Using Decision Problems in Public Key Cryptography

Abstract

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. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. 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.