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We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D > 0. Chaitin's halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through… (More)

A theory of one-tape linear-time Turing machines is quite different from its polynomial-time counterpart. This paper discusses the computational complexity of one-tape Turing machines of various machine types (deterministic, nondeterministic, reversible, alternating, prob-abilistic, counting, and quantum Turing machines) that halt in time O(n), where the… (More)

We develop a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities, such as free energy, energy, statistical mechanical entropy, and specific heat, into algorithmic information theory. We investigate the properties of these quantities by means of program-size complexity from the point… (More)

In 1975 Chaitin introduced his Ω number as a concrete example of random real. The real Ω is defined based on the set of all halting inputs for an optimal prefix-free machine U , which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the… (More)

We proposed the concept, piece in hand (soldiers in hand) matrix and have developed the framework based on the concept so far. The piece in hand matrix is a general concept which can be applicable to any type of multivariate public key cryptosystems to enhance their security. In this paper, we make improvements in the PH matrix method as follows. (i) In the… (More)

In this paper we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding. Especially, it is discovered… (More)

A New digital signature scheme based on Stepwise Triangular Scheme (STS) is proposed. The proposed trapdoor has resolved the vulnerability of STS and secure against both Gröbner Bases and Rank Attacks. In addition, as a basic trapdoor, it is more efficient than the existing systems. With the efficient implementation, the Multivariate Public Key… (More)

, we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T , such as free energy F (T), energy E(T), and statistical mechanical entropy S(T), into the theory. These quantities are real functions of real argument T > 0. We then discovered that, in the… (More)

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol. 5407, pp. 422–440, 2009], where we introduced the notion of thermodynamic… (More)