For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to @math ? And for which sets A does there exist such a mapping that respects the lexicographical ordering within A? Both cases are types of perfect, minimal hash functions. The complexity-theoretic versions of these notions are known as compression functions and ranking functions. The present paper defines and studies the recursion-theoretic versions of compression and ranking functions, and in particular studies the question of which sets have, or lack, such functions. Thus, this is a case where, in contrast to the usual direction of notion transferal, notions from complexity theory are inspiring notions, and an investigation, in computability theory. We show that the rankable and compressible sets broadly populate the 1-truth-table degrees, and we prove that every nonempty coRE cylinder is recursively compressible.
 In dieser Habilitationsschrift werden Struktur und Eigenschaften von Komplexitatsklassen wie P und NP untersucht, vor allem im Hinblick auf: Zertifikatkomplexitat, Einwegfunktionen, Heuristiken gegen NP-Vollstandigkeit und Zahlkomplexitat. Zum letzten Punkt werden speziell untersucht: (a) die Komplexitat von Zahleigenschaften von Schaltkreisen, (b) Separationen von Zahlklassen mit Immunitat und (c) die Komplexitat des Zahlens der Losungen von ,,tally NP-Problemen. Diffie and Hellman [2] propose the use of the exponential function in a finite field for cryptographic purposes. The proposal is based on the conjecture that the inverse function, the logarithm, is not feasibly computable. We show that a proof of this conjecture would have important consequences for theoretical computer science, even under the assumption that P @math NP. We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including: • In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*. • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. • In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments.We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse.