You are here:

# Hill Ciphers over Near-FieldsARTICLE

## Mark Farag

Mathematics and Computer Education Volume 41, Number 1, ISSN 0730-8639

## Abstract

Hill ciphers are linear codes that use as input a "plaintext" vector [p-right arrow above] of size n, which is encrypted with an invertible n x n matrix E to produce a "ciphertext" vector [c-right arrow above] = E [middle dot] [p-right arrow above]. Informally, a near-field is a triple [left angle bracket]N; +, *[right angle bracket] that satisfies all the axioms of a field with the possible exception of one distributive law and the commutativity of *. Formally, a (left) near-field [left angle bracket]N; +, [right angle bracket] is a nonempty set N together with binary operations + and * for which [left angle bracket]N, +[right angle bracket] is a group with identity element denoted by 0[subscript N], [left angle bracket]N; [right angle bracket] is a monoid, [left angle bracket]N / [left curley bracket]0[subscript N][right curley bracket], [right angle bracket] is a group, and for any a,b,c [is a member of] N, a(b+c) = ab + ac holds. Right near-fields may be defined analogously by substituting the right distributive law for the left distributive law. This paper discusses matrices over near-fields, explains coding matrices over near-fields, and presents three projects in coding matrices over near-fields.

## Citation

Farag, M. (2007). Hill Ciphers over Near-Fields. Mathematics and Computer Education, 41(1), 46-54. Retrieved August 7, 2020 from .

This record was imported from ERIC on December 3, 2015. [Original Record]

ERIC is sponsored by the Institute of Education Sciences (IES) of the U.S. Department of Education.

Copyright for this record is held by the content creator. For more details see ERIC's copyright policy.