4,199 research outputs found
Combinatorial group theory and public key cryptography
After some excitement generated by recently suggested public key exchange
protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent
opinion now that the conjugacy search problem is unlikely to provide sufficient
level of security if a braid group is used as the platform. In this paper we
address the following questions: (1) whether choosing a different group, or a
class of groups, can remedy the situation; (2) whether some other "hard"
problem from combinatorial group theory can be used, instead of the conjugacy
search problem, in a public key exchange protocol. Another question that we
address here, although somewhat vague, is likely to become a focus of the
future research in public key cryptography based on symbolic computation: (3)
whether one can efficiently disguise an element of a given group (or a
semigroup) by using defining relations.Comment: 12 page
Using decision problems in public key cryptography
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
Group theory in cryptography
This paper is a guide for the pure mathematician who would like to know more
about cryptography based on group theory. The paper gives a brief overview of
the subject, and provides pointers to good textbooks, key research papers and
recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor
typographical changes. To appear in Proceedings of Groups St Andrews 2009 in
Bath, U
Post Quantum Cryptography from Mutant Prime Knots
By resorting to basic features of topological knot theory we propose a
(classical) cryptographic protocol based on the `difficulty' of decomposing
complex knots generated as connected sums of prime knots and their mutants. The
scheme combines an asymmetric public key protocol with symmetric private ones
and is intrinsecally secure against quantum eavesdropper attacks.Comment: 14 pages, 5 figure
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