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Integer Modified Sine-Cosine Transforms Type VII. A construction Method and Separable Directional Adaptive Transforms for Intra Prediction with 8 x 8 Chroma Blocks in Image/Video Coding

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CYBERNETICS AND SYSTEMS ANALYSIS 2021年第1期57卷 155-164页

作者： Hnativ, L. O. Luts, V. K. Natl Acad Sci Ukraine VM Glushkov Inst Cybernet Kiev Ukraine

A matrix method for constructing a modified order-8 integer sine-cosine transform type VII is proposed. Based on the method, two order-8 integer modified sine-cosine transforms type VII are constructed and algorithms for fast computing of these transforms are developed, which require only integer operations. These algorithms are of low multiplicative complexity, which is 7 and 10.5 times less, and they require 23.3 and 44.2% less additional operations than the well-known algorithm of the discrete sine transform type VII. These transforms have higher coding gain performance for quality and compression ratio compared to the well-known sine transforms. Algorithms for fast computing of 2D separate directional integer cosine and modified sine-cosine type VII adaptive transforms for intra prediction with 8 x 8 chroma blocks are developed. These algorithms have low multiplicative complexity, which is 6.6 and 16.5 times less than that of the well-known algorithms.

关键词： discrete cosine transform discrete sine transform discrete sine-cosine transform integer cosine transform integer sine transform integer modified sine-cosine transform separate directional adaptive transform scaled transform multiplicative complexity intra prediction video coding H 264 H 265

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USING DUALITY FOR THE SYNTHESIS OF AN OPTIMAL ALGORITHM INVOLVING MATRIX MULTIPLICATION

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INFORMATION PROCESSING LETTERS 1981年第2期13卷 48-49页

作者： MAKAROV, OM Institute of Biology of the South Seas 2 Nakimov Street Sevastopol U.S.S.R.

The multiplicative complexity of computing a set of bilinear forms over a noncommutative ring is considered. An expression is derived to represent this complexity. Problems are presented, showing how the expression can be computed in the class of bilinear algorithms. In the computations, matrices of dimensions serve as the basis for representing the problems. Mathematical notation.

关键词： An optimal algorithm duality multiplicative complexity

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Short non-interactive cryptographic proofs

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JOURNAL OF CRYPTOLOGY 2000年第4期13卷 449-472页

作者： Boyar, J Damgård, I Peralta, R Univ So Denmark Dept Math & Comp Sci Odense Denmark Univ Aarhus BRICS Dept Comp Sci DC-8000 Aarhus C Denmark Yale Univ Dept Comp Sci New Haven CT 06520 USA

We show how to produce short proofs of theorems such that a distrusting Verifier can be convinced that the theorem is true yet obtains no information about the proof itself. We assume the theorem is represented by a boolean circuit, of size m gates, which is satisfiable if and only if the theorem holds. We use bit commitments of size k and bound the probability of false proofs going undetected by 2(-r). We obtain non-interactive zero-knowledge proofs of size O(mk(log m + r)) bits. In the random oracle model, we obtain non-interactive proofs of size O(m(log m + r) + rk) bits. By simulating a random oracle, we obtain non-interactive proofs which are short enough to be used in practice. We call the latter proofs "discreet.".

关键词： cryptographic proofs non-interactive proofs discreet proofs circuit complexity multiplicative complexity

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The complexity of bivariate power series arithmetic

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THEORETICAL COMPUTER SCIENCE 2003年第1-3期295卷 65-83页

作者： Bläser, M Med Univ Lubeck Inst Theoret Informat D-23560 Lubeck Germany

Inspired by Schonhage's discussion in the Proc. 11th Applied Algebra and Error Correcting Codes Conference (AAECC), Lecture Notes in Comput. Sci., Springer, Berlin, Vol. 948, 1995 pp. 70, we study the multiplicative complexity of the multiplication, squaring, inversion, and division of bivariate power series modulo the "triangular" and "quadratic" ideals (Xd+1,(XY)-Y-d,Xd-1 Y-2,..., Yd+1) and (Xd+1, Yd+1), respectively. For multiplication, we obtain the lower bounds 5/4 d(2) -O(d) and 2 1/3 d(2) - O(d) for the triangular and quadratic case, respectively, opposed to the upper bounds 3/2 d(2) + O(d) and 3d(2) + O(d). For squaring, we prove the lower bounds 7/8 d(2) - O(d) and 1 3/5 d(2) - O(d). As upper bounds, we have d(2) + O(d) and 2 1/2 d(2) + O(d) for the triangular and quadratic case, respectively. Concerning inversion, the obtained lower bounds coincide with those of squaring. As upper bounds, we show 3 5/6 d(2) +O(d) and 8 1/3 d(2) + O(d), respectively. The lower bounds for division are those of multiplication. The upper bounds follow from combining the bounds for inversion and multiplication. All of the above lower bounds hold over arbitrary fields (in the case of multiplication and division) and over fields of characteristic distinct from two (in the case of squaring and inversion), respectively. All upper bounds are valid for fields that "support FFTs", that is, fields that have characteristic zero and contain all roots of unity. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： multiplicative complexity bivariate power series multiplication division

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Beyond the Alder-Strassen bound

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THEORETICAL COMPUTER SCIENCE 2005年第1期331卷 3-21页

作者： Bläser, M Med Univ Lubeck Inst Theoret Informat D-23560 Lubeck Germany

We prove a lower bound of 5/2 n(2) - 3n for the multiplicative complexity of n x n-matrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity C (A) of the multiplication in A is bounded from below by 5/2 dim A - 3(n(1) + ... + n(t)) if the decomposition of A congruent to A(1) x ... x A(t) into simple algebras A(tau) congruent to D-tau(tau)tau(n)xn contains only noncommutative factors, that is, the division algebra D-tau is noncommutative or ntau greater than or equal to 2. We also deal with the complexity of multiplication in algebras with nonzero radical. We present an example that shows that our methods in the semisimple case cannot be applied directly to this problem. We exhibit lower bound techniques for C (A) that yield bounds still significantly above the Alder-Strassen bound. The main application is the lower bound C (T-n (k)) greater than or equal to (2 1/8 - o(1)) dim T-n (k) for the multiplicative complexity of multiplication in the algebra Tn (k) of upper triangular n x n-matrices. (C) 2004 Elsevier B.V. All rights reserved.

关键词： associative algebra lower bound multiplicative complexity

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ON MULTIPLICATION OF POLYNOMIALS MODULO A POLYNOMIAL

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SIAM JOURNAL ON COMPUTING 1980年第2期9卷 225-229页

作者： WINOGRAD, S

The multiplicative complexity of the direct product of algebras $A_p $ of polynomials modulo a polynomial P is studied. In particular, we show that if P and Q are irreducible polynomials then the multiplicative complexity of $A_{\text{P}} \times A_{\text{Q}} $ is $2\deg ({\text{P}})\deg ({\text{Q}}) - {\text{k}}$, where k is the number of factors of P in the field extended by a root of ${\text{Q}}$.

关键词： complexity of computations multiplicative complexity product of polynomials

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Optimizing S-Box Implementations for Several Criteria Using SAT Solvers 23rd

Optimizing S-Box Implementations for Several Criteria Using ...

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23rd International Conference on Fast Software Encryption (FSE)

作者： Stoffelen, Ko Radboud Univ Nijmegen Digital Secur Nijmegen Netherlands

ISBN: (纸本)9783662529935;9783662529928

We explore the feasibility of applying SAT solvers to optimizing implementations of small functions such as S-boxes for multiple optimization criteria, e.g., the number of nonlinear gates and the number of gates. We provide optimized implementations for the S-boxes used in Ascon, ICEPOLE, Joltik/Piccolo, Keccak/Ketje/Keyak, LAC, Minalpher, PRIMATEs, Prost, and RECTANGLE, most of which are candidates in the secound round of the CAESAR competition. We then suggest a new method to optimize for circuit depth and we make tooling publicly available to find efficient implementations for several criteria. Furthermore, we illustrate with the 5-bit S-box of PRIMATEs how multiple optimization criteria can be combined.

关键词： S-box SAT solvers Implementation optimization multiplicative complexity Circuit depth complexity Shortest linear straightline program

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Design of Fast Transforms for High-Resolution Image and Video Coding

Design of Fast Transforms for High-Resolution Image and Vide...

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Conference on Applications of Digital Image Processing XXXII

作者： Reznik, Yuriy A. Chivukula, Ravi K. Qualcomm Inc 5775 Morehouse Dr San Diego CA 92121 USA

ISBN: (纸本)9780819477330

We review design of 4-, 8-, and 16-point transforms currently used in image and video coding standards, and compare them with fast implementations of Discrete Cosine Transform of various other sizes (including non-dyadic even and odd numbers) in the range of 2-64. We show that among such transforms there exist few that offer better complexity/coding gain tradeoffs than current dyadic-sized transforms. In our construction and analysis we utilize an array of known techniques (such as Heideman's mapping between DCT and DFT, Winograd short length DFT modules, prime-factor-and common-factor algorithms), and also offer a new factorization scheme for even-sized scaled transforms.

关键词： discrete cosine transform DCT FFT factorization multiplicative complexity scaled transform video coding standards H.261 JPEG MPEG-1 MPEG-2 MPEG-4 H.263 H.264.

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Generalized Polynomial Decomposition for S-boxes with Application to Side-Channel Countermeasures 19th

Generalized Polynomial Decomposition for S-boxes with Applic...

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19th International Conference on Cryptographic Hardware and Embedded Systems (CHES)

作者： Goudarzi, Dahmun Rivain, Matthieu Vergnaud, Damien Vivek, Srinivas CryptoExperts Paris France PSL Res Univ ENS CNRS INRIA Paris France Univ Bristol Bristol Avon England

ISBN: (纸本)9783319667874;9783319667867

Masking is a widespread countermeasure to protect implementations of block-ciphers against side-channel attacks. Several masking schemes have been proposed in the literature that rely on the efficient decomposition of the underlying s-box(es). We propose a generalized decomposition method for s-boxes that encompasses several previously proposed methods while providing new trade-offs. It allows to evaluate n lambda-bit to m lambda-bit s-boxes for any integers n, m, lambda >= 1 by seeing it a sequence of m n-variate polynomials over F-2 lambda and by trying to minimize the number of multiplications over F-2 lambda.

关键词： S-box decomposition multiplicative complexity Side-channel countermeasure Masking Software implementation Block-cipher

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Fast Computing of Discrete Cosine and Sine Transforms of Types VI and VII

Fast Computing of Discrete Cosine and Sine Transforms of Typ...

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Conference on Applications of Digital Image Processing XXXIV

作者： Chivukula, Ravi K. Reznik, Yuriy A. Qualcomm Inc San Diego CA 92122 USA

ISBN: (纸本)9780819487452

We propose fast algorithms for computing Discrete Sine and Discrete Cosine Transforms (DCT and DST) of types VI and VII. Particular attention is paid to derivation of fast algorithms for computing DST-VII of lengths 4 and 8, which are currently under consideration for inclusion in ISO/IEC/ITU-T High Efficiency Video Coding (HEVC) standard.

关键词： Discrete Cosine Transform Discrete Sine Transform DCT DST DST-VII factorization multiplicative complexity video coding HEVC

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