However, this algorithm is not cryptographically secure; an attacker who determines which bit of pi (i.e. the state of the algorithm) is currently in use will be able to calculate all preceding bits as well.

Most PRNGs are not suitable for use as CSPRNGs and will fail on both counts. First, while most PRNGs outputs appear random to assorted statistical tests, they do not resist determined reverse engineering. Specialized statistical tests may be found specially tuned to such a PRNG that shows the random numbers not to be truly random. Second, for most PRNGs, when their state has been revealed, all past random numbers can be retrodicted, allowing an attacker to read all past messages, as well as future ones.

CSPRNGs are designed explicitly to resist this type of cryptanalysis.