Introduction to Algebra

Kostrikin, Α. I.

doi:10.1007/978-1-4615-7286-2

Cited by 77 publications

(64 citation statements)

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“…The set K n has a natural ring structure as the direct sum of n copies of the field K, as well as a natural vector space structure as a vector space of dimension n over K. A set with such structure is known as an algebra [8]. More formally, we have the following definition.…”

Section: State Space Algebrasmentioning

confidence: 99%

“…A representation of an n-dimensional algebra A is formally defined as an algebra homomorphism from A to a subalgebra of M l (K) [8], where M l (K) denotes the set of l × l matrices over the field K. Thus a representation of the algebra A identifies A with an n-dimensional subalgebra of the l × l matrices. If the algebra homomorphism is an isomorphism, then we may identify A with this n-dimensional subalgebra of the l × l matrices.…”

Section: Regular Representations Of State Space Algebrasmentioning

confidence: 99%

“…Clearly, there are many ways in to define a representation. One standard technique is the regular representation, which is the algebra homomorphism ν : A → M n (K) that maps a ∈ A to the matrix corresponding to the linear transformation z → az (z a K-vector of length n) [8].…”

Section: Regular Representations Of State Space Algebrasmentioning

confidence: 99%

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An Algebraic Framework for Cipher Embeddings

Cid

Murphy

Robshaw

2005

Cryptography and Coding

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Section: State Space Algebrasmentioning

confidence: 99%

Section: Regular Representations Of State Space Algebrasmentioning

confidence: 99%

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An Algebraic Framework for Cipher Embeddings

Cid

Murphy

Robshaw

2005

Cryptography and Coding

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“…To complete the security proof, we require some lemmas (presented by Beaumont [Bea65] and Kostrikin [Kos82] PROOF: Presented by Kostrikin [Kos82].…”

Section: A Linear Algebra Lemmasmentioning

confidence: 99%

Verifiable Secret Redistribution for Threshold Sharing Schemes

Wong¹,

Wang²,

Wing³

2002

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We present a new protocol for the verifiable redistribution of secrets from (m,n) to (m ,n ) access structures for threshold sharing schemes. Our protocol enables the addition or removal of shareholders and also guards against mobile adversaries that cause permanent damage. We observe that existing protocols either cannot be readily extended to allow redistribution between different access structures, or have vulnerabilities that allow faulty old shareholders to corrupt the shares of new shareholders. Our primary contribution is that, in our protocol, new shareholders can verify the validity of their shares after redistribution between different access structures. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of DARPA or the U.S. Government. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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“…An equivalent result is that a polynomial over a unique factorization domain of characteristic zero has a repeated nonconstant factor if and only if its discriminant is zero. It is also well known that if K is a field and F an extension of K, then a polynomial f ∈ K[X] has c ∈ F for a multiple root if and only if f (c) = f (c) = 0 (see [6]). For an account on the theory of separable polynomials over commutative rings, the reader is referred to DeMeyer [1], [2], Janusz [5] Nagahara [8], [9], Harrison and McKenzie [4], and McKenzie [7].…”

Section: Introductionmentioning

confidence: 99%