In 2003, Leonid A. Levin presented the idea of a combinatorial complete one-way function and a sketch of the proof that Tiling represents such a function. In this paper, we present two new one-way functions based on semi-Thue string rewriting systems and a version of the Post Correspondence Problem and prove their completeness. Besides, we present an alternative proof of Levin's result. We also discuss the properties a combinatorial problem should have in order to hold a complete one-way function.
Deep Dive into New Combinatorial Complete One-Way Functions.
In 2003, Leonid A. Levin presented the idea of a combinatorial complete one-way function and a sketch of the proof that Tiling represents such a function. In this paper, we present two new one-way functions based on semi-Thue string rewriting systems and a version of the Post Correspondence Problem and prove their completeness. Besides, we present an alternative proof of Levin’s result. We also discuss the properties a combinatorial problem should have in order to hold a complete one-way function.
arXiv:0802.2863v1 [cs.CC] 20 Feb 2008
Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 457-466
www.stacs-conf.org
NEW COMBINATORIAL COMPLETE ONE-WAY FUNCTIONS
ARIST KOJEVNIKOV 1 AND SERGEY I. NIKOLENKO 1
1 St.Petersburg Department of V. A. Steklov Institute of Mathematics
Fontanka 27, St.Petersburg, Russia, 191023
URL: http://logic.pdmi.ras.ru/{~arist,~sergey}/
Abstract. In 2003, Leonid A. Levin presented the idea of a combinatorial complete
one-way function and a sketch of the proof that Tiling represents such a function.
In
this paper, we present two new one-way functions based on semi-Thue string rewriting
systems and a version of the Post Correspondence Problem and prove their completeness.
Besides, we present an alternative proof of Levin’s result. We also discuss the properties
a combinatorial problem should have in order to hold a complete one-way function.
1. Introduction
In computer science, complete objects play an extremely important role. If a certain
class of problems has a complete representative, one can shift the analysis from the whole
class (where usually nothing can really be proven) to this certain, well-specified complete
problem.
Examples include Satisfiability and Graph Coloring for NP (see [GJ79] for a
survey) or, which is more closely related to our present work, Post Correspondence and
Matrix Transformation problems for DistNP [Gur91, BG95].
However, there are problems that are undoubtedly complete for their complexity classes
but do not actually cause such a nice concept shift because they are too hard to analyze.
Such problems usually come from diagonalization procedures and require enumeration of
all Turing machines or all problems of a certain complexity class.
Our results lie in the field of cryptography. For a long time, little has been known
about complete problems in cryptography.
While “conventional” complexity classes got
their complete representatives relatively soon, it had taken thirty years since the definition
of a public-key cryptosystem [DH76] to present a complete problem for the class of all
public-key cryptosystems [HKN+05, GHP06]. However, this complete problem is of the
“bad” kind of complete problems, requires enumerating all Turing machines and can hardly
be put to any use, be it practical implementation or theoretical complexity analysis.
1998 ACM Subject Classification: E.3, F.1.1, F.1.3.
Key words and phrases: cryptography, complete problem, one-way function.
Supported in part by INTAS (YSF fellowship 05-109-5565) and RFBR (grants 05-01-00932, 06-01-00502).
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A. Kojevnikov and S. I. Nikolenko
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Creative Commons Attribution-NoDerivs License
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A. KOJEVNIKOV AND S. I. NIKOLENKO
Before tackling public-key cryptosystems, it is natural to ask about a seemingly simpler
object: one-way functions (public-key cryptography is equivalent to the existence of a trap-
door function, a particular case of a one-way function). The first big step towards useful
complete one-way functions was taken by Leonid A. Levin who provided a construction of
the first known complete one-way function [Lev87] (see also [Gol99]).
The construction uses a universal Turing machine U to compute the following function:
funi(desc(M), x) = (desc(M), M(x)),
where desc(M) is the description of a Turing machine M. If there are one-way functions
among M’s (and it is easy to show that if there are any, there are one-way functions that
run in, say, quadratic time), then funi is a (weak) one-way function.
As the reader has probably already noticed, this complete one-way function is of the
“useless” kind we’ve been talking about. Naturally, Levin asked whether it is possible to
find “combinatorial” complete one-way functions, functions that would not depend on enu-
merating Turing machines or giving their descriptions as input. For 15 years, the problem
remained open and then was resolved by Levin himself [Lev03]. Levin devised a clever trick
of having determinism in one direction and indeterminism in the other.
Having showed that a modified Tiling problem is in fact a complete one-way function,
Levin asked to find other combinatorial complete one-way functions.
In this work, we
answer this open question. We take Levin’s considerations further to show how a complete
one-way function may be derived from string-rewriting problems shown to be average-case
complete in [Wan95] and a variation of the Post Correspondence Problem. Moreover, we
discuss the general properties a combinatorial problem should enjoy in order to contain a
complete one-way function by similar arguments.
2. Distributional Accessibility problem for semi-Thue systems
Consider a finite alphabet A.
An ordered pair of strings ⟨g, h⟩over A is called a
rewriting rule (sometimes also called a production). We write these pairs as g →h because
we interpret them as rewriting rules for other strings. Namely, for two strings u, v we write
u ⇒g→h v if u = agb, v = ahb for some a, b ∈A∗. A set of rewriting rules is called a
semi
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