Publication 2005 Issue No. 3 - March Abstract - Searching Worst Cases of a One-Variable Function Using Lattice Reduction
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Searching Worst Cases of a One-Variable Function Using Lattice Reduction
March 2005 (vol. 54 no. 3)
pp. 340-346
 ASCII Text x Damien Stehl?, Vincent Lef?vre, Paul Zimmermann, "Searching Worst Cases of a One-Variable Function Using Lattice Reduction," IEEE Transactions on Computers, vol. 54, no. 3, pp. 340-346, March, 2005.
 BibTex x @article{ 10.1109/TC.2005.55,author = {Damien Stehl? and Vincent Lef?vre and Paul Zimmermann},title = {Searching Worst Cases of a One-Variable Function Using Lattice Reduction},journal ={IEEE Transactions on Computers},volume = {54},number = {3},issn = {0018-9340},year = {2005},pages = {340-346},doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2005.55},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - Searching Worst Cases of a One-Variable Function Using Lattice ReductionIS - 3SN - 0018-9340SP340EP346EPD - 340-346A1 - Damien Stehl?, A1 - Vincent Lef?vre, A1 - Paul Zimmermann, PY - 2005KW - Computer arithmeticKW - multiple precision arithmeticKW - special function approximations.VL - 54JA - IEEE Transactions on ComputersER -
We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem—i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith's work on the integer small value problem—for polynomials with integer coefficients—using lattice reduction. For floating-point numbers with a mantissa less than N and a polynomial approximation of degree d, our algorithm finds all worst cases at distance less than N^{\frac{-d^2}{2d+1}} from a machine number in time O(N^{{\frac{d+1}{2d+1}}+\varepsilon}). For d=2, a detailed study improves on the O(N^{2/3+\varepsilon}) complexity from Lefèvre's algorithm to O(N^{4/7+\varepsilon}). For larger d, our algorithm can be used to check that there exist no worst cases at distance less than N^{-k} in time O(N^{1/2+\varepsilon}).

[1] D. Boneh and G. Durfee, “Cryptanalysis of RSA with Private Key $d$ Less than $n^{0. 292}$ ,” Proc. Eurocrypt '99, pp. 1-11, 1999.
[2] D. Boneh, G. Durfee, and N. Howgrave-Graham, “Factoring $n=p^rq$ for Large $r$ ,” Proc. Eurocrypt '99, pp. 326-337, 1999.
[3] D. Coppersmith, “Finding a Small Root of a Bivariate Integer Equation: Factoring with High Bits Known,” Proc. Eurocrypt '96, pp. 178-189, 1996.
[4] D. Coppersmith, “Finding a Small Root of a Univariate Modular Equation,” Proc. Eurocrypt '96, pp. 155-165, 1996.
[5] D. Coppersmith, “Finding Small Solutions to Small Degree Polynomials,” Proc. Cryptography and Lattices Conf. (CALC '01), pp. 20-31, 2001.
[6] D. Defour, F. de Dinechin, and J.-M. Muller, “Correctly Rounded Exponential Function in Double Precision Arithmetic,” Proc. SPIE 46th Ann. Meeting, Int'l Symp. Optical Science and Technology, 2001.
[7] D. Defour, G. Hanrot, V. Lefèvre, J.-M. Muller, N. Revol, and P. Zimmermann, “Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic,” Numerical Algorithms, vol. 37, nos. 1-4, pp. 367-375, 2004, .
[8] N. Elkies, “Rational Points Near Curves and Small Nonzero $|x^3-y^2|$ via Lattice Reduction,” Proc. Algorithmic Number Theory Symp. (ANTS-IV), pp. 33-63, 2000.
[9] G. Gonnet, “A Note on Finding Difficult Values to Evaluate Numerically,” http://www.kluweronline.com/issn/1017-1398http:/ /www.inf.ethz.ch/personal/gonnet/ FPAccuracyNastyValues.ps, 2002.
[10] T. Granlund, GNU MP: The GNU Multiple Precision Arithmetic Library, 4.1.2 ed., 2002, http://www.swox.se/gmp#DOC.
[11] “IEEE Standard For Binary Floating-Point Arithmetic,” Technical Report ANSI-IEEE Standard 754-1985, 1985.
[12] C.S. Iordache and D.W. Matula, “Infinitely Precise Rounding for Division, Square Root, and Square Root Reciprocal,” Proc. 14th IEEE Symp. Computer Arithmetic, pp. 233-240, 1999.
[13] T. Lang and J.-M. Muller, “Bounds on Runs of Zeros and Ones for Algebraic Functions,” Proc. 15th IEEE Symp. Computer Arithmetic (ARITH 15), N. Burgess and L. Ciminiera, eds., pp. 13-20, 2001.
[14] V. Lefèvre, “Moyens Arithmétiques pour un Calcul Fiable,” thèse de doctorat, École Normale Supérieure de Lyon, 2000.
[15] V. Lefèvre and J.-M. Muller, “Worst Cases for Correct Rounding of the Elementary Functions in Double Precision,” Proc. 15th IEEE Symp. Computer Arithmetic (ARITH 15), N. Burgess and L. Ciminiera, eds., pp. 111-118, 2001.
[16] A.K. Lenstra, H.W. Lenstra, and L. Lovász, “Factoring Polynomials with Rational Coefficients,” Mathematische Annalen, vol. 261, pp. 515-534, 1982.
[17] L. Lovász, “An Algorithmic Theory of Numbers, Graphs and Convexity,” SIAM Lecture Series, vol. 50, 1986.
[18] D. Stehlé, V. Lefèvre, and P. Zimmermann, “Worst Cases and Lattice Reduction,” Proc. 16th IEEE Symp. Computer Arithmetic, pp. 142-147, 2003.
[19] D. Stehlé, “Breaking Littlewood's Cipher,” Cryptologia, vol. 28, no. 4, pp. 341-357, 2004.
[20] A. Ziv, “Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit,” ACM Trans. Math. Software, vol. 17, no. 3, pp. 410-423, 1991.

Index Terms:
Computer arithmetic, multiple precision arithmetic, special function approximations.
Citation:
Damien Stehl?, Vincent Lef?vre, Paul Zimmermann, "Searching Worst Cases of a One-Variable Function Using Lattice Reduction," IEEE Transactions on Computers, vol. 54, no. 3, pp. 340-346, March 2005, doi:10.1109/TC.2005.55