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Braid Group Cryptography
, 2008
"... 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, ..."
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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 publickey 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 noncommutative groups).
Groups with two generators having unsolvable word problem and presentations of Mihailova subgroups
 SHENZHEN UNIVERSITY SHENZHEN CITY 518060, CHINA XUCHEN@TOM.COM
, 2014
"... A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2 × F2 with finite number of generators are given. Where Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. One then can use the ..."
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A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2 × F2 with finite number of generators are given. Where Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. One then can use the presentation to create entities’ private keys in a public key cryptsystem.
On the number of mth roots of permutations
 AUSTRALASIAN JOURNAL OF COMBINATORICS VOLUME 52 (2012), PAGES 41–54
, 2012
"... Let m be a fixed positive integer. It is wellknown that a permutation σ of {1,...,n} may have one, many, or no mth roots. In this article we provide an explicit expression and a generating function for the number of mth roots of σ. Let pm(n) be the probability that a random npermutation has an mth ..."
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Let m be a fixed positive integer. It is wellknown that a permutation σ of {1,...,n} may have one, many, or no mth roots. In this article we provide an explicit expression and a generating function for the number of mth roots of σ. Let pm(n) be the probability that a random npermutation has an mth root. We also include a proof of the fact that pm(jq) = pm(jq +1)=·· · = pm(jq +(q − 1)), j =0, 1,..., when m is a power of a prime number q.
Polynomial time solutions of computational problems in noncommutativealgebraic cryptography
, 2013
"... By introducing extra shields on Shpilrain and Ushakov’s KoLeelike protocol based on the decomposition problem of group elements we propose two new key exchange schemes and then a number of public key cryptographic protocols. We show that these protocols are free of known attacks. Particularly, if ..."
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By introducing extra shields on Shpilrain and Ushakov’s KoLeelike protocol based on the decomposition problem of group elements we propose two new key exchange schemes and then a number of public key cryptographic protocols. We show that these protocols are free of known attacks. Particularly, if the entities taking part in our protocols create their private keys composed by the generators of the Mihailova subgroups of Bn, we show that the safety of our protocols are very highly guarantied by the insolvability of subgroup membership problem of the Mihailova subgroups.