Efficiency improvement techniques for private intersection-sum protocol using Bloom filter

A private set intersection protocol is one of the secure multi-party computation protocols, and allows participants to compute the intersection of their sets without revealing them to each other. Ion et al. proposed the private intersection-sum protocol (PI-Sum). The PI-Sum is one of the two-party private set intersection protocol. In the PI-Sum, two parties (say Alice and Bob) have the private sets A and B. Moreover, Bob additionaly has a rational integer associated with each element of B. The PI-Sum allows Bob to obtain the sum of the rational integers associated with the elements of $$A \cap B$$ A ∩ B . This paper proposes the efficiency improvement techniques for the PI-Sum. The proposed techniques are based on Bloom filters which are probabilistic data structures. More precisely, this paper proposes three protocols which are modifications of the PI-Sum. The proposed protocols are more efficient than the PI-Sum.

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

On Power Integral Bases for Certain Pure Number Fields Defined by x 36 − m

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x36 − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

On monogenity of certain pure number fields defined by xpr − m

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

On power integral bases for certain pure number fields defined by x24 – m

Let K = ℚ(α) be a number field generated by a complex root α of a monic irreducible polynomial f(x) = x24 – m, with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.

On certain pure sextic fields related to a problem of Hasse

An algebraic number ring is monogenic, or one-generated, if it has the form [Formula: see text] for a single algebraic integer [Formula: see text]. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if [Formula: see text] is a square-free rational integer, [Formula: see text] and [Formula: see text], then the pure sextic field [Formula: see text] is not monogenic. Our results are illustrated by examples.

A Variant of Lehmer’s Conjecture, II: The CM-case

Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

Rational Integer Invariants of Regular Cyclic Actions

Let g : M2n → M2n be a smooth map of period m ≥ 2 which preserves orientation. Suppose that the cyclic action defined by g is regular and that the normal bundle of the fixed point set F has a g-equivariant complex structure. Let F ⋔ F be the transverse self-intersection of F with itself. If the g-signature Sign(g, M) is a rational integer and n < ϕ(m), then there exists a choice of orientations such that Sign(g, M) = Sign F = Sign(F ⋔ F).

Values of zeta functions and class number 1 criterion for the simplest cubic fields

Let K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m2 + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function ζK(s) at s = −1 and get class number 1 criterion for the simplest cubic fields.