No abstract

We study random number generation using a biased source motivated by previous works on this topic, mainly, von Neumman (1951), Elias (1972), Knuth and Yao (1976) and Peres (1992). We study the problem in two cases: first, when the source distribution is unknown, and second, when the source distribution is known. In the first case, we characterize the functions that use a discrete random source of unknown distribution to simulate a target discrete random variable with a given rational distribution. We identify the functions that minimize the ratio of source inputs to target outputs. We show that these optimal functions are efficiently computable. In the second case, we prove that it is impossible to construct an optimal tree algorithm recursively, using algebraic decision procedures. Our model of computation is sufficiently general to encompass previously known algorithms for this problem. ABSTRACTWe study random number generation using a biased source motivated by previous works on this topic, mainly, von Neumman (1951), Elias (1972), Knuth and Yao (1976) and Peres (1992). We study the problem in two cases: first, when the source distribution is unknown, and second, when the source distribution is known. In the first case, we characterize the functions that use a discrete random source of unknown distribution to simulate a target discrete random variable with a given rational distribution. We identify the functions that minimize the ratio of source inputs to target outputs. We show that these optimal functions are efficiently computable. In the second case, we prove that it is impossible to construct an optimal tree algorithm recursively, using algebraic decision procedures. Our model of computation is sufficiently general to encompass previously known algorithms for this problem. My thanks go to many colleagues of mine for the fun time spent together, suggestions, advices and discussions that sometimes helped my research. I would like to thank es-

All binary plane curves of degree less than 7 are examined for curves with a large number of F q rational points on their smooth model, for q = 2 m , m = 3, 4, ..., 11. Previous results are improved, and many new curves are found meeting or close to Serre's, Lauter's, and Ihara's upper bounds.