Word Processing in Groups

Epstein, D. B. A.; Paterson, Mike; Cannon, J. W.; Holt, Derek F.; Levy, Silvio; Thurston, William P.

doi:10.1201/9781439865699

Cited by 866 publications

(717 citation statements)

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“…In late sixties, Garsides solved the word problem after exploring the properties of the semigroup of positive words in [16] and his idea was improved by Thurston [14], Elrifai-Morton [13] and BirmanKo-Lee [6]. They showed that there is a fast algorithm to compute the canonical form, which is unique for their results briefly.…”

Section: Describing Braids Using Permutationsmentioning

confidence: 99%

“…For a canonical factor A corresponding to a permutation π ∈ Σ n , S(A) = {i | π(i) > π(i + 1)} [14] and similarly F (A) = {i | π −1 (i) > π −1 (i + 1)}. We note that S(A) is just the descent set [33] defined in the combinatorics.…”

Section: Describing Braids Using Permutationsmentioning

confidence: 99%

“…The adversary may try to use a mathematical solution to the conjugacy problem by Garside [16], Thurston [14], Elrifai-Morton [13] and Birman-Ko-Lee [6]. But the known algorithms find an element a ∈ B +r , not in LB .…”

Section: Attack Using the Super Summit Setmentioning

confidence: 99%

“…So the cardinality ofΣ n is n!. The permutation braid can be characterized by the property that every crossing is positive and any pair of strands crosses at most once [13,14].…”

Section: Describing Braids Using Permutationsmentioning

confidence: 99%

“…In this section, we will briefly introduce the notion of braids and give evidence that the braid groups can also play important roles in cryptography. The general reference for braid theory is the Birman's book [5] and for the word problem and conjugacy problem, see [6,13,14,16]. This section is composed as follows: §2.1 is the definition of the braid groups.…”

Section: A Cryptographic Aspect Of the Braid Groupsmentioning

confidence: 99%

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New Public-Key Cryptosystem Using Braid Groups

Lee

Cheon

et al. 2000

Advances in Cryptology — CRYPTO 2000

273

351

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Abstract. The braid groups are infinite non-commutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers. (ii) The group operations can be performed efficiently. (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives. The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups. The efficiency of our systems is demonstrated by their speed and information rate. The security of our systems is based on topological, combinatorial and group-theoretical problems that are intractible according to our current mathematical knowledge. The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design.

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Section: Describing Braids Using Permutationsmentioning

confidence: 99%

Section: Describing Braids Using Permutationsmentioning

confidence: 99%

Section: Attack Using the Super Summit Setmentioning

confidence: 99%

“…So the cardinality ofΣ n is n!. The permutation braid can be characterized by the property that every crossing is positive and any pair of strands crosses at most once [13,14].…”

Section: Describing Braids Using Permutationsmentioning

confidence: 99%