Cryptographic protocols based on inner product spaces and group theory with a special focus on the use of Nielsen transformations
چکیده
The topic of this thesis is established in the area of mathematical cryptology, more precisely in group based cryptology. We give extensions of cryptographic protocols, develop new crypto- graphic protocols concerning the mathematical background and give modifications of them. In addition cryptographic analysis as well as examples are given. The focus lays on the development of new cryptographic protocols using non-commutative groups and of techniques, which are typ- ically studied in combinatorial group theory. Automorphisms on finitely generated free groups are used, which can be generated by Nielsen transformations or Whitehead-Automorphisms. With the help of the Whitehead-Automorphisms we develop an approach for choosing automor- phisms randomly of the automorphism group Aut(F ), with F a finitely generated free group. Altogether twelve cryptographic protocols are explained. Among these are two extensions of a (n, t)-secret sharing protocol, which is introduced by C. S. Chum, B. Fine, G. Rosenberger and X. Zhang. Both extensions depend on the Closest Vector Theorem in a real inner product space. The first one (Protocol 1) is a symmetric key cryptosystem and the second one is a challenge and response system (Protocol 2), which can be used by a variation as a two-way authentication. Furthermore, the HKKS-key exchange protocol by M. Habbeb, D. Kahrobaei, C. Koupparis and V. Shpilrain, which uses semidirect products of (semi)groups, is extended to an ElGamal like public key cryptosystem (Protocol 3) and to a signature protocol (Protocol 4). There is an ongoing research about the HKKS-key exchange protocol with linear algebra at- tacks as well as research about suitable platforms, which also affects the ElGamal like public key cryptosystem and the signature protocol. A short overview of the research is given in this thesis. Furthermore, a purely combinatorial secret sharing scheme (Protocol 5) is introduced, which uses a share distribution method explained by D. Panagopoulos for a (n, t)-secret sharing scheme. We show that this share distribution method is a special case of a multiple assignment scheme introduced by M. Ito, A. Saito and T. Nishizeki. Furthermore, the introduced combinatorial secret sharing protocol is shown to be similar to a variation of a secret sharing protocol ex- plained by J. Benaloh and J. Leichter. The idea of enhancing the combinatorial secret sharing scheme by using automorphisms on finitely generated free groups leads to two new secret sharing schemes. In addition a comparison to Shamir’s secret sharing scheme is given. The first one is a secret sharing scheme using a finitely generated abstract free group F , a finitely generated free subgroup in SL(2,Q) and Nielsen transformations (Protocol 6). Protocol 6 is the basis for Protocol 7-12, which are also based on combinatorial group theory. The other secret sharing scheme (Protocol 7) uses a finitely generated free group F = 〈X | 〉, a Nielsen reduced set U 6= X and a Nielsen equivalent set V to U and gives therefore the final input for the newly developed cryptographic Protocols 8-12, which are the main result in this thesis. Two new private key cryptosystems with similar modifications (Protocol 8 and Protocol 9) were de- veloped, another new private key cryptosystem (Protocol 10), a new ElGamal like public key cryptosystem (Protocol 11) and a new challenge and response system (Protocol 12), which all use the combinatorial group theory and automorphisms on finitely generated free groups. Depending on the protocols the security is based on a linear congruence generator, the discrete logarithm problem in cyclic subgroups of the automorphism group of a finitely generated free group, the unknown algorithmic solution of the (constructive) membership problem in matrix groups over the rational numbers or Hilbert’s Tenth Problem.