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New upper bounds for kissing numbers from semidefinite programming

Christine Bachoc, Frank Vallentin
2008 Journal of The American Mathematical Society  
Schrijver derived new upper bounds for binary codes using semidefinite programming.  ...  In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions.  ...  Schrijver in [21] to the unit sphere whereas he obtains new upper bounds for binary codes from an SDP.  ... 
doi:10.1090/s0894-0347-07-00589-9 fatcat:j7qeeofflbcfjbpeonwxbs2hra

New upper bounds on the linear complexity

P. Caballero-Gil
2000 Computers and Mathematics with Applications  
ln this work, the general upper bound on the linear complexity given by Key is improved for certain families of nonlinear filter functions.  ...  Also, a new class of cyclotomic cosets whose degeneration is relatively easy to prove in several conditions is introduced and analysed. (~)  ...  Since our concern is with binary sequences, most of the expressions discussed in this work will be over GF (2) , the finite field with two elements• Let (L, k)c denote the C th common divisor of L and  ... 
doi:10.1016/s0898-1221(99)00331-4 fatcat:nikeyl4qdrgltmpqfsafqdls5q

New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

Maria Dostert, Cristóbal Guzmán, Fernando Mário de Oliveira Filho, Frank Vallentin
2017 Discrete & Computational Geometry  
The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.  ...  In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the l^p_3-norm) and of Platonic and Archimedean solids having  ...  New upper bounds for the density of translative packings of three-dimensional convex bodies 27  ... 
doi:10.1007/s00454-017-9882-y fatcat:ee62luw4obc7zhwmm6jfucjscu

New Characterizations of Matrix Φ-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity [article]

Hao-Chung Cheng, Min-Hsiu Hsieh
2015 arXiv   pre-print
This allows us to upper bound the Holevo quantity of a classical-quantum ensemble that undergoes a special Markov evolution.  ...  We also provide a new proof for the matrix Efron-Stein inequality.  ...  Let X be uniformly distributed over X ≡ {0, 1} n (an n-dimensional binary hypercube) and f : X → M + d be an arbitrary matrix-valued function.  ... 
arXiv:1506.06801v1 fatcat:ipg73aq5mjfbbgv7p5xzendgpi

Polynomial Multiplication over Finite Fields Using Field Extensions and Interpolation

Murat Cenk, Cetin Kaya Koc, Ferruh Ozbudak
2009 2009 19th IEEE Symposium on Computer Arithmetic  
A method for polynomial multiplication over finite fields using field extensions and polynomial interpolation is introduced.  ...  The proposed method uses polynomial interpolation as Toom-Cook method together with field extensions. Furthermore, the proposed method can be used when Toom-Cook method cannot be applied directly.  ...  Any polynomial multiplication formula over F q can be used for finite field multiplication because element of finite fields can be represented by polynomials.  ... 
doi:10.1109/arith.2009.11 dblp:conf/arith/CenkKO09 fatcat:3h77fmisrrblbn5jjrcu7oo2ja

Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count [article]

Iggy van Hoof
2020 arXiv   pre-print
In this paper we look at multiplication of 2 binary polynomials of degree at most n-1, modulo an irreducible polynomial of degree n with 2n input and n output qubits, without ancillary qubits, assuming  ...  This paper introduces a new algorithm that uses the same space, but by utilizing space-efficient variants of Karatsuba multiplication methods it requires only O(n^log_2(3)) Toffoli gates at the cost of  ...  Acknowledgements The author thanks Tanja Lange for her insights into quantum algorithms and classical finite field operations, Tanja Lange and Gustavo Banegas for their advice and supervision both on this  ... 
arXiv:1910.02849v2 fatcat:ti2pxvhncrebze4zde6hrj6dd4

Binary Representations of Finite Fields and Their Application to Complexity Theory

Jürg Ganz
1996 Finite Fields and Their Applications  
Binary representations of finite fields are defined as an injective mapping from a finite field to l-tuples with components in ͕0, 1͖ where 0 and 1 are elements of the field itself.  ...  This permits one to study the algebraic complexity of a particular binary representation, i.e., the minimum number of additions and multiplications in the field needed to compute the binary representation  ...  Recently, this representation has also been used in [4] to devise an efficient factoring algorithm for polynomials over every finite field.  ... 
doi:10.1006/ffta.1996.0022 fatcat:pevl73mnfndhrbugi2urgia6s4

A new architecture for a parallel finite field multiplier with low complexity based on composite fields

C. Paar
1996 IEEE transactions on computers  
The Karatsuba-Ofman algorithm is investigated and applied to the multiplication of polynomials over GF(2 n ).  ...  In this paper a new bit-parallel structure for a multiplier with low complexity in Galois elds is introduced. The multiplier operates over composite elds GF((2 n ) m ), with k = nm.  ...  1, m = 2 t , over GF(2 n ) can be upper bounded by: T T xor (2dlog 2 ne + 3 log 2 m) + T and : (19) Reduction Modulo the Primitive Polynomial This section describes the second step of the eld multiplication  ... 
doi:10.1109/12.508323 fatcat:a3osulkwavhspljl4rwlfs7qne

Complexity of computation in finite fields

S. B. Gashkov, I. S. Sergeev
2013 Journal of Mathematical Sciences  
Аннотация Review of some works about the complexity of implementation of arithmetic operations in finite fields by boolean circuits.  ...  Multiplication in General Finite Fields Let M q,f (n) be the total number of operations over GF (q) (or the complexity over GF (q)) required for multiplication of polynomials modulo f , deg f = n.  ...  Cantor's method [53] for polynomial multiplication over finite fields. The asymptotic complexity of this method is slightly greater than FFT's (e.g.  ... 
doi:10.1007/s10958-013-1350-5 fatcat:q52hckwy2zaclorf6u3wt52gcy

Computational linear algebra over finite fields [article]

Jean-Guillaume Dumas, Clément Pernet (INRIA Grenoble Rhône-Alpes / LIG Laboratoire d'Informatique de Grenoble)
2012 arXiv   pre-print
We present here algorithms for efficient computation of linear algebra problems over finite fields.  ...  ij = a ij + δ i * a kj ), as is the case with matrices over finite fields  ...  Tiny finite fields The practical efficiency of matrix multiplication depends highly on the representation of field elements.  ... 
arXiv:1204.3735v1 fatcat:a3j26roivfd55fsf3sk7ahxgk4

Polynomial evaluation over finite fields: new algorithms and complexity bounds [article]

Michele Elia, Joachim Rosenthal, Davide Schipani
2011 arXiv   pre-print
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough.  ...  In the following we describe a method to evaluate polynomials with coefficients over a finite field F p s , and estimate its complexity in terms of field multiplications and sums.  ...  A polynomial P (x) over the binary field is simply decomposed into a sum of two polynomials by collecting odd and even powers of x as P (x) = P 1,0 (x 2 ) + xP 1,1 (x 2 ) = P 1,0 (x) 2 + xP 1,1 (x) 2 .  ... 
arXiv:1102.4772v2 fatcat:bus7gq5cpbhmbdcs3agdid3g7e

On the complexity of multiplication in finite fields

A. Lempel, G. Seroussi, S. Winograd
1983 Theoretical Computer Science  
We derive a new linear lower bound, and we describe an algorithm leading to a quasi-linear upper bound.  ...  In this paper we study the bilinear complexity of multiplying two arbitrary elements from an nth degree extension (I, of a finite field .F, and the related problem of multiplying, over F, two polynomials  ...  Statement of main results In this paper, we derive new lower and upper boun.::i for the bilinear complexity of rp and of r over finite fields. In Section 3 we prove the following theorem: Theorem 1.  ... 
doi:10.1016/0304-3975(83)90108-1 fatcat:2e57my2ejrhpja4hmjetn6laie

On Repeated Squarings in Binary Fields [chapter]

Kimmo U. Järvinen
2009 Lecture Notes in Computer Science  
In this paper, we discuss the problem of computing repeated squarings (exponentiations to a power of 2) in finite fields with polynomial basis.  ...  Repeated squarings have importance, especially, in elliptic curve cryptography where they are used in computing inversions in the field and scalar multiplications on Koblitz curves.  ...  Squaring in Binary Fields A binary field, F 2 m , is generated from the ring of polynomials over F 2 , F 2 [x], with an irreducible polynomial, p(x), with a degree m by setting F 2 m : F 2 [x]/p(x).  ... 
doi:10.1007/978-3-642-05445-7_21 fatcat:mytvzs6mqvf4vh4tx3gjq5tuay

Towers of Function Fields over Non-prime Finite Fields

Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth
2015 Moscow Mathematical Journal  
Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places.  ...  Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity A( ), for = p n with p prime and n > 3 odd. A modular interpretation of the towers is given as well.  ...  ., hence our lower bound is only around 6 % below the Drinfeld-Vlȃduţ upper bound, for large odd-degree extensions of the binary field F 2 .  ... 
doi:10.17323/1609-4514-2015-15-1-1-29 fatcat:7zzaeteo3rhnpk6iipidhvkmfi

Counting points on abelian surfaces over finite fields with Elkies's method [article]

Jean Kieffer
2022 arXiv   pre-print
If A is defined over a number field, we count points on A modulo sufficiently many primes in Otilde(log6 q) binary operations on average.  ...  We generalize Elkies's method, an essential ingredient in the SEA algorithm to count points on elliptic curves over finite fields of large characteristic, to the setting of p.p. abelian surfaces.  ...  To this date, Schoof's polynomial-time algorithm [42, 40] remains the central approach to point counting for abelian varieties of dimension 2 or more over finite fields of large characteristic, and much  ... 
arXiv:2203.02009v1 fatcat:btl3mliq2bd2haxezylouscfem
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