There are several public key establishment protocols as well as complete
public key cryptosystems based on allegedly hard problems from combinatorial
(semi)group theory known by now. Most of these problems are search problems,
i.e., they are of the following nature: given a property P and the information
that there are objects with the property P, find at least one particular object
with the property P. So far, no cryptographic protocol based on a search
problem in a non-commutative (semi)group has been recognized as secure enough
to be a viable alternative to established protocols (such as RSA) based on
commutative (semi)groups, although most of these protocols are more efficient
than RSA is.
In this paper, we suggest to use decision problems from combinatorial group
theory as the core of a public key establishment protocol or a public key
cryptosystem. By using a popular decision problem, the word problem, we design
a cryptosystem with the following features: (1) Bob transmits to Alice an
encrypted binary sequence which Alice decrypts correctly with probability "very
close" to 1; (2) the adversary, Eve, who is granted arbitrarily high (but
fixed) computational speed, cannot positively identify (at least, in theory),
by using a "brute force attack", the "1" or "0" bits in Bob's binary sequence.
In other words: no matter what computational speed we grant Eve at the outset,
there is no guarantee that her "brute force attack" program will give a
conclusive answer (or an answer which is correct with overwhelming probability)
about any bit in Bob's sequence.Comment: 12 page
