Group-based cryptography is a use of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
to construct
cryptographic primitive Cryptographic primitives are well-established, low-level cryptography, cryptographic algorithms that are frequently used to build cryptographic protocols for computer security systems. These routines include, but are not limited to, one-way hash fun ...
s. A group is a very general algebraic object and most cryptographic schemes use groups in some way. In particular
Diffie–Hellman key exchange
Diffie–Hellman (DH) key exchangeSynonyms of Diffie–Hellman key exchange include:
* Diffie–Hellman–Merkle key exchange
* Diffie–Hellman key agreement
* Diffie–Hellman key establishment
* Diffie–Hellman key negotiation
* Exponential ke ...
uses finite
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s. So the term ''group-based cryptography'' refers mostly to
cryptographic protocol
A cryptographic protocol is an abstract or concrete Communications protocol, protocol that performs a information security, security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol desc ...
s that use infinite
non-abelian group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ...
s such as a
braid group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
.
Examples
* Shpilrain–Zapata public-key protocols
* Magyarik–Wagner public key protocol
*
Anshel–Anshel–Goldfeld key exchange
Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols, it doe ...
* Ko–Lee et al. key exchange protocol
See also
*
Non-commutative cryptography Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative. One of the earliest applications of a non-c ...
References
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Further reading
* Paul, Kamakhya; Goswami, Pinkimani; Singh, Madan Mohan. (2022)
"ALGEBRAIC BRAID GROUP PUBLIC KEY CRYPTOGRAPHY" Jnanabha', Vol. 52(2) (2022), 218-223. ISSN 0304-9892 (Print) ISSN 2455-7463 (Online)
External links
Cryptography and Braid Groups page(archived version 7/17/2017)
Theory of cryptography
Braid groups
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