Narrow Results

In this paper, we give a concrete method to construct cellular algebras from matrix algebras by specifying certain fixed matrices for the data of inflations. In particular, orthogonal matrices can be chosen for such data.

Consider the real space 𝔻_U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dim_ℝ (𝔻_U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻_U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator I_U on 𝕄_N, and where D(U) is the multiplicity of 1 in the spectrum of I_U. This characterization allows us to establish the dependence of D(U⁽¹⁾ ⊗ … ⊗U^(r)) on D(U^(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of I_U we study the ‘permute and enphase’ symmetries and the equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is divided in two parts — the present one and the second appearing in the next issue of OSID [1].

Consider the real space 𝔻_U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dim_ℝ (𝔻_U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻_U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator ℐ_U on 𝕄_N, and where D(U) is the multiplicity of 1 in the spectrum of ℐ_U. This characterisation allows us to establish the dependence of D(U⁽¹⁾ ⊗ … ⊗ U^(r)) on D(U^(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of ℐ_U we study the ‘permute and enphase’ symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is published as two papers — the first part [1] and the current second one.

Cryptographic key is the central problem in cryptographic techniques. The key based on password protection is not enough secure because of the low entropy in user chosen passwords that can be exploited to launch password-guessing attacks. And the length of cryptographic key generated from user's biometric features directly is limited. Differ from prior methods, a novel scheme of cryptographic key generation based on biometric features and secret sharing is proposed. Biometric feature vectors is transformed through orthogonal matrix, and mapped to integer spaces. Then, a steady integer vector is obtained, and can be utilized to bind with cryptographic key by Shamir's secret sharing scheme. This method may realize the security of key storing, and can produce the key of the arbitrary length.