Polynomial Evaluation and Interpolation on Special Sets of Points
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Polynomial Evaluation Engl2
Alexander Rostovtsev, Alexey Bogdanov and Mikhail Mikhaylov St. Petersburg state Polytechnic University [email protected] Secure evaluation of polynomial using privacy ring homomorphisms Method of secure evaluation of polynomial y = F(x1, …, xk) over some rings on untrusted computer is proposed. Two models of untrusted computer are considered: passive and active. In passive model untrusted computer correctly computes polyno- mial F and tries to know secret input (x1, …, xk) and output y. In active model un- trusted computer tries to know input and output and tries to change correct output y so that this change cannot be determined. Secure computation is proposed by using one-time privacy ring homomorphism /n ® /n[z]/(f(z)), n = pq, generated by trusted computer. In the case of active model secret check point v = F(u1, …, uk) is used. Trusted computer generates poly- nomial f(z) = (z - t)(z + t), t Î /n and input Xi(z) Î /n[z]/(f(z)) such that Xi(t) º xi (mod n) for passive model, and f(z) = (z - t1)(z - t2)(z - t3), ti Î /n and input Xi(z) Î /n[z]/(f(z)) such that Xi(t1) º xi (mod n), Xi(t2) º ui (mod n) for active model. Untrusted computer computes function Y(z) = F(X1(z), …, Xk(z)) in the ring /n[z]/(f(z)). For passive model trusted computer determines secret output y º Y(t) (mod n). For active model trusted computer checks that Y(t2) º v (mod n), then deter- mines correct output y º Y(t1) (mod n). -
Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants
Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants Andreas Björklund1, Petteri Kaski2, and Ryan Williams3 1 Department of Computer Science, Lund University, Lund, Sweden 2 Department of Computer Science, Aalto University, Helsinki, Finland 3 Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA Abstract We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q − 1, our first data structure relies on (d + 1)n+2 tabulated values of P to produce the value of P at any of the qn points using O(nqd2) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q − 1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s+1)n+s tabulated values to produce the value of P at any point using O(nqssq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (2004), Saraf and Sudan (2008), and Dvir (2009) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for com- puting integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (2011) that captures numerous fundamental algebraic and combin- atorial invariants such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. -
Factoring Polynomials Over Finite Fields
Factoring Polynomials over Finite Fields More precisely: Factoring and testing irreduciblity of sparse polynomials over small finite fields Richard P. Brent MSI, ANU joint work with Paul Zimmermann INRIA, Nancy 27 August 2009 Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 1 / 64 Outline Introduction I Polynomials over finite fields I Irreducible and primitive polynomials I Mersenne primes Part 1: Testing irreducibility I Irreducibility criteria I Modular composition I Three algorithms I Comparison of the algorithms I The “best” algorithm I Some computational results Part 2: Factoring polynomials I Distinct degree factorization I Avoiding GCDs, blocking I Another level of blocking I Average-case complexity I New primitive trinomials Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 2 / 64 Polynomials over finite fields We consider univariate polynomials P(x) over a finite field F. The algorithms apply, with minor changes, for any small positive characteristic, but since time is limited we assume that the characteristic is two, and F = Z=2Z = GF(2). P(x) is irreducible if it has no nontrivial factors. If P(x) is irreducible of degree r, then [Gauss] r x2 = x mod P(x): 2r Thus P(x) divides the polynomial Pr (x) = x − x. In fact, Pr (x) is the product of all irreducible polynomials of degree d, where d runs over the divisors of r. Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 3 / 64 Counting irreducible polynomials Let N(d) be the number of irreducible polynomials of degree d. Thus X r dN(d) = deg(Pr ) = 2 : djr By Möbius inversion we see that X rN(r) = µ(d)2r=d : djr Thus, the number of irreducible polynomials of degree r is ! 2r 2r=2 N(r) = + O : r r Since there are 2r polynomials of degree r, the probability that a randomly selected polynomial is irreducible is ∼ 1=r ! 0 as r ! +1. -
The Fast Fourier Transform and Applications to Multiplication
The Fast Fourier Transform and Applications to Multiplication Analysis of Algorithms Prepared by John Reif, Ph.D. Topics and Readings: - The Fast Fourier Transform • Reading Selection: • CLR, Chapter 30 Advanced Material : - Using FFT to solve other Multipoint Evaluation Problems - Applications to Multiplication Nth Roots of Unity • Assume Commutative Ring (R,+,·, 0,1) • ω is principal nth root of unity if – ωk ≠ 1 for k = 1, …, n-1 – ωn = 1, and n-1 ∑ω jp = 0 for 1≤ p ≤ n j=0 2πi/n • Example: ω = e for complex numbers Example of nth Root of Unity for Complex Numbers ω = e 2 π i / 8 is the 8th root of unity Fourier Matrix #1 1 … 1 $ % n−1 & %1 ω … ω & M (ω) = %1 ω 2 … ω 2(n−1) & n % & % & % n−1 (n−1)(n−1) & '1 ω … ω ( ij so M(ω)ij = ω for 0 ≤ i, j<n #a0 $ % & given a =% & % & 'a n-1 ( Discrete Fourier Transform T Input a column n-vector a = (a0, …, an-1) Output an n-vector which is the product of the Fourier matrix times the input vector DFTn (a) = M(ω) x a "f0 # $ % = $ % where $ % &fn-1 ' n-1 f = a ik i ∑ kω k=0 Inverse Fourier Transform -1 -1 DFTn (a) = M(ω) x a 1 Theorem M(ω)-1 = ω -ij ij n proof We must show M(ω)⋅M(ω)-1 = I 1 n-1 1 n-1 ∑ωikω -kj = ∑ω k(i-j) n k=0 n k=0 $0 if i-j ≠ 0 = % &1 if i-j = 0 n-1 using identity ∑ω kp = 0, for 1≤ p < n k=0 Fourier Transform is Polynomial Evaluation at the Roots of Unity T Input a column n-vector a = (a0, …, an-1) T Output an n-vector (f0, …, fn-1) which are the values polynomial f(x)at the n roots of unity "f0 # DFT (a) = $ % where n $ % $ % &fn-1 ' i fi = f(ω ) and n-1 f(x) = a x j ∑ j -
A Casual Primer on Finite Fields
A very brief introduction to finite fields Olivia Di Matteo December 10, 2015 1 What are they and how do I make one? Definition 1 (Finite fields). Let p be a prime number, and n ≥ 1 an integer. A finite field n n n of order p , denoted by Fpn or GF(p ), is a collection of p objects and two binary operations, addition and multiplication, such that the following properties hold: 1. The elements are closed under addition modulo p, 2. The elements are closed under multiplication modulo p, 3. For all non-zero elements, there exists a multiplicative inverse. 1.1 Prime dimensions Nothing much to see here. In prime dimension p, the finite field Fp is very simple: Fp = Zp = f0; 1; : : : ; p − 1g: (1) 1.2 Power of prime dimensions and field extensions Fields of prime-power dimension are constructed by extending a field of smaller order using a primitive polynomial. See section 2.1.2 in [1]. 1.2.1 Primitive polynomials Definition 2. Consider a polynomial n q(x) = a0 + a1x + ··· + anx ; (2) having degree n and coefficients ai 2 Fq. Such a polynomial is called monic if an = 1. Definition 3. A polynomial n q(x) = a0 + a1x + ··· + anx ; ai 2 Fq (3) is called irreducible if q(x) has positive degree, and q(x) = u(x)v(x); (4) 1 and either u(x) or v(x) a constant polynomial. In other words, the equation n q(x) = a0 + a1x + ··· + anx = 0 (5) has no solutions in the field Fq. Example 1 (Irreducible polynomial). -
Type-II Optimal Polynomial Bases
Type-II Optimal Polynomial Bases Daniel J. Bernstein1 and Tanja Lange2 1 Department of Computer Science (MC 152) University of Illinois at Chicago, Chicago, IL 60607{7053, USA [email protected] 2 Department of Mathematics and Computer Science Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands [email protected] Abstract. In the 1990s and early 2000s several papers investigated the relative merits of polynomial-basis and normal-basis computations for F2n . Even for particularly squaring-friendly applications, such as implementations of Koblitz curves, normal bases fell behind in performance unless a type-I normal basis existed for F2n . In 2007 Shokrollahi proposed a new method of multiplying in a type-II normal basis. Shokrol- lahi's method efficiently transforms the normal-basis multiplication into a single multiplication of two size-(n + 1) polynomials. This paper speeds up Shokrollahi's method in several ways. It first presents a simpler algorithm that uses only size-n polynomials. It then explains how to reduce the transformation cost by dynamically switching to a `type-II optimal polynomial basis' and by using a new reduction strategy for multiplications that produce output in type-II polynomial basis. As an illustration of its improvements, this paper explains in detail how the multiplication over- head in Shokrollahi's original method has been reduced by a factor of 1:4 in a major cryptanalytic computation, the ongoing attack on the ECC2K-130 Certicom challenge. The resulting overhead is also considerably smaller than the overhead in a traditional low-weight-polynomial-basis ap- proach. This is the first state-of-the-art binary-elliptic-curve computation in which type-II bases have been shown to outperform traditional low-weight polynomial bases. -
Arxiv:0911.1393V5 [Cs.CC] 1 Jul 2013 Author’S Addresses: C
0 Most Tensor Problems are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant of a 4-tensor is NP-, #P-, and VNP-hard. We shall argue that our results provide another view of the boundary separating the computa- tional tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra General Terms: Tensors, Decidability, Complexity, Approximability Additional Key Words and Phrases: Numerical multilinear algebra, tensor rank, tensor eigenvalue, tensor singular value, tensor spectral norm, system of multilinear equations, hyperdeterminants, symmetric ten- sors, nonnegative definite tensors, bivariate matrix polynomials, NP-hardness, #P-hardness, VNP-hardness, undecidability, polynomial time approximation schemes ACM Reference Format: Hillar, C. J., and Lim, L.-H. 2012. Most tensor problems are NP-hard J. ACM 0, 0, Article 0 (June 2013), 38 pages. DOI = 10.1145/0000000.0000000 http://doi.acm.org/10.1145/0000000.0000000 1. -
Time Complexity Analysis of Cloud Data Security: Elliptical Curve and Polynomial Cryptography
International Journal of Computer Sciences and Engineering Open Access Research Paper Vol.-7, Issue-2, Feb 2019 E-ISSN: 2347-2693 Time Complexity Analysis of Cloud Data Security: Elliptical Curve and Polynomial Cryptography D.Pharkkavi1*, D. Maruthanayagam2 1Sri Vijay Vidyalaya College of Arts & Science, Dharmapuri, Tamilnadu, India 2PG and Research Department of Computer Science, Sri Vijay Vidyalaya College of Arts & Science, Dharmapuri, Tamilnadu, India *Corresponding Author: [email protected] DOI: https://doi.org/10.26438/ijcse/v7i2.321331 | Available online at: www.ijcseonline.org Accepted: 10/Feb/2019, Published: 28/Feb/2019 Abstract- Encryption becomes a solution and different encryption techniques which roles a significant part of data security on cloud. Encryption algorithms is to ensure the security of data in cloud computing. Because of a few limitations of pre-existing algorithms, it requires for implementing more efficient techniques for public key cryptosystems. ECC (Elliptic Curve Cryptography) depends upon elliptic curves defined over a finite field. ECC has several features which distinguish it from other cryptosystems, one of that it is relatively generated a new cryptosystem. Several developments in performance have been found out during the last few years for Galois Field operations both in Normal Basis and in Polynomial Basis. On the other hand, there is still some confusion to the relative performance of these new algorithms and very little examples of practical implementations of these new algorithms. Efficient implementations of the basic arithmetic operations in finite fields GF(2m) are need for the applications of coding theory and cryptography. The elements in GF(2m) know how to be characterized in a choice of bases. -
Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases
Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases S. Karamia, M. Ahmadnasabb, M. Hadizadehd, A. Amiraslanic,d aDepartment of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran bDepartment of Mathematics, University of Kurdistan, Sanandaj, Iran cSchool of STEM, Department of Mathematics, Capilano University, North Vancouver, BC, Canada dFaculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran Abstract Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In this work, we present formulas and techniques for polynomial multiplications expressed in a variety of well-known polynomial bases without any change of basis. In particular, we take into consideration degree-graded polynomial bases including, but not limited to orthogonal polynomial bases and non-degree-graded polynomial bases including the Bernstein and Lagrange bases. All of the described polynomial multiplication formulas and tech- niques in this work, which are mostly presented in matrix-vector forms, preserve the basis in which the polynomials are given. Furthermore, using the results of direct multiplication of polynomials, we devise techniques for intra-basis polynomial division in the polynomial bases. A generalization of the well-known \long division" algorithm to any degree-graded polynomial basis is also given. The proposed framework deals with matrix-vector computations which often leads to well-structured matrices. Finally, an application of the presented techniques in constructing the Galerkin repre- sentation of polynomial multiplication operators is illustrated for discretization of a linear elliptic problem with stochastic coefficients. -
Hardware and Software Normal Basis Arithmetic for Pairing Based
Hardware and Software Normal Basis Arithmetic for Pairing Based Cryptography in ? Characteristic Three R. Granger, D. Page and M. Stam Department of Computer Science, University of Bristol, MerchantVenturers Building, Wo o dland Road, Bristol, BS8 1UB, United Kingdom. fgranger, page, [email protected] Abstract. Although identity based cryptography o ers a number of functional advantages over conventional public key metho ds, the compu- tational costs are signi cantly greater. The dominant part of this cost is the Tate pairing which, in characteristic three, is b est computed using the algorithm of Duursma and Lee. However, in hardware and constrained environments this algorithm is unattractive since it requires online com- putation of cub e ro ots or enough storage space to pre-compute required results. We examine the use of normal basis arithmetic in characteristic three in an attempt to get the b est of b oth worlds: an ecient metho d for computing the Tate pairing that requires no pre-computation and that may also be implemented in hardware to accelerate devices such as smart-cards. Since normal basis arithmetic in characteristic three has not received much attention b efore, we also discuss the construction of suitable bases and asso ciated curve parameterisations. 1 Intro duction Since it was rst suggested in 1984 by Shamir [29], the concept of identity based cryptography has b een an attractive target for researchers b ecause of the p oten- tial for simplifying conventional approaches to public key based systems. The central idea is that the public key for a user is simply their identity and is hence implicitly known to all other users. -
Lecture 8: Stream Ciphers - LFSR Sequences
Lecture 8: Stream ciphers - LFSR sequences Thomas Johansson T. Johansson (Lund University) 1 / 42 Introduction Symmetric encryption algorithms are divided into two main categories, block ciphers and stream ciphers. Block ciphers tend to encrypt a block of characters of a plaintext message using a fixed encryption transformation A stream cipher encrypt individual characters of the plaintext using an encryption transformation that varies with time. A stream cipher built around LFSRs and producing one bit output on each clock = classic stream cipher design. T. Johansson (Lund University) 2 / 42 A stream cipher z , z ,... keystream 1 2 generator m , m , . .? c , c ,... 1 2 - 1 2 - m z = z1, z2,... keystream key K T. Johansson (Lund University) 3 / 42 A stream cipher Design goal is to efficiently produce random-looking sequences that are as “indistinguishable” as possible from truly random sequences. Recall the unbreakable Vernam cipher. For a synchronous stream cipher, a known-plaintext attack (or chosen-plaintext or chosen-ciphertext) is equivalent to having access to the keystream z = z1, z2, . , zN . We assume that an output sequence z of length N from the keystream generator is known to Eve. T. Johansson (Lund University) 4 / 42 Type of attacks Key recovery attack: Eve tries to recover the secret key K. Distinguishing attack: Eve tries to determine whether a given sequence z = z1, z2, . , zN is likely to have been generated from the considered stream cipher or whether it is just a truly random sequence. Distinguishing attack is a much weaker attack T. Johansson (Lund University) 5 / 42 Distinguishing attack Let D(z) be an algorithm that takes as input a length N sequence z and as output gives either “X” or “RANDOM”. -
Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF(2 M)
Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF (2m) using polynomial bases by Fan Zhou B.Eng., Zhejiang University, 2013 A Report Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF ENGINEERING in the Department of Electrical and Computer Engineering c Fan Zhou, 2018 University of Victoria All rights reserved. This report may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF (2m) using polynomial bases by Fan Zhou B.Eng., Zhejiang University, 2013 Supervisory Committee Dr. Fayez Gebali, Supervisor (Department of Electrical and Computer Engineering) Dr. Watheq El-Kharashi, Departmental Member (Department of Electrical and Computer Engineering) iii ABSTRACT Finite field arithmetic is important for the field of information security. The inversion operation consumes most of the time and resources among all finite field arithmetic operations. In this report, two main classes of algorithms for inversion are studied. The first class of inverters is Extended Euclidean based inverters. Extended Euclidean Algorithm is an extension of Euclidean algorithm that computes the greatest common divisor. The other class of inverters is based on Fermat's little theorem. This class of inverters is also called multiplicative based inverters, because, in these algorithms, the inversion is performed by a sequence of multiplication and squaring. This report represents a literature review of inversion algorithm and implements a multiplicative based inverter and an Extended Euclidean based inverter in MATLAB. The experi- mental results show that inverters based on Extended Euclidean Algorithm are more efficient than inverters based on Fermat's little theorem.