Public Key Encryption in Non-Abelian Groups

In this paper, we propose a brand new public key encryption scheme in the Lie group that is a non-abelian group. In particular, we firstly investigate the intractability assumptions in the Lie group, including the non-abelian factoring assumption and non-abelian inserting assumption. After that, by using the FO technique, a CCA secure public key encryption scheme in the Lie group is proposed. At last, we present the security proof in the random oracle based on the non-abelian inserting assumption.

[1]  Neal R. Wagner,et al.  A Public Key Cryptosystem Based on the Word Problem , 1985, CRYPTO.

[2]  Delaram Kahrobaei,et al.  A CCA secure cryptosystem using matrices over group rings , 2014, ArXiv.

[3]  Delaram Kahrobaei,et al.  Decision and Search in Non-Abelian Cramer-Shoup Public Key Cryptosystem , 2009, Groups Complex. Cryptol..

[4]  Martin Rötteler,et al.  Quantum algorithms: A survey of some recent results , 2006, Informatik - Forschung und Entwicklung.

[5]  Christof Zalka,et al.  Shor's discrete logarithm quantum algorithm for elliptic curves , 2003, Quantum Inf. Comput..

[6]  Jung Hee Cheon,et al.  New Public-Key Cryptosystem Using Braid Groups , 2000, CRYPTO.

[7]  B. Eick,et al.  POLYCYCLIC GROUPS: A NEW PLATFORM FOR CRYPTOLOGY? , 2004, math/0411077.

[8]  Spyros S. Magliveras,et al.  A Public Key Cryptosystem Based on Non-abelian Finite Groups , 2008, Journal of Cryptology.

[9]  G. Marchuk.  Methods of Numerical Mathematics , 1982 .

[10]  Tran van Trung,et al.  Public key cryptosystem MST3: cryptanalysis and realization , 2010, J. Math. Cryptol..

[11]  Delaram Kahrobaei,et al.  Public Key Exchange Using Semidirect Product of (Semi)Groups , 2013, ACNS.

[12]  Vladimir Shpilrain,et al.  Thompson's Group and Public Key Cryptography , 2005, ACNS.

[13]  Douglas R. Stinson,et al.  New Approaches to Designing Public Key Cryptosystems Using One-Way Functions and Trapdoors in Finite Groups , 2001, Journal of Cryptology.

[14]  Nasir D. Memon,et al.  Algebraic properties of cryptosystem PGM , 1992, Journal of Cryptology.

[15]  Markus Bläser.  Noncommutativity makes determinants hard , 2012, Inf. Comput..

[16]  Andrea Caranti,et al.  The Round Functions of Cryptosystem PGM Generate the Symmetric Group , 2006, Des. Codes Cryptogr..

[17]  B. Hall.  Lie Groups, Lie Algebras, and Representations , 2003 .

[18]  Simon R. Blackburn,et al.  Cryptanalysis of the MST3 Public Key Cryptosystem , 2009, IACR Cryptol. ePrint Arch..

[19]  Tatsuaki Okamoto,et al.  Secure Integration of Asymmetric and Symmetric Encryption Schemes , 1999, Journal of Cryptology.

[20]  N. Jacobson,et al.  Lectures in Abstract Algebra : vol. III, Theory of Fields and Galois Theory. By N. Jacobson. Pp. xi, 323. 76s. (Van Nostrand) , 1966, The Mathematical Gazette.

[21]  G. Higman.  Suzuki $2$-groups , 1963 .

[22]  Spyros S. Magliveras,et al.  Properties of Cryptosystem PGM , 1989, CRYPTO.

[23]  Delaram Kahrobaei,et al.  Public key exchange using matrices over group rings , 2013, Groups Complex. Cryptol..

[24]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[25]  Marius van der Put,et al.  Galois Theory of Difference Equations , 1997 .

[26]  Maria Isabel Gonzalez Vasco,et al.  A note on the security of MST3 , 2010, Des. Codes Cryptogr..

[27]  Spyros S. Magliveras,et al.  ON THE SECURITY OF A REALIZATION OF CRYPTOSYSTEM MST3 , 2008 .