Kohtaro Tadaki

Proposal of a signature scheme based on STS trapdoor

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 Grobner 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 Cryptosystems (MPKC) signature public key has the signature longer than the message by less than 25 %, for example.

A Statistical Mechanical Interpretation of Instantaneous Codes

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 that the temperature 1 corresponds to the average codeword length of an instantaneous code in this statistical mechanical interpretation of noiseless source coding scheme. This correspondence is also verified by the investigation using box-counting dimension. Using the notion of temperature and statistical mechanical arguments, some information-theoretic relations can be derived in the manner which appeals to intuition.

Proposal for Piece in Hand Matrix: General Concept for Enhancing Security of Multivariate Public Key Cryptosystems

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 polynomial time due to the trapdoor structure. In this paper, we introduce a new concept, piece in hand (soldiers in hand) matrix, which brings the computational complexity of cryptanalysis of multivariate public key cryptosystems close to exponential time by adding random polynomial terms to original cryptosystems. This is a general concept which can be applicable to any type of multivariate public key cryptosystems for the purpose of enhancing their security. As an implementation of the concept, we propose the linear PH matrix method with random variables. In 2003 Faugere and Joux broke the first HFE challenge (80 bits), where HFE is one of the major variants of multivariate public key cryptosystem, by computing a Grobner basis of the public key of the cryptosystem. We show, in an experimental manner, that the linear PH matrix method with random variables can enhance the security of HFE even against the Grobner basis attack. In what follows, we consider the strength of the linear PH matrix method against other possible attacks.

Fixed Point Theorems on Partial Randomness

In our former work [K. Tadaki, Local Proceedings of CiE 2008,pp. 425---434, 2008], we developed a statistical mechanicalinterpretation of algorithmic information theory by introducing thenotion of thermodynamic quantities, such as free energyF (T ), energy E (T ), andstatistical mechanical entropy S (T ), into thetheory. We then discovered that, in the interpretation, thetemperature T equals to the partial randomness of thevalues of all these thermodynamic quantities, where the notion ofpartial randomness is a stronger representation of the compressionrate by program-size complexity. Furthermore, we showed that thissituation holds for the temperature itself as a thermodynamicquantity. Namely, the computability of the value of partitionfunction Z (T ) gives a sufficient condition forT ⊆ (0,1) to be a fixed point on partial randomness.In this paper, we show that the computability of each of all thethermodynamic quantities above gives the sufficient condition also.Moreover, we show that the computability of F (T )gives completely different fixed points from the computability ofZ (T ).

Theory of One Tape Linear Time Turing Machines

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, probabilistic, counting, and quantum Turing machines) that halt in time O(n), where the running time of a machine is defined as the height of its computation tree. We also address a close connection between one-tape linear-time Turing machines and finite state automata.

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitins halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitins Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H (s) is defined as –log2m (s) without reference to the concept of program-size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitins Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed. In what follows, we introduce an operator version (s) of H (s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase Transition and Strong Predictability

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp.425–434, 2008], where we introduced the notion of thermodynamic quantities into AIT. These quantities are real functions of temperature T > 0. The values of all the thermodynamic quantities diverge when T exceeds 1. This phenomenon corresponds to phase transition in statistical mechanics. In this paper we introduce the notion of strong predictability for an infinite binary sequence and then apply it to the partition function Z(T), which is one of the thermodynamic quantities in AIT. We then reveal a new computational aspect of the phase transition in AIT by showing the critical difference of the behavior of Z(T) between T = 1 and T < 1 in terms of the strong predictability for the base-two expansion of Z(T).

Chaitin Ω Numbers and Halting Problems

Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach. , vol. 22, pp. 329---340, 1975] introduced his Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer 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 base-two expansion of Ω solve the halting problem of the optimal computer for all binary inputs of length at most n . In the present paper we investigate this property from various aspects. It is known that the base-two expansion of Ω and the halting problem are Turing equivalent. We consider elaborations of both the Turing reductions which constitute the Turing equivalence. These elaborations can be seen as a variant of the weak truth-table reduction, where a computable bound on the use function is explicitly specified. We thus consider the relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems.

Robustness of statistical mechanical interpretation of algorithmic information theory

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008], where we introduced the thermodynamic quantities into AIT. In this paper, we reveal a certain sort of the robustness of statistical mechanical interpretation of AIT. The thermodynamic quantities in AIT are originally defined based on the set of all programs, i.e., all halting inputs, for an optimal prefix-free machine, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that we can recover the original properties of the thermodynamic quantities in AIT if we replace all programs by all minimal-size programs in the definitions of the thermodynamic quantities in AIT. The results of this paper illustrate the generality and validity of the statistical mechanical interpretation of AIT.

A knapsack cryptosystem based on multiple knapsacks

In this paper, we propose a knapsack cryptosystem based on three knapsacks. Although one of the three secret knapsacks is superincreasing, the other two are non-superincreasing. On the encryption, a ciphertext is formed by multiplying the two non-superincreasing knapsacks together and then adding it to the superincreasing knapsack. Due to this structure of the cipher text, our knapsack cryptosystem is thought to be secure against all existing attacks, i.e., the low density attack and Shamir attack