Analysis and Construction of Nonlinear Correctors Used in True Random Number Generators

The problem of true random number generators (TRNGs) traces back to von Neumann’s 1951 work that aims to simulate an unbiased coin by using a biased coin with unknown probability. The core component in a TRNG is the corrector which is a post-processing function used to reduce or eliminate statistical weaknesses of physical random number generators. Note that an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -resilient function is an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -corrector. Hence, a natural question is how to construct an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -corrector which is not <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -resilient? In this paper, a framework concerning the construction of nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -correctors with algebraic degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m+1$ </tex-math></inline-formula> is proposed based on an equidistant linear code. We show that the derived correctors are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t-1)$ </tex-math></inline-formula> -resilient, but not <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -resilient. Given the importance of equidistant linear codes, we discuss how to get such a code with relatively flexible length, and how to get a pair of disjoint equidistant linear codes. In addition, the parameters comparison with linear correctors is given. It is shown that our method achieves the same correction order compared to the optimal linear method. As far as we know, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -correctors we constructed also possess the best-known correction order compared with the known nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m)$ </tex-math></inline-formula> resilient functions. The algebraic degree and nonlinearity of the constructed correctors are also analyzed. Through a pair of disjoint equidistant linear codes, the nonlinearity of the nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -correctors can be improved. The results show that our <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m,t)$ </tex-math></inline-formula> -correctors also possess the best algebraic degree and nonlinearity for fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m)$ </tex-math></inline-formula> .