Group-based cryptography

Short description: 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

• Shpilrain–Zapata public-key protocols
• Magyarik–Wagner public key protocol
• Anshel–Anshel–Goldfeld key exchange
• Ko–Lee et al. key exchange protocol

See also

• Non-commutative cryptography

References

• Myasnikov, A.G.; Shpilrain, V.; Ushakov, A. (2008). Group-based Cryptography. Advanced Courses in Mathematics – CRM Barcelona. Birkhauser. ISBN 9783764388270. https://books.google.com/books?id=mEa3BAAAQBAJ&pg=PR7. 
• 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. 196. Springer. pp. 19–36. doi:10.1007/3-540-39568-7_3. ISBN 978-3-540-39568-3. https://doi.org/10.1007/3-540-39568-7_3. 
• Anshel, I.; Anshel, M.; Goldfeld, D. (1999). "An algebraic method for public-key cryptography". Math. Res. Lett. 6 (3): 287–291. doi:10.4310/MRL.1999.v6.n3.a3. https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0003/MRL-1999-0006-0003-a003.pdf. 
• 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. 1880. Springer. pp. 166–183. doi:10.1007/3-540-44598-6_10. ISBN 978-3-540-44598-2. https://link.springer.com/chapter/10.1007/3-540-44598-6_10. 
• Shpilrain, V.; Zapata, G. (2006). "Combinatorial group theory and public key cryptography". Appl. Algebra Eng. Commun. Comput. 17 (3–4): 291–302. doi:10.1007/s00200-006-0006-9. 

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)