Abstract
Up to now there has been an algorithm for the calculation of Minkowski-reduced lattice bases to dimensionn=6 or at most ton=7. A new algorithm is presented which is practicable for greater dimensions and requires less computation time.
Zusammenfassung
Bisher gab es nur einen Algorithmus für die Berechnung von Minkowski-reduzierten Gitterbasen bis zur Dimensionn=6 oder höchstens bisn=7. Es wird nun ein neues Verfahren angegeben, das auch für höhere Dimensionen praktikabel ist und wenig Rechenzeit benötigt.
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References
Afflerbach, L.: Minkowskische Reduktionsbedingungen für positiv definite quadratische Formen in 5 Variablen. Mh. Math.94, 1–8 (1982).
Afflerbach, L.: The Sub-Lattice Structure of Linear Congruential Generators. FB Math. Technische Hochschule Darmstadt, Preprint-Nr. 821 (1984).
Beyer, W. A., Roof, R. B., Williamson, D.: The lattice structure of multiplicative congruential pseudo-random vectors. Math. Comput.25, 345–360 (1971).
Beyer, W. A.: Lattice structure and reduced bases of random vectors generated by linear recurrences. In: Applications of Number Theory to Numerical Analysis (Zaremba, S. K., ed.), 361–370 (1972).
Dieter, U., Ahrens, J. H.: Uniform Random Numbers. Institut f. Math. Stat. Technische Hochschule Graz (1974).
Dieter, U.: How to calculate shortest vectors in a lattice. Math. Comput.29, 827–833 (1975).
Knuth, D. E.: The Art of Computer Programming, Vol. II. 2nd ed. Reading (Mass.), Menlo Park (Cal.), London, Amsterdam, Don Mills (Ont.), Sydney: Addison-Wesley (1981).
Marsaglia, G.: Random numbers fall mainly in the planes. Proc. Nat. Acad. Sci.61, 25–28 (1968).
Marsaglia, G.: Regularities in congruential random number generators. Numerische Math.16, 8–10 (1970).
Marsaglia, G.: The structure of linear congruential sequences. In: Applications of Number Theory to Numerical Analysis (Zaremba, S. K., ed.), 249–285 (1972).
Minkowski, H.: Gesammelte Abhandlungen 1, pp. 145–148, 153–156, 217–218. Leipzig-Berlin: Teubner 1911.
Minkowski, H.: Gesammelte Abhandlungen 2, pp. 78–80. Leipzig-Berlin: Teubner 1911.
Pohst, M.: On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications. ACM SIGSAM Bulletin15, 37–44 (1981).
Ryshkov, S. S.: On the reduction theory of positive quadratic forms. Soviet Math. Dokl.12, 946–950 (1971).
Ryshkov, S. S.: The theory of Hermite-Minkowski reduction of positive definite quadratic forms. J. Soviet Math.6, 651–671 (1976).
Tammela, P. P.: On the reduction theory of positive quadratic forms. Soviet Math. Dokl.14, 651–655 (1973).
Tammela, P. P.: The Hermite-Minkowski domain of reduction of positive definite quadratic forms in six variables. J. Soviet Math.6, 677–688 (1976).
Tammela, P. P.: The Minkowski's fundamental reduction domain of positive quadratic forms in seven variables. Zapiski nauc. Sem. Leningrad. Otd. mat. Inst. Steklov67, 108–143 (1977).
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Afflerbach, L., Grothe, H. Calculation of Minkowski-reduced lattice bases. Computing 35, 269–276 (1985). https://doi.org/10.1007/BF02240194
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DOI: https://doi.org/10.1007/BF02240194