Hypersurfaces in Weighted Projective Spaces Over Finite Fields with Applications to Coding Theory

AbstractThe problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.Supplementary materials are available with this article.

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The modular counterparts of Cayley's hyperdeterminants

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031890 ◽

1998 ◽

Vol 57 (3) ◽

pp. 479-492 ◽

Cited By ~ 9

Author(s):

David G. Glynn

Keyword(s):

Finite Fields ◽

Invariant Theory ◽

Zero Point ◽

Projective Spaces ◽

Modular Invariant ◽

Geometrical Meaning ◽

Historical Aspects ◽

Field Characteristic ◽

Finite Projective Spaces

LetHbe a hypersurface of degreeminPG(n, q),q=ph,pprime.(1) Ifm<n+ 1,Hhas 1 (modp) points.(2) Ifm=n+ 1,Hhas 1 (modp) points ⇔Hp−1has no termWe show some applications, including the generalised Hasse invariant for hypersurfaces of degreen+ 1 inPG(n, F), various porperties of finite projective spaces, and in particular ap-modular invariant detpof any (n+ 1)r+2= (n+ 1)×…×(n+ 1) array on hypercubeAover a field characteristicp. This invariant is multiplicative in that detp(AB) = detp(B), whenever the product (or convolution of the two arraysAandBis defined, and both arrays are not 1-dimensional vectors. (IfAis (n+ 1)r+2andBis (n+ 1)s+2, thenABis (n+ 1)r+s+2.) The geometrical meaning of the invariant is that over finite fields of characteristicpthe number of projections ofAfromr+ 1 points in any givenr+ 1 directions of the array to a non-zero point in the final direction is 0 (modp). Equivalently, the number of projections ofAfromrpoints in any givenrdirections to a non-singular (n+ 1)2matrix is 0 (modp). Historical aspects of invariant theory and connections with Cayley's hyperdeterminant Det for characteristic 0 fields are mentioned.

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GCD of Aunu Binary Polynomials of Cardinality Seven Using Extended Euclidean Algorithm

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH ◽

10.47191/ijmcr/v9i9.02 ◽

2021 ◽

Vol 09 (09) ◽

Author(s):

Ibrahim A. A. ◽

Keyword(s):

Finite Fields ◽

Coding Theory ◽

Error Correcting Codes ◽

Euclidean Algorithm ◽

Greatest Common Divisor ◽

Algebraic Structures ◽

Extended Euclidean Algorithm ◽

Encryption Schemes ◽

Theory Error

Finite fields is considered to be the most widely used algebraic structures today due to its applications in cryptography, coding theory, error correcting codes among others. This paper reports the use of extended Euclidean algorithm in computing the greatest common divisor (gcd) of Aunu binary polynomials of cardinality seven. Each class of the polynomial is permuted into pairs until all the succeeding classes are exhausted. The findings of this research reveals that the gcd of most of the pairs of the permuted classes are relatively prime. This results can be used further in constructing some cryptographic architectures that could be used in design of strong encryption schemes.

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