Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers

We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[ζ], i.e. the integers extended with ζ, a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n 2) for n bit input. This is an improvement from the known results based on the Euclidean algorithm, and taking time O(n · M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols.

[1]  J. Stein.  Computational problems associated with Racah algebra , 1967 .

[2]  C. Crépeau,et al.  On the Computational Collapse of Quantum Information , 2003 .

[3]  H. Lenstra,et al.  Euclidean number fields 1 , 1979 .

[4]  Franz Lemmermeyer,et al.  THE EUCLIDEAN ALGORITHM IN ALGEBRAIC NUMBER FIELDS , 2004 .

[5]  Arnold Schönhage,et al.  Schnelle Berechnung von Kettenbruchentwicklungen , 1971, Acta Informatica.

[6]  Walter Penney,et al.  A ``Binary'' System for Complex Numbers , 1965, JACM.

[7]  Franz Lemmermeyer,et al.  Reciprocity Laws: From Euler to Eisenstein , 2000 .

[8]  Olivier Danvy.  A New One-Pass Transformation into Monadic Normal Form , 2003, CC.

[9]  Vladimiro Sassone,et al.  Jeeg: temporal constraints for the synchronization of concurrent objects , 2005, Concurr. Pract. Exp..

[10]  Jonathan P. Sorenson,et al.  Efficient Algorithms for Computing the Jacobi Symbol , 1996, J. Symb. Comput..

[11]  Jeffrey Shallit,et al.  A binary algorithm for the Jacobi symbol , 1993, SIGS.

[12]  A. Weilert.  Fast Computation of the Biquadratic Residue Symbol , 2002 .

[13]  Ivan Damgård,et al.  Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers , 2005, J. Symb. Comput..

[14]  Claus Brabrand,et al.  The metafront System: Extensible Parsing and Transformation , 2003, LDTA@ETAPS.

[15]  Renate Scheidler,et al.  A public-key cryptosystem utilizing cyclotomic fields , 1995, Des. Codes Cryptogr..

[16]  André Weilert,et al.  (1+i)-ary GCD Computation in Z[i] as an Analogue to the Binary GCD Algorithm , 2000, J. Symb. Comput..

[17]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[18]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[19]  Olivier Danvy,et al.  Tagging, Encoding, and Jones Optimality , 2003, ESOP.

[20]  Ivan Damgård,et al.  An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates , 2003, FCT.

[21]  H. C. Williams,et al.  Computation of the Solution of x 3 + Dy 3 = 1 , 1977 .

[22]  André Weilert,et al.  Asymptotically Fast GCD Computation in Z[i] , 2000, ANTS.

[23]  Donald E. Knuth,et al.  A imaginary number system , 1960, Commun. ACM.

[24]  Hugh C. Williams,et al.  An M³ Public-Key Encryption Scheme , 1985, CRYPTO.

[25]  Peter Kornerup,et al.  Redundant Radix Representations of Rings , 1999, IEEE Trans. Computers.

[26]  Vladimiro Sassone,et al.  Deriving Bisimulation Congruences: 2-Categories Vs Precategories , 2003, FoSSaCS.

[27]  Ivan Damgård,et al.  Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers , 2003 .