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Polynomial Functions on Finite Commutative Rings
"... . Every function on a finite residue class ring D=I of a Dedekind domain D is induced by an integervalued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a ..."
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. Every function on a finite residue class ring D=I of a Dedekind domain D is induced by an integervalued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all
Majorization of the Critical Points of a Polynomial by Its Zeros
"... Abstract. Let z1,..., zn be the zeros of a polynomial f(z) and let ζ1,..., ζn be those of zf ′(z). Suppose that for both polynomials the zeros are labelled in order of nonincreasing modulus. We show that k∑ ν=1 ζν  ≤ k∑ ν=1 zν  , k = 1,..., n, which means that the moduli of the zeros of f(z) we ..."
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Abstract. Let z1,..., zn be the zeros of a polynomial f(z) and let ζ1,..., ζn be those of zf ′(z). Suppose that for both polynomials the zeros are labelled in order of nonincreasing modulus. We show that k∑ ν=1 ζν  ≤ k∑ ν=1 zν  , k = 1,..., n, which means that the moduli of the zeros of f
Invertible Polynomial Representation for Private Set Operations
"... Abstract. In many private set operations, a set is represented by a polynomial over a ring Zσ for a composite integer σ, where Zσ is the message space of some additive homomorphic encryption. While it is useful for implementing set operations with polynomial additions and multiplications, a polynomi ..."
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polynomial representation has a limitation due to the hardness of polynomial factorizations over Zσ. That is, it is hard to recover a corresponding set from a resulting polynomial over Zσ if σ is not a prime. In this paper, we propose a new representation of a set by a polynomial over Zσ, in which σ is a
Volume I: Computer Science and Software Engineering
, 2013
"... Algebraic algorithms deal with numbers, vectors, matrices, polynomials, formal power series, exponential and differential polynomials, rational functions, algebraic sets, curves and surfaces. In this vast area, manipulation with matrices and polynomials is fundamental for modern computations in Sc ..."
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Algebraic algorithms deal with numbers, vectors, matrices, polynomials, formal power series, exponential and differential polynomials, rational functions, algebraic sets, curves and surfaces. In this vast area, manipulation with matrices and polynomials is fundamental for modern computations
Reviewed by Joel Berman
"... On Hagemann’s and Herrmann’s characterization of strictly affine complete algebras. (English summary) Algebra Universalis 44 (2000), no. 12, 105–121. An algebra A is called strictly kaffine complete if for every finite T ⊆ A k and for every congruence preserving function f: T → A there is a kary ..."
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On Hagemann’s and Herrmann’s characterization of strictly affine complete algebras. (English summary) Algebra Universalis 44 (2000), no. 12, 105–121. An algebra A is called strictly kaffine complete if for every finite T ⊆ A k and for every congruence preserving function f: T → A there is a k
APractical Latticebased Digital Signature Schemes
"... Digital signatures are an important primitive for building secure systems and are used in most real world security protocols. However, almost all popular signature schemes are either based on the factoring assumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case o ..."
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Digital signatures are an important primitive for building secure systems and are used in most real world security protocols. However, almost all popular signature schemes are either based on the factoring assumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case of classical cryptanalytic advances or progress on the development of quantum computers the hardness of these closely related problems might be seriously weakened. A potential alternative approach is the construction of signature schemes based on the hardness of certain lattices problems which are assumed to be intractable by quantum computers. Due to significant research advancements in recent years, latticebased schemes have now become practical and appear to be a very viable alternative to numbertheoretic cryptography. In this paper we focus on recent developments and the current stateoftheart in latticebased digital signatures and provide a comprehensive survey discussing signature schemes with respect to practicality. Additionally, we discuss future research areas that are essential for the continued development of latticebased cryptography.
ALGEBRAIC ALGORITHMS1
, 2012
"... This is a preliminary version of a Chapter on Algebraic Algorithms in the up ..."
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This is a preliminary version of a Chapter on Algebraic Algorithms in the up
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