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By Ascheuer N., Junger M., Reinelt G.

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For practical purposes, the smooth numbers method will remain preferable until quantum computers become available. Acknowledgments This article was written while I held the 2000–2001 HP-MSRI Visiting Research Professorship. , and Ulrich Vollmer for answering my questions, and Bart de Smit for providing numerical assistance. A special word of thanks is due to Hugh Williams, whose version [2002] of the same story contains many details omitted in mine. 20 HENDRIK W. LENSTRA, JR. References [Archimedes 1999] Archimedes, The cattle problem, in English verse by S.

Ap ;bp / ; if a D p ap ; b D p bp : p p p The Greeks would have been more comfortable with the following geometric definition of the GCD: if a > b > 0 consider a rectangle of width a and height b. Remove a b-by-b square from one end of the rectangle. Continue removing maximal subsquares (squares with one side being a side of the rectangle) until the remaining rectangle is a square. , their GCD. For instance, if a D 73 and b D 31, then the removal of two maximal subsquares leaves an 11-by-31 rectangle, the removal of two further maximal squares leaves an 11-by-9 rectangle, the removal of one maximal subsquare leaves a 2-by-9 rectangle, the removal of four maximal subsquares leaves a 2by-1 rectangle, and the removal of one maximal subsquare leaves a unit square.

With overwhelming likelihood a nonresidue will be found quickly, and thus quadratic nonresidues can be found quickly with a probabilistic polynomialtime algorithm. However, no deterministic polynomial-time algorithm is known. Nothing seems to be better than testing a D 2; 3; : : : until arriving at a nonresidue. log1C" p/. p 1=4 /. The simple-minded (deterministic) algorithm for finding a nonresidue could take exponential time. log2 p/ [Bach 1990]. P ROBLEM 8. , a divisor m such that 1 < m < n.

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