A ring \(R = (S,+,\times)\) is the extension of a group \((S,+)\) with an additional operation ×, subject to the following additional axioms:
Commutativity of +: For all \(x,y \in S\), \(x + y = y + x\).
Closure of ×: For all \(x,y \in S\), \(x \times y \in S\).
Associativity of ×: For all \(x,y,z \in S\), \((x \times y) \times z = x \times (y \times z)\).
Distributivity of × over +: For all \(x,y,z \in S\), \(x \times (y + z) =(x \times y) + (x \times z)\) and \((x + y) \times z = (x \times z) + (y \times z)\).
In other words, a ring can be viewed as the extension of a commutative additive group with a multiplication operation. The rings of interest in cryptography generally also have an identity element:
Identity of ×: There exists a multiplicative identity element, denoted 1, such that for all \(x \in S\), \(x \times 1 = 1 \times x = 1\).
Let \(S ^{\ast}\) denote the elements that have a multiplicative inverse; these are sometimes called the unitsof the ring. (The additive...
This is a preview of subscription content, log in via an institution to check access.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 International Federation for Information Processing
About this entry
Cite this entry
Kaliski, B. (2005). Ring. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_359
Download citation
DOI: https://doi.org/10.1007/0-387-23483-7_359
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23473-1
Online ISBN: 978-0-387-23483-0
eBook Packages: Computer ScienceReference Module Computer Science and Engineering