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existence of minimal logarithmic signatures for classical groups
Title
The existence of minimal logarithmic signatures for classical groups.
Creator
Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
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Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172943
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)
the minimal logarithmic signature conjecture
Title
On the minimal logarithmic signature conjecture.
Creator
Singhi, Nidhi., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups,...
Show moreThe minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BN-pair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS.
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Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172946
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)