Security analysis of pseudo-random number generators with input

Abstract

A pseudo-random number generator (PRNG) is a deterministic algorithm that produces numbers whose distribution is indistinguishable from uniform. A formal security model for PRNGs with input was proposed in 2005 by Barak and Halevi (BH). This model involves an internal state that is refreshed with a (potentially biased) external random source, and a cryptographic function that outputs random numbers from the continually internal state. In this work we extend the BH model to also include a new security property capturing how it should accumulate the entropy of the input data into the internal state after state compromise. This property states that a good PRNG should be able to eventually recover from compromise even if the entropy is injected into the system at a very slow pace, and expresses the real-life expected behavior of existing PRNG designs. Unfortunately, we show that neither the model nor the specific PRNG construction proposed by BH meet this new property, despite meeting a weaker robustness notion introduced by BH. From a practical side, we give a precise assessment of the Linux PRNGs, /dev/random and /dev/urandom. In particular, we show attacks proving that these PRNGs are not robust according to our definition, due to vulnerabilities in their entropy estimator and their internal mixing function. Finally, we propose a simple PRNG construction that is provably robust in our new and stronger adversarial model and we show that it is more efficient than the Linux PRNGs. We therefore recommend to use this construction whenever a PRNG with input is used for cryptography.