On the Andrews-Curtis equivalence

Myasnikov, Alexei; Shpilrain, Vladimir

doi:10.1090/conm/296/05074

Cited by 19 publications

(18 citation statements)

References 23 publications

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“…Hence, the inequality (3.1) could not be satisfied. On the other hand, as was found out by Myasnikov [17], see also [5], [18], the AC-conjecture holds for the pair (a 2 b −3 , abab −1 a −1 b −1 ). Therefore, the pair (a 2 b −3 , abab −1 a −1 b −1 ) is not minimal and gives a counterexample to [2,Conjecture 4].…”

Section: One More Conjecture Of Andrews and Curtissupporting

confidence: 58%

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On Conjectures of Andrews and Curtis

Ivanov

2018

Proc. Amer. Math. Soc.

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It is shown that the original Andrews-Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis [2] made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews-Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews-Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.

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Section: One More Conjecture Of Andrews and Curtissupporting

confidence: 58%

“…Recall that there is another, more general, version of the AC-conjecture, called the AC-conjecture with stabilizations, see [4], [6], [18], in which a fourth type of operations, called stabilizations, is allowed.…”

Section: Introductionmentioning

confidence: 99%

On Conjectures of Andrews and Curtis

Ivanov

2018

Proc. Amer. Math. Soc.

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“…Experimental results with instances of the TTP protocol generated using |z| = 50 (which is almost three times greater than the suggested value) showed 100% success rate. They indicate that the attack may fail when the length of z is large relative to the length of ∆ 2 (for more details, see [101,Section 3.4…”

Section: The Attacksmentioning

confidence: 99%

Braid Group Cryptography

Garber

2009

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them.This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of cryptosystems which are based on other non-commutative groups).

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“…After that, she has to apply several (T3)s and (T4 ′ )s to mix the new generators with the old ones. We note that there are many non-trivial presentations of the trivial group to choose from; for example, in [16], there are given several infinite series of such presentations in the special case where t = q (so-called balanced presentations). Without this restriction, there are even more choices; in particular, Alice can just add arbitrary relators to a balanced presentation of the trivial group, thus adding to the confusion of the adversary.…”

Section: Pool Of Group Presentationsmentioning

confidence: 99%

Using Decision Problems in Public Key Cryptography

Shpilrain

Zapata²

2009

Groups – Complexity – Cryptology

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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.

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On the Andrews-Curtis equivalence

Cited by 19 publications

References 23 publications

On Conjectures of Andrews and Curtis

On Conjectures of Andrews and Curtis

Braid Group Cryptography

Using Decision Problems in Public Key Cryptography

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