Primitive Points Research Articles

Elliptic Curve Cryptosystems

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over ${\text {GF}}({2^n})$. We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

Elliptic curve cryptosystems

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF ( 2 n ) {\text {GF}}({2^n}) . We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

Primitive ideals and integral points on cones

Primitive ideals and integral points on cones

Non-nodal interpolation — an improved technique for the uniform B-spline method

The uniform B-spline method, when used to interpolate unevenly spaced 2D or 3D points, may result in a bad configuration. The undesirable result is caused by the distributional unevenness of the B-splinefs nodes. The non-nodal interpolating method separates the nodes from the primitive data points by assigning each latter a proper parameter and gives goods results. An experimental package has been developed, which posseses some noticeable capabilities and can be run in a minicomputer.

Primitive integral points on the surface of an elliptic cone

Primitive integral points on the surface of an elliptic cone

Primitive points on elliptic curves

Primitive points on elliptic curves

A modified form of Siegel's mean-value theorem

The Minkowski–Hlawka theorem† asserts that, if S is any n-dimensional star body, with the origin o as centre, and with volume less than 2ζ(n), then there is a lattice of determinant 1 which has no point other than o in S. One of the methods used to prove this theorem splits up into three stages, (a) A function ρ(x) is considered, and it is shown that some suitably defined mean value of the sumtaken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integralover the whole space. (b) By taking ρ(x) to be equal, or approximately equal, towhere σ(x) is the characteristic function of S, and μ(r) is the Möbius function, it is shown that a corresponding mean value of the sumwhere Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to

A theorem of Minkowski on the product of two linear forms

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the regioncontains a pair of generating points of Λ; and that, unless the lattice Λ is of the special formwhere both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,inscribed in the region (1). With simple geometrical arguments, he showed that(i) Пλ always contains a point of Λ, other than 0,(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.

Societies and Academies

LONDONMathematical Society, March 8.-Mr. C. W. Merrineld? F.R.S., vice-president, in the chair.-The following communi"cations were made:-On a new view of the Pascal hexagram, by Mr. T. Cotterill. In a system of co-planar points, the number of intersections of two chords is a multiple of 3. In the case of the hexagram the forty-five points thus derived are divided into four sets of triangles-(i) The three intersections of the chords joining four points form a triad self-conjugate to the conies through the four points. (2) Any three non-conterminous chords intersect in three points, forming a diagonal triangle. In each of these two cases, a derived point determines uniquely its corresponding triad, the number of triads being fifteen. (3) An inscribed triangle determines an opposite inscribed triangle; the three intersections of the pairs of sides supposed to correspond form a triangle, the intersections of two inscribed triangles, the nine intersections of the two triangles forming an ennead. (4) The three intersections of the opposite sides of a hexagon of the system form a Pascal triangle. The number of triangles in each of the two last cases is sixty; to each triangle of one set corresponding a triangle of the other, as well as a triad of the second set, the nine points forming three triads of the first set. Denoting, then, the primitive points by italics and fifteen of the derived points (no two of which are conjugate) by Greek letter?, we obtain all the derived points by accenting once and twice the Greek letters to form self-conjugate triads. Tables are then formed in matrices of the nine chords joining the vertices of two opposite triangles and their eighteen intersections, found to consist of six triangles of each of the second and fourth sets. To these corresponds a matrix containing the nine intersections of the two triangles. In the case of a conic hexagram, the properties of the sixty points of intersection of chords with the tangents at the conic points are then examined.-On a class of integers expressible as the sum of two integral squares, by Mr. T. Muir. [The class of integers considered included those whose square root, when expressed as a continued fraction, has two middle terms in the cycle of partial denominators. A general expression was given for all such integers, and an equivalent expression in the form of the sum of two squares.]-Some properties of the double-theta functions, by Prof. Cay ley, F.R.S. (founded on papers by Goepel and Rosenhain).-A property of an envelope, by Mr. J. J. Walker.