Optimal FIR filter design

The design of Finite Impulse Response (FIR) digital filters that considers both phase and magnitude specifications is investigated. This dissertation is divided into two parts. In Part I we present our implementation of an algorithm for the design of minimum phase filters. In Part II we investigate the design of FIR filters in the complex domain and develop a new powerful design method for digital FIR filters with arbitrary specification of magnitude and phase;Part I considers the design of minimum-phase filters. The method presented uses direct factorization of the transfer function of a companion Parks-McClellan linear-phase filter of twice the length of the desired minimum-phase filter. The minimum-phase filter is derived with excision of half the zeros of the companion linear-phase filter. The zeros of the prototype filter are found using Laguerre\u27s method. We will present our implementation of the design method, and describe some practical aspects and problems associated with the design of minimum-phase filters;Part II investigates the design of optimal Chebychev FIR filters in the complex domain. The design of FIR filters with arbitrary specification of magnitude and phase is formulated into a problem of complex approximation. The method developed is capable of designing filters with real or complex coefficients. Complex impulse response designs are an extension of the real coefficient case based on a proper selection of the approximating basis functions;The minimax criterion is used and the complex Chebychev approximation is posed as a minimization problem in linear optimization. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang (14). The relaxation of the linear-phase constraint results in a reduction of the number of coefficients compared to linear-phase designs. Linear-phase filters are a special case of our filter design approach. We examine the design of frequency selective filters with or without the conjugate symmetry, the design of one-sided, two-sided, narrowband and fullband Hilbert Transformers and differentiators

BCH 부호를 이용한 FrodoKEM의 성능 개선 및 동형 비교를 위한 합성함수에 의한 부호 함수의 미니맥스 근사

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 전기·정보공학부, 2020. 8. 노종선.In this dissertation, two main contributions are given as; Performance improvement of FrodoKEM using Gray and error-correcting codes (ECCs). Optimal minimax polynomial approximation of sign function by composite polynomial for homomorphic comparison. First, modification of FrodoKEM using Gray codes and ECCs is studied. Lattice-based scheme is one of the most promising schemes for post-quantum cryptography (PQC). Among many lattice-based cryptosystems, FrodoKEM is a well-known key-encapsulation mechanism (KEM) based on (plain) learning with errors problems and is advantageous in that the hardness is based on the problem of unstructured lattices. Many lattice-based cryptosystems adopt ECCs to improve their performance, such as LAC, Three Bears, and Round5 which were presented in the NIST PQC Standardization Round 2 conference. However, for lattice-based cryptosystems that do not use ring structures such as FrodoKEM, it is difficult to use ECCs because the number of transmitted symbols is small. In this dissertation, I propose a method to apply Gray and ECCs to FrodoKEM by encoding the bits converted from the encrypted symbols. It is shown that the proposed method improves the security level and/or the bandwidth of FrodoKEM, and 192 message bits, 50\% more than the original 128 bits, can be transmitted using one of the modified Frodo-640's. Second, an optimal minimax polynomial approximation of sign function by a composite polynomial is studied. The comparison function of the two numbers is one of the most commonly used operations in many applications including deep learning and data processing systems. Several studies have been conducted to efficiently evaluate the comparison function in homomorphic encryption schemes which only allow addition and multiplication for the ciphertext. Recently, new comparison methods that approximate sign function using composite polynomial in the homomorphic encryption, called homomorphic comparison operation, were proposed and it was proved that the methods have optimal asymptotic complexity. In this dissertation, I propose new optimal algorithms that approximate the sign function in the homomorphic encryption by using composite polynomials of the minimax approximate polynomials, which are constructed by the modified Remez algorithm. It is proved that the number of required non-scalar multiplications and depth consumption for the proposed algorithms are less than those for any methods that use a composite polynomial of component polynomials with odd degree terms approximating the sign function, respectively. In addition, an optimal polynomial-time algorithm for the proposed homomorphic comparison operation is proposed by using dynamic programming. As a result of numerical analysis, for the case that I want to minimize the number of non-scalar multiplications, the proposed algorithm reduces the required number of non-scalar multiplications and depth consumption by about 33% and 35%, respectively, compared to those for the previous work. In addition, for the case that I want to minimize the depth consumption, the proposed algorithm reduces the required number of non-scalar multiplications and depth consumption by about 10% and 47%, respectively, compared to those for the previous work.이 학위 논문에서는, 다음 두 가지 내용이 연구되었다. FrodoKEM을 그레이 부호 및 오류정정부호를 사용하여 개선 동형 비교 연산을 위해 합성 다항식을 사용한 부호 함수의 최적 미니맥스 다항식 근사 먼저, 그레이 부호 및 오류정정부호를 사용하여 FrodoKEM을 변형시키는 방법이 연구되었다. 격자기반암호는 가장 유망한 포스트 양자 암호 스킴이다. 많은 격자기반암호 시스템 중에서 FrodoKEM은 learning with errors (LWE) 문제에 기반을 둔 잘 알려진 키-캡슐화 메커니즘 (KEM) 이며 구조를 갖지 않은 격자 문제에 기반을 둔 어려움을 가진다는 장점이 있다. NIST 포스트 양자 암호 표준화 라운드 2에 발표된 LAC, Three Bears, Round5와 같이 성능 개선을 위해 오류정정부호를 사용하는 많은 암호 시스템들이 있다. 그러나 FrodoKEM과 같이 링 구조를 사용하지 않는 격자기반 암호 시스템에서는 전송되는 심볼 개수가 작기 때문에 오류정정부호를 사용하기 어렵다. 나는 암호화된 심볼로부터 변환된 비트들을 부호화하여 오류정정부호와 그레이 부호를 FrodoKEM에 적용하는 방법을 제안하였다. 제안한 알고리즘은 FrodoKEM의 보안성 레벨 혹은 데이터전송량을 향상하고 기존 128비트보다 50\% 많은 192비트가 변형된 Frodo-640에서 전송될 수 있음을 보여주었다. 두 번째로, 합성 다항식을 사용한 부호 함수의 최적 미니맥스 다항식 근사가 연구되었다. 두 숫자의 비교 함수는 딥러닝 및 데이터 처리 시스템을 포함한 많은 응용에서 가장 많이 사용되는 연산 중 하나이다. 암호문 상에서의 덧셈과 곱셈만 지원하는 동형 암호에서 비교 함수를 효율적으로 계산하는 몇몇 연구가 진행되었다. 동형 암호에서 합성 다항식을 사용하여 부호 함수를 근사하는 비교 방법은 동형 비교 연산이라고 불리는데 최근 새로운 동형 비교 연산 방법이 제안되었고 그 방법이 최적 점근적 복잡도를 가진다는 것이 증명되었다. 본 논문에서 나는 미니맥스 근사다항식의 합성함수를 사용하여 동형암호에서 부호 함수를 근사하는 새로운 최적 알고리즘을 제안한다. 미니맥스 근사 다항식은 modified Remez 알고리즘에 의해 얻을 수 있다. 제안하는 알고리즘은 임의의 부호 함수를 근사하는 홀수 차수 항들을 가진 다항식의 합성 다항식을 사용하는 방법보다 더 적은 넌스칼라 곱 및 뎁스 소모를 사용한다는 것이 증명되었다. 또한, 제안한 동형 비교 연산에 대한 다이나믹 프로그래밍을 사용한 최적 다항시간 알고리즘이 제안되었다. 수치 분석 결과, 넌스칼라 곱 개수를 최소로 할 때, 제안하는 알고리즘은 필요한 넌스칼라 곱 개수와 뎁스 소모를 기존 방법의 필요한 넌스칼라 곱 개수 및 뎁스 소모보다 각각 33%, 35%정도 감소시킨다. 또한, 뎁스 소모를 최소로 할 때, 제안하는 알고리즘은 필요한 넌스칼라 곱 개수와 뎁스 소모를 기존 방법의 필요한 넌스칼라 곱 개수 및 뎁스 소모보다 각각 10%, 47%정도 감소시킨다.1 Introduction 1 1.1 Background 1 1.2 Overview of Dissertation 3 1.3 Notations 5 2 Preliminaries 6 2.1 NIST Post-Quantum Cryptography Standardization 6 2.1.1 Background 6 2.1.2 Categories for Security Level 7 2.1.3 List of Algorithms in NIST PQC Round 2 8 2.2 Public-Key Encryption and Key-Encapsulation Mechanism 10 2.3 Lattice-Based Cryptogaphy 13 2.3.1 Learning with Errors Problem 13 2.3.2 Overview of FrodoPKE Algorithm 14 2.3.3 Parameters of FrodoKEM 17 2.4 BCH and Gray Codes 18 2.5 Fully Homomorphic Encryption 20 2.5.1 Homomorphic Encryption 20 2.5.2 Comparison Operation in Fully Homomorphic Encryption 21 2.6 Approximation Theory 22 2.7 Algorithms for Minimax Approximation 24 3. Improvement of FrodoKEM Using Gray and BCH Codes 29 3.1 Modification of FrodoKEM with Gray and Error-Correcting Codes 33 3.1.1 Viewing FrodoPKE as a Digital Communication System 33 3.1.2 Error-Correcting Codes for FrodoPKE 34 3.1.3 Gray Coding 36 3.1.4 IND-CCA Security of Modified FrodoKEM 38 3.1.5 Evaluation of DFR 40 3.1.6 Error Dependency 43 3.2 Performance Improvement of FrodoKEM Using Gray and BCH Codes 43 3.2.1 Improving the Security Level of FrodoKEM 43 3.2.2 Increasing the Message Size of Frodo-640 47 3.2.3 Reducing the Bandwidth of Frodo-640 50 4. Homomorphic Comparison Using Optimal Composition of Minimax Approximate Polynomials 54 4.1 Introduction 54 4.1.1 Previous Works 55 4.1.2 My Contributions 56 4.2 Approximation of Sign Function by Using Optimal Composition of Minimax Approximate Polynomials 58 4.2.1 New Approximation Method for Sine Function Using Composition of the Minimax Approximate Polynomials 58 4.2.2 Optimality of Approximation of the Sign Function by a Minimax Composite Polynomial 64 4.2.3 Achieving Polynomial-Time Algorithm for New Approximation Method by Using Dynamic Programming 68 4.3 Numerical Results 80 4.3.1 Computation of the Required Non-Scalar Multiplications and Depth Consumption 81 4.3.2 Comparisons 81 5. Conclusions 88 Abstract (In Korean) 97Docto

Design &implementation of complex-valued FIR digital filters with application to migration of seismic data

One-dimensional (I-D) and two-dimensional (2-D) frequency-space seismic migration FIR digital filter coefficients are of complex values when such filters require special space domain as well as wavenumber domain characteristics. In this thesis, such FIR digital filters are designed using Vector Space Projection Methods (VSPMs), which can satisfy the desired predefined filters' properties, for 2-D and three-dimensional (3-D) seismic data sets, respectively. More precisely, the pure and the relaxed projection algorithms, which are part of the VSPM theory, are derived. Simulation results show that the relaxed version of the pure algorithm can introduce significant savings in terms of the number of iterations required. Also, due to some undesirable background artifacts on migrated sections, a modified version of the pure algorithm was used to eliminate such effects. This modification has also led to a significant reduction in the number of computations when compared to both the pure and relaxed algorithms. We further propose a generalization of the l-D (real/complex-valued) pure algorithm to multi-dimensional (m-D) complex-valued FIR digital filters, where the resulting frequency responses possess an approximate equiripple nature. Superior designs are obtained when compared with other previously reported methods. In addition, we also propose a new scheme for implementing the predesigned 2-D migration FIR filters. This realization is based on Singular Value Decomposition (SVD). Unlike the existing realization methods which are used for this geophysical application, this cheap realization via SVD, compared with the true 2-D convolution, results in satisfactory wavenumber responses. Finally, an application to seismic migration of 2-D and 3-D synthetic sections is shown to confirm our theoretical conclusions. The proposed resulting migration FIR filters are applied also to the challenging SEGIEAGE Salt model data. The migrated section (image) outperformed images obtained using other FIR filters and with other standard migration techniques where difficult structures contained in such a challenging model are imaged clearly

Certification of Real Inequalities -- Templates and Sums of Squares

We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like $\cos$, $\arctan$, $\exp$, etc. Our general framework is to use different approximation methods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations. In particular, we combine the ideas of the maxplus estimators (originally introduced in optimal control) and of the linear templates (originally introduced in static analysis by abstract interpretation). The nonlinear templates control the complexity of the semialgebraic relaxations at the price of coarsening the maxplus approximations. In that way, we arrive at a new - template based - certified global optimization method, which exploits both the precision of sums of squares relaxations and the scalability of abstraction methods. We analyze the performance of the method on problems from the global optimization literature, as well as medium-size inequalities issued from the Flyspeck project.Comment: 27 pages, 3 figures, 4 table

Minimax Approximation of Sign Function by Composite Polynomial for Homomorphic Comparison

The comparison operation for two numbers is one of the most commonly used operations in several applications, including deep learning. Several studies have been conducted to efficiently evaluate the comparison operation in homomorphic encryption schemes, termed homomorphic comparison operation. Recently, Cheon et al. (Asiacrypt 2020) proposed new comparison methods that approximate the sign function using composite polynomial in homomorphic encryption and proved that these methods have optimal asymptotic complexity. In this paper, we propose a practically optimal method that approximates the sign function by using compositions of minimax approximate polynomials. It is proved that this approximation method is optimal with respect to depth consumption and the number of non-scalar multiplications. In addition, a polynomial-time algorithm that determines the optimal compositions of minimax approximate polynomials for the proposed homomorphic comparison operation is proposed by using dynamic programming. The numerical analysis demonstrates that the proposed homomorphic comparison operation reduces running time by approximately 45\% (resp. 41\%) on average, compared with the previous algorithm if running time (resp. depth consumption) is to be minimized. In addition, when $N$ is $2^{17}$, and the precision parameter $\alpha$ is 20, the previous algorithm does not achieve 128-bit security, while the proposed algorithm achieves 128-bit security due to small depth consumption