Transformations of Cryptographic Schemes Through Interpolation Techniques Stamatios-Aggelos N. Alexandropoulos, Gerasimos C. Meletiou, Dimitrios S. Triantafyllou, and Michael N. Vrahatis Abstract The problem of transforming cryptographic schemes using interpolation techniques is studied. Firstly, explicit forms for the discrete logarithm and the Diffie–Hellman cryptographic functions are given. Subsequently, the inverse Aitken and Neville interpolation methods for the discrete logarithm and the Lucas logarithm problems are presented. Next, the representation of cryptographic functions through polynomials or algebraic functions as well as a special case of discrete logarithm problem is given. Finally, a study of cryptographic functions using factorization of matrices is analyzed. Keywords: Public key cryptography • Discrete logarithm • Diffie Hellman mapping • Polynomial interpolation techniques • Matrix factorization 1 Introduction A basic task of cryptography is the transformation or encryption, of a given message into another one which appears meaningful only to the intended recipient after the process of decryption. Messages and cryptograms are represented as elements of finite algebraic structures. Encryption and decryption processes are functions over finite structures especially over finite fields. It is well known that, in a finite field GF.q/, where q is a prime power, every function can be represented as a polynomial through the Lagrangian interpolation. S.-A.N. Alexandropoulos () • M.N. Vrahatis Computational Intelligence Laboratory (CILab), Department of Mathematics, University of Patras, GR-26110 Patras, Greece e-mail: alekst@master.math.upatras.gr; vrahatis@math.upatras.gr G.C. Meletiou A.T.E.I. of Epirus, P.O. 110, GR-47100 Arta, Greece e-mail: gmelet@teiep.gr D.S. Triantafyllou Hellenic Army Academy (SSE), University of Military Education, GR-16673 Vari, Attica, Greece e-mail: dtriant@sse.gr © Springer International Publishing Switzerland 2015 N.J. Daras, M.Th. Rassias (eds.), Computation, Cryptography, and Network Security, DOI 10.1007/978-3-319-18275-9_1 1 1