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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)
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)
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)
It is widely believed to take exponential time to find a solution of a system of random multivariate polynomials because of the NP-completeness of such a task. On the other hand, in most of multivariate public key cryptosystems proposed so far, the computational complexity of cryptanalysis is apt to be polynomial time due to the trapdoor structure. In this(More)
—The statistical mechanical interpretation of algorith-mic information theory (AIT, for short) was introduced and developed by our former works [K. we introduced the notion of thermodynamic quantities, such as partition function Z(T), free energy F (T), energy E(T), and statistical mechanical entropy S(T), into AIT. We then discovered that, in the(More)