Hill Ciphers over Near-Fields
Mathematics and Computer Education Volume 41, Number 1, ISSN 0730-8639
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.
Farag, M. (2007). Hill Ciphers over Near-Fields. Mathematics and Computer Education, 41(1), 46-54.