As a part of the construction of an information theory based on general probabilistic theories, we propose and investigate the several distinguishability measures and "entropies" in general probabilistic theories. As their applications, no-cloning theorems, information-disturbance theorems are reformulated, and a bound of the accessible informations is discussed in any general probabilistic theories, not resorting to quantum theory. We also propose the principle of equality for pure states which makes general probabilistic theories to be more realistic, and discuss the role of entropies as a measure of pureness.

It has been proven that the code lengths of Tardos's collusion-secure fingerprinting codes are of theoretically minimal order with respect to the number of adversarial users (pirates). However, the code lengths can be further reduced as some preceding studies have revealed. In this article we improve a recent discrete variant of Tardos's codes, and give a security proof of our codes under an assumption weaker than the original Marking Assumption. Our analysis shows that our codes have significantly shorter lengths than Tardos's codes. For example, when c = 8, our code length is about 4.94% of Tardos's code in a practical setting and about 4.62% in a certain limit case. Our code lengths for large c are asymptotically about 5.35% of Tardos's codes.Communicated by H. van Tilborg.A part of this work was presented at 17th Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-17),

Abstract. We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signedeliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs. IntroductionLet V = V ℓ be an ℓ-dimensional vector space over a field K of characteristic zero, {x 1 , . . . , x ℓ } a basis for the dual vector space V * and S := Sym(Vis homogeneous of degree p if f i is zero or homogeneous of degree p for each i.A hyperplane arrangement A (or simply an arrangement ) is a finite collection of affine hyperplanes in V . If each hyperplane in A contains the origin, we say that A is central. In this article we assume that all arrangements are central unless otherwise specified. A multiplicity m on an arrangement A is a map m : A → Z ≥0 and a pair (A, m) is called a multiarrangement. Let |m| denote the sum of the multiplicities H∈A m(H). When m ≡ 1, (A, m) is the same as the hyperplane arrangement A and sometimes called a simple arrangement. For each hyperplane H ∈ A fix a linear form α H ∈ V * such that ker(α H ) = H. The first main object in this article is the logarithmic derivation module D(A, m) of (A, m) defined byis free, then there exists a homogeneous free basis {θ 1 , . . . , θ ℓ } for D(A, m). Then we define the exponents of a free multiarrangement (A, m) by exp(A, m) := (deg(θ 1 ), . . . , deg(θ ℓ )). The exponents are independent of a choice of a basis. When m ≡ 1, the logarithmic derivation module and exponents are denoted by D(A) and exp(A). When we fix a simple arrangement A, we say that a multiplicity m on A is free (resp. non-free) if a multiarrangement (A, m) is free (resp. non-free).A fundamental object of study in hyperplane arrangements is the arrangement of all reflecting hyperplanes of a Coxeter group, called a Coxeter arrangement. [26]. In this article we generalize the study of free multiplicities on the braid arrangement.A braid arrangement A ℓ , or the Coxeter arrangement of type A ℓ is defined as. By using the primitive derivation introduced in [16], free multiplicities on Coxeter arrangements are studied by , Terao [21], Yoshinaga [23], and the first author and Yoshinaga [6]. Combining these results, we have a characterization of the freeness of quasi-constant multiplicities m on a Coxeter arrangement, i.e., multiplicities such that max H,H ′ ∈A |m(H) − m(H ′ )| ≤ 1. However, it is known that if max H,H ′ ∈A |m(H) − m(H ′ )| = 2 then the same method using the primitive derivation does not work. Also, to determine explicitly which multiplicity makes (A, m) free is a difficult problem. Our aim is to consider these multiplicities on the braid arrangement and classify their freeness completely. In fact...

In this paper, we construct a fully homomorphic encryption (FHE) scheme over integers with the message space Z Q for any prime Q. Even for the binary case Q = 2, our decryption circuit has a smaller degree than that of the previous scheme; the multiplicative degree is reduced from O(λ(log λ)2 ) to O(λ), where λ is the security parameter. We also extend our FHE scheme to a batch FHE scheme.