Some applications of arithmetic groups in cryptography
-
Delaram Kahrobaei
and
Keivan Mallahi-Karai
Abstract
In this paper, we will offer a new symmetric-key cryptographic scheme which is based on the existence of exponentially distorted subgroups in arithmetic groups.
Aside from this, we will also provide new examples of distorted subgroups in
Funding source: Office of Naval Research
Award Identifier / Grant number: N00014-15-1-2164
Funding source: National Science Foundation
Award Identifier / Grant number: CCF-1564968
Funding statement: Delaram Kahrobaei is partially supported by a PSC-CUNY grant from the CUNY Research Foundation, the City Tech Foundation, and ONR (Office of Naval Research) grant N00014-15-1-2164. Delaram Kahrobaei has also partially supported by an NSF travel grant CCF-1564968 to IHP in Paris.
References
[1] I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), no. 3–4, 287–291. 10.4310/MRL.1999.v6.n3.a3Search in Google Scholar
[2] E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126. 10.2307/1969218Search in Google Scholar
[3] G. Baumslag, B. Fine and X. Xu, Cryptosystems using linear groups, Appl. Algebra Engrg. Comm. Comput. 17 (2006), no. 3–4, 205–217. 10.1007/s00200-006-0003-zSearch in Google Scholar
[4] A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1969. Search in Google Scholar
[5] B. Cavallo, G. Di Crescenzo, D. Kahrobaei and V. Shpilrain, Efficient and secure delegation of group exponentiation to a single server, Radio Frequency Identification—RFIDsec 2015, Lecture Notes in Comput. Sci. 9440, Springer, Cham (2015), 156–173. 10.1007/978-3-319-24837-0_10Search in Google Scholar
[6] I. Chatterji, D. Kahrobaei and N. Y. Lu, Cryptosystems using subgroup distortion, Theoret. Appl. Inform. 29 (2017), 14–24. 10.20904/291-2014Search in Google Scholar
[7] G. Di Crescenzo, D. Kahrobaei, M. Khodjaeva and V. Shpilrain, Efficient and secure delegation to a single malicious server: Exponentiation over non-abelian groups, Mathematical Software—ICMS 2018, Lecture Notes in Comput. Sci. 10931, Springer, Cham (2018), 137–146. 10.1007/978-3-319-96418-8_17Search in Google Scholar
[8] K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J.-s. Kang and C. Park, New public-key cryptosystem using braid groups, Advances in Cryptology—CRYPTO 2000, Lecture Notes in Comput. Sci. 1880, Springer, Berlin (2000), 166–183. 10.1007/3-540-44598-6_10Search in Google Scholar
[9] A. Lubotzky, S. Mozes and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Publ. Math. Inst. Hautes Études Sci. (2000), no. 91, 5–53. 10.1007/BF02698740Search in Google Scholar
[10] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3) 17, Springer, Berlin, 1991. 10.1007/978-3-642-51445-6Search in Google Scholar
[11]
T. R. Riley,
Navigating in the Cayley graphs of
[12] A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477. 10.1070/IM1977v011n02ABEH001709Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Closure properties in the class of multiple context-free groups
- Cryptanalysis of a hash function, and the modular subset sum problem
- Some applications of arithmetic groups in cryptography
- An improved version of the AAG cryptographic protocol
- Conjugacy search problem and the Andrews–Curtis conjecture
Articles in the same Issue
- Frontmatter
- Closure properties in the class of multiple context-free groups
- Cryptanalysis of a hash function, and the modular subset sum problem
- Some applications of arithmetic groups in cryptography
- An improved version of the AAG cryptographic protocol
- Conjugacy search problem and the Andrews–Curtis conjecture
