scholarly journals Integral Cayley Graphs Generated by Distance Sets in Vector Spaces over Finite Fields

10.37236/2730 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Ngoc Dai Nguyen ◽  
Minh Hai Nguyen ◽  
Duy Hieu Do ◽  
Anh Vinh Le
Keyword(s):  
Finite Fields ◽  
Euclidean Distance ◽  
Cayley Graphs ◽  
Vector Spaces ◽  
Sufficient Condition ◽  
Finite Rings ◽  
Distance Sets

Si Li and the fourth listed author (2008) considered unitary graphs attached to the vector spaces over finite rings using an analogue of the Euclidean distance. These graphs are shown to be integral when the cardinality of the ring is odd or the dimension is even. In this paper, we show that the statement also holds for the remaining case: the cardinality of the ring is even and the dimension is odd, by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields to be integral.

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

scholarly journals Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

2014 ◽  
Vol 57 (4) ◽  
pp. 834-844
Author(s):  
Doowon Koh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Algebraic Varieties ◽  
Test Functions ◽  
Radial Functions ◽  
Zero Radius ◽  
Intersection Points ◽  
Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

On groupoids representable by vector spaces over finite fields

1973 ◽  
Vol 24 (1) ◽  
pp. 14-20 ◽  
Author(s):  
A. Mitschke ◽  
H. Werner
Keyword(s):  
Finite Fields ◽  
Vector Spaces
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