Pseudo-random number generation with β-encoders

The β-encoder is an analog circuit that converts an input signal x ∈ [0,1] into a finite bit stream {b_i}. The bits {b_i} are correlated and therefore are not immediately suitable for random number generation, but they can be used to generate bits {a_i} that are (nearly) uniformly distributed. In this paper, we study two such methods. In the first part the bits {a_i} are defined as the digits of the base-2 representation of the original input x. Under the assumption that there is no noise in the amplifier we then study a question posed by Jitsumatsu and Matsumura on how many bits b₁, …,b_m are needed to correctly determine the first n bits a₁, …, a_n. In the second part, we show this method fails for random amplification factors. Nevertheless, even in this case, nearly uniformly distributed bits can still be generated from b₁, …, b_m using modern cryptographic techniques.