First Implementation of Cryptographic Protocols Based on Algebraic Number Fields

We show for the first time how to implement cryptographic protocols based on class groups of algebraic number fields of degree > 2. We describe how the involved objects can be represented and how the arithmetic in class groups can be realized efficiently. To speed up the arithmetic we present our new method for multiplication of ideals. Furthermore we show how to generate cryptographically suitable algebraic number fields. Besides,w e give a numerical example and analyse our run times.

[1]  Johannes Buchmann,et al.  LiDIA : a library for computational number theory , 1995 .

[2]  S. Louboutin.  The Exponent 2-Class-Group Problem for Non-Galois Over Q Quartic Fields That Are Quadratic Extensions of Imaginary Quadratic Fields , 1994 .

[3]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[4]  Hans-Joachim Stender.  Eine formel für Grundeinheiten in reinen algebraischen Zahlkörpern dritten, vierten und sechsten grades , 1975 .

[5]  Henri Cohen,et al.  Class Groups of Number Fields: Numerical Heuristics , 1987 .

[6]  J. Pollard.  A monte carlo method for factorization , 1975 .

[7]  J. Buchmann.  A generalization of Voronoi's unit algorithm II , 1985 .

[8]  Burton S. Kaliski.  Advances in Cryptology - CRYPTO '97 , 1997 .

[9]  Johannes Buchmann,et al.  On the period length of the generalized Lagrange algorithm , 1987 .

[10]  J. Buchmann.  The computation of the fundamental unit of totally complex quartic orders , 1987 .

[11]  Johannes A. Buchmann,et al.  On some computational problems in finite abelian groups , 1997, Math. Comput..

[12]  K. McCurley,et al.  A rigorous subexponential algorithm for computation of class groups , 1989 .

[13]  S. Lang.  Algebraic Number Theory , 1971 .

[14]  Nigel P. Smart,et al.  How Secure Are Elliptic Curves over Composite Extension Fields? , 2001, EUROCRYPT.

[15]  Stéphane Louboutin,et al.  Class-number problems for cubic number fields , 1995, Nagoya Mathematical Journal.

[16]  Sachar Paulus,et al.  A One Way Function Based on Ideal Arithmetic in Number Fields , 1997, CRYPTO.

[17]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[18]  Henri Cohen,et al.  Heuristics on class groups of number fields , 1984 .

[19]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1991, STOC '91.

[20]  H. M. Stark,et al.  Some effective cases of the Brauer-Siegel Theorem , 1974 .

[21]  Nigel P. Smart,et al.  The Discrete Logarithm Problem on Elliptic Curves of Trace One , 1999, Journal of Cryptology.

[22]  Stefan Neis.  Reducing Ideal Arithmetic to Linear Algebra Problems , 1998, ANTS.

[23]  H. Lenstra,et al.  A rigorous time bound for factoring integers , 1992 .

[24]  G. Frey,et al.  A remark concerning m -divisibility and the discrete logarithm in the divisor class group of curves , 1994 .

[25]  Damian Weber,et al.  Computing Discrete Logarithms with the General Number Field Sieve , 1996, ANTS.

[26]  Michael Pohst,et al.  Algorithmic algebraic number theory , 1989, Encyclopedia of mathematics and its applications.

[27]  Bodo Möller,et al.  Security of Cryptosystems Based on Class Groups of Imaginary Quadratic Orders , 2000, ASIACRYPT.

[28]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[29]  Henri Cohen,et al.  Étude heuristique des groupes de classes des corps de nombres. , 1990 .

[30]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[31]  Claus-Peter Schnorr,et al.  Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems , 1991, FCT.

[32]  John A. Howell,et al.  Spans in the module (Zm ) s , 1986 .

[33]  Johannes Buchmann,et al.  Cryptographic Protocols Based on Intractability of Extracting Roots and Computing Discrete Logarithms , 1999 .

[34]  Arjen K. Lenstra,et al.  Algorithms in Number Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[35]  Johannes A. Buchmann,et al.  A Signature Scheme Based on the Intractability of Computing Roots , 2002, Des. Codes Cryptogr..

[36]  E. Bach.  Explicit bounds for primality testing and related problems , 1990 .