Group-based cryptography

Application of group theory to cryptography

Group-based cryptography is a use of groups to construct cryptographic primitives. A group is a very general algebraic object and most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite non-abelian groups such as a braid group.

Examples

Myasnikov, A.G.; Shpilrain, V.; Ushakov, A. (2008). Group-based Cryptography. Advanced Courses in Mathematics – CRM Barcelona. Birkhauser. ISBN 9783764388270.
Myasnikov, A.G.; Shpilrain, V.; Ushakov, A. (2011). Non-commutative cryptography and complexity of group-theoretic problems. Amer. Math. Soc. Surveys and Monographs. ISBN 9780821853603.
Magyarik, M.R.; Wagner, N.R. (1985). "A Public Key Cryptosystem Based on the Word Problem". Advances in Cryptology—CRYPTO 1984. Lecture Notes in Computer Science. Vol. 196. Springer. pp. 19–36. doi:10.1007/3-540-39568-7_3. ISBN 978-3-540-39568-3.
Anshel, I.; Anshel, M.; Goldfeld, D. (1999). "An algebraic method for public-key cryptography" (PDF). Math. Res. Lett. 6 (3): 287–291. CiteSeerX 10.1.1.25.8355 . doi:10.4310/MRL.1999.v6.n3.a3.
Ko, K.H.; Lee, S.J.; Cheon, J.H.; Han, J.W.; Kang, J.; Park, C. (2000). "New public-key cryptosystem using braid groups". Advances in Cryptology—CRYPTO 2000. Lecture Notes in Computer Science. Vol. 1880. Springer. pp. 166–183. CiteSeerX 10.1.1.85.5306 . doi:10.1007/3-540-44598-6_10. ISBN 978-3-540-44598-2.
Shpilrain, V.; Zapata, G. (2006). "Combinatorial group theory and public key cryptography". Appl. Algebra Eng. Commun. Comput. 17 (3–4): 291–302. arXiv:math/0410068 . CiteSeerX 10.1.1.100.888 . doi:10.1007/s00200-006-0006-9. S2CID 2251819.

v t e Cryptography
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Mathematics	Cryptographic hash function Block cipher Stream cipher Symmetric-key algorithm Authenticated encryption Public-key cryptography Quantum key distribution Quantum cryptography Post-quantum cryptography Message authentication code Random numbers Steganography
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