Abstract
We first give an extension of F[x]-lattice basis reduction algorithm to the polynomial ring R[x] where F is a field and R an arbitrary integral domain. So a new algorithm is presented for synthesizing minimum length linear recurrence (or minimal polynomials) for the given multiple sequences over R. Its computational complexity is O(N 2) multiplications in R where N is the length of each sequence. A necessary and sufficient conditions for the uniqueness of minimal polynomials are given. The set of all minimal polynomials is also described.
Research supported by NSF under grants No. 19931010 and G 1999035803.
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References
Fitzpatrick, P., Norton, G. H.: The Berlekamp-Massey algorithm and linear recurring sequences over a factorial domain. AAECC, 6 (1995) 309–323
Lu, P. Z., Liu, M. L.: Gröbner basis for characteristic ideal of linear recurrence sequence over UFD. Science in China (Series A), vol. 28, No. 6. 508–519 (1998)
Norton, G. H.: On shortest linear recurrences. J. Symb. Comp. 27 (1999) 325–349
Schmidt, W. M.: Construction and estimation of bases in function fields. J. Number Theory 39, no. 2, 181–224 (1991)
Wang, L. P., Zhu, Y. F.: F[x]-lattice basis reduction algorithm and multisequence synthesis. Science in China, in press
Lenstra, A. K., Lenstra, H. W., Lovasz, L.: Factoring polynomials with rational coefficients. Math. Ann., vol. 261 (1982) 515–534
Lagarias, J. C., Odlyzko, A. M.: Soluting low density subset problems. Pro. 24th Annual IEEE Symp. on Found of Comp. Science (1983) 1–10
Coppersmith, D.: Small solutions to polynomial equations and low exponent RSA vulnerabilities. Journal of Cryptology 10 (1997) 233–260
Berlekamp, E. R.: Algebraic Coding Theory. New York: McGrawHill (1968)
Massey, J. L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory, vol. IT-15, no. 1, 122–127 (1969)
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© 2001 Springer-Verlag Berlin Heidelberg
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Wang, Lp., Zhu, Yf. (2001). Multisequence Synthesis over an Integral Domain. In: Silverman, J.H. (eds) Cryptography and Lattices. CaLC 2001. Lecture Notes in Computer Science, vol 2146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44670-2_15
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DOI: https://doi.org/10.1007/3-540-44670-2_15
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