A proper mathematical formulation of the Monte Carlo method is presented. Based on it, a Monte Carlo integration method called Random Weyl sampling is proved to be secure, i.e., it is justifiable in rigorous mathematics.

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We consider a family of ergodic transformations on the real line R preserving Cauchy laws. A dualistic nature between the ergodic transformation and the associated transformation of the scale parameter of a Cauchy law is proven to be hold, which provides a systematic view of explicit mixing property with the ergodic transformation having the Cauchy law as the limiting distribution.

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We perform an entropy test, combined with a diagnostic test for the degree of visible determinism, for the complexity in chaotic time series. A time series is coarse-grained into a binary sequence, partitioned into segments of D binary digits, and transformed into a string of “alphabets” binary-coded in D bits. Using the probability density function estimated from the histogram representing the frequencies of appearance of the 2^D alphabets in the string, we calculate the information entropy referred to as the string entropy. Case studies are conducted for numerical time series generated by the logistic map, the tent map, the Lorenz equations, and the augmented Lorenz equations. We discuss the performances of the time series as sequences of pseudorandom numbers for chaotic cryptography.

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Vector Stream Cipher (VSC) is a stream cipher based on the chaos theory. The algorithm for generating stream keys is very simple and the encryption speed is very fast. Some theoretical attacks for VSC have been reported so far since the invention of VSC in 2004. In this paper, we improve the security of VSC and design a new cipher system “Vector Stream Cipher 2.0” so that the theoretical attacks cannot work. We show that the encryption speed of VSC 2.0 keeps more than 85% of that of VSC, and key-stream of VSC 2.0 has good randomness. The main result of this paper is that our proposed VSC 2.0 is shown to have provable security for attacks with linear masking. Because there is few cryptography based on the chaos theory which has proven security, VSC 2.0 is a rare example.

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A β encoder is an analog-to-digital (A/D) converter, proposed by Daubechies et al. in 2002, that outputs a truncated sequence of β expansion of an input value x. It is known that the conventional pulse code modulation (PCM) that outputs the binary expansion of x is sensitive to the offset of the threshold voltage, while a β encoder is robust to such an offset. We propose an algorithm that calculates the binary expansion of an interval that is identified by an output sequence from a β encoder. Such a method is referred to as a β-ary to binary converter. We generate sequences of random numbers, using a hardware β encoder followed by the β-ary to binary converter. The randomness of the generated binary random numbers is verified by the National Institute of Standards and Technology (NIST) statistical test suite.

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