Niederreiter cryptosystem: Difference between revisions

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In [[geometric algebra]], a '''blade''' is a generalization of the concept of [[Scalar (mathematics)|scalars]] and [[Euclidean vector|vector]]s to include ''simple'' [[bivector]]s, [[trivector]]s, etc. Specifically, a {{mvar|k}}-blade is any object that can be expressed as the [[exterior product]] (informally ''wedge product'') of {{mvar|k}} vectors, and is of ''[[graded algebra|grade]]'' {{mvar|k}}.
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In detail:<ref name= Rodrigues>{{cite book |title=Invariants for pattern recognition and classification |author=Marcos A. Rodrigues |chapter=§1.2 Geometric algebra: an outline |url=http://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3 |page=3 ''ff'' |isbn=981-02-4278-6 |year=2000 |publisher=World Scientific}}</ref>
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*A 0-blade is a [[scalar (mathematics)|scalar]]. The [[inner product]]{{relevance-inline|date=March 2013|reason=The vector space is not necessarily an inner product space, eh?}} or dot product of two vectors {{math|'''a'''}} and {{math|'''b'''}} is a 0-blade and is denoted as:
:<math>\mathbf{a} \cdot \mathbf{b} .</math>
*A 1-blade is a [[vector space|vector]].  Every vector is simple.
*A 2-blade is a ''simple'' [[bivector]]. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades.  A 2-blade may be expressed as the wedge product of two vectors {{math|'''a'''}} and {{math|'''b'''}}:
:<math>\mathbf{a} \wedge \mathbf{b} .</math>
*A 3-blade is a simple trivector, that is, it may expressed as the wedge product of three vectors {{math|'''a'''}}, {{math|'''b'''}}, and {{math|'''c'''}}:
:<math>\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c}. </math>
*In a space of dimension {{mvar|n}}, a blade of grade {{math|''n'' − 1}} is called a ''[[pseudovector]]''.<ref name=Baylis01>{{cite book |url=http://books.google.com/books?id=oaoLbMS3ErwC&pg=PA100&dq=%22pseudovectors+%28grade+n+-+1+elements%29%22&cd=1#v=onepage&q=%22pseudovectors%20%28grade%20n%20-%201%20elements%29%22&f=false |page=100 |author=William E Baylis |title=Lectures on Clifford (geometric) algebras and applications |isbn=0-8176-3257-3 |year=2004 |chapter=§4.2.3 Higher-grade multivectors in ''C''ℓ<sub>''n''</sub>: Duals |publisher=Birkhäuser}}</ref>
*The highest grade element in a space is called a ''[[pseudoscalar]]'', and in a space of dimension {{mvar|n}} is an {{mvar|n}}-blade.<ref name=Vince>{{cite book |url=http://books.google.com/books?id=3VxZqfm3I_MC&pg=PA85&dq=pseudoscalar+%22highest+grade%22&cd=1#v=onepage&q=pseudoscalar%20%22highest%20grade%22&f=false |title=Geometric algebra for computer graphics |author=John A. Vince |page=85 |isbn=1-84628-996-3 |year=2008 |publisher=Springer}}</ref>
*In a space of dimension {{mvar|n}}, there are {{math|1=''k''(''n'' − ''k'') + 1}} dimensions of freedom in choosing a {{mvar|k}}-blade, of which one dimension is an overall scaling multiplier.<ref>For Grassmannians (including the result about dimension) a good book is: {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523  | year=1994}}. The proof of the dimensionality is actually straightforward. Take {{mvar|k}} vectors and wedge them together <math>v_1\wedge\cdots\wedge v_k</math> and perform elementary column operations on these (factoring the pivots out) until the top {{math|''k'' &times; ''k''}} block are elementary basis vectors of <math>\mathbb{F}^k</math>.  The wedge product is then parametrized by the product of the pivots and the lower {{math|1=''k'' &times; (''n'' − ''k'')}} block.</ref>
 
In a {{mvar|n}}-dimensional spaces, there are blades of grade 0 through {{mvar|n}}. A vector subspace of finite dimension {{mvar|k}} may be represented by the {{mvar|k}}-blade formed as a wedge product of all the elements of a basis for that subspace.<ref name=Hestenes>{{cite book |title=New foundations for classical mechanics: Fundamental Theories of Physics | author=David Hestenes |url=http://books.google.com/books?id=AlvTCEzSI5wC&pg=PA54 | page=54 | isbn=0-7923-5302-1 | year=1999 | publisher=Springer}}</ref>
 
==Examples==
For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as [[pseudoscalar]]s, in that they are one-dimensional objects distinct from regular scalars.
 
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, but in three-dimensions, areas have an orientation, so while 2-blades are area elements, they are oriented. 3-blades (trivectors) represent volume elements and in three-dimensional space, these are scalar-like—i.e., 3-blades in three-dimensions form a one-dimensional vector space.
 
==See also==
*[[Multivector]]
*[[Exterior algebra]]
*[[Geometric algebra]]
*[[Clifford algebra]]
 
==Notes==
<references/>
 
==General references==
*{{cite book |title=Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics |author=[[David Hestenes]], Garret Sobczyk |chapter=Chapter 1: Geometric algebra |page=1 ''ff'' |isbn=90-277-2561-6 |year=1987 |publisher=Springer |url=http://books.google.com/books?id=yyjLeKEdt20C&printsec=frontcover#v=onepage&q=&f=false}}
*{{cite book |title=Geometric algebra for physicists |author=Chris Doran and Anthony Lasenby |isbn=0-521-48022-1 |year=2003 |publisher=Cambridge University Press |url=http://books.google.com/books?id=nZ6MsVIdt88C&printsec=frontcover&dq=%22geometric+algebra+for+physicists%22&cd=1#v=onepage&q=&f=false }}
*[http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/GA-SIG/Conformal%20Geometry%20Papers/Cambridge/Covarient%20Approach%20to%20Geometry%20Using%20Geometric%20Algebra.pdf A Lasenby, J Lasenby & R Wareham] (2004) ''A covariant approach to geometry using geometric algebra'' Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
*{{cite book |title=Computer algebra and geometric algebra with applications |year=2005 |page=329 ''ff'' |author=R Wareham, J Cameron, & J Lasenby |chapter=Applications of conformal geometric algebra to computer vision and graphics |url=http://books.google.com/books?id=uxofVAQE3LoC&pg=PA330#v=onepage&q=&f=false |editor= Hongbo Li, Peter J. Olver, Gerald Sommer |isbn=3-540-26296-2 |publisher=Springer}}
 
==External links==
*[http://www.jaapsuter.com/paper/ga_primer.pdf A Geometric Algebra Primer], especially for computer scientists.
 
[[Category:Geometric algebra]]
[[Category:Vector calculus]]

Latest revision as of 10:41, 6 October 2014

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