Problems of Information Transmission, Vol. 39, No. 1, 2003, pp. 92–103. Translated from Problemy Peredachi Informatsii, No. 1, 2003, pp. 103–117.
Original Russian Text Copyright
2003 by Levin.
The Tale of One-Way Functions
L. A. Levin
All the king’s horses, and all the king’s men,
Couldn’t put Humpty together again.
Abstract—The existence of one-way functions (owf) is arguably the most important problem
in computer theory. The article discusses and reﬁnes a number of concepts relevant to this
problem. For instance, it gives the ﬁrst combinatorial complete owf, i.e., a function which is
one-way if any function is. There are surprisingly many subtleties in basic deﬁnitions. Some
of these subtleties are discussed or hinted at in the literature and some are overlooked. Here,
a uniﬁed approach is attempted.
1. INTRODUCTION I: INVERTING FUNCTIONS
From time immemorial, humanity has got frequent, often cruel, reminders that many things are
easier to do than to reverse. When the foundations of mathematics started to be seriously analyzed,
this experience immediately found a formal expression.
1.1. An Odd Axiom
Over a century ago, George Cantor reduced all the great variety of mathematical concepts to just
one—the concept of sets—and derived all mathematical theorems from just one axiom scheme—
Cantor’s Postulate. For each set-theoretical formula A(x), it postulates the existence of a set
containing those and only those x satisfying A. This axiom looked a triviality, almost a deﬁnition,
but was soon found to yield more than Cantor wanted, including contradictions. To salvage its
great promise, Zermelo, Fraenkel, and others pragmatically replaced Cantor’s Postulate with a
collection of its restricted cases, limiting the types of allowed properties A. The restrictions turned
out to cause little inconvenience and precluded (so far) any contradictions; the axioms took their
ﬁrm place in the foundations of mathematics.
In 1904, Zermelo noticed that one more axiom was needed to derive all known mathematics, the
(in)famous Axiom of Choice: every function f has an inverse g such that f(g(x)) = x for x in the
range of f . It was accepted reluctantly; to this day, proofs dependent on it are being singled out.
Its strangeness was not limited to going beyond Cantor’s Postulate—it brought paradoxes! Allow
me a simple illustration based on the ability of the axiom of choice to enable a symmetric choice
of an arbitrary integer.
Consider the additive group T = R/Z of reals mod 1 as points x ∈ [0, 1) on a circle; take its
⊂ T of decimal fractions a/10
.Letf(x) be the (countable) coset x + Q
, i.e., f
projects T onto its factor group T/Q
.Anyinverseg of f then selects one representative from
each coset. Denote by G = g(f(T)) the image of such a g;theneachx ∈ T is brought into G by
This research was partially conducted by the author for the Clay Mathematics Institute and supported
by the Institut des Hautes
Etudes Scientiﬁques and NSF, grant no. CCR-9820934.
2003 MAIK “Nauka/Interperiodica”