A Multi-level Blocking Distinct Degree Factorization Algorithm

We give a new algorithm for performing the distinct-degree factorization of a polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our approach is that a finer level of blocking replaces multiplications by squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval polynomials when P(x) is sparse. As an application we give a fast algorithm to search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2), while producing a certificate that can be checked in less time than the full search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over all trinomials of given degree r. Under a plausible assumption about the distribution of factors of trinomials, the new algorithm has complexity O(r^2 (log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r. Our implementation achieves a speedup of greater than a factor of 560 over the naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our program, we have found two new primitive trinomials of degree 24036583 over GF(2) (the previous record degree was 6972593)

Reducing the Depth of Quantum FLT-Based Inversion Circuit

Works on quantum computing and cryptanalysis has increased significantly in the past few years. Various constructions of quantum arithmetic circuits, as one of the essential components in the field, has also been proposed. However, there has only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor's algorithm for Elliptic Curve Discrete Logarith Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for binary finite field. In particular, we propose follow a complete waterfall approach to translate the Itoh-Tsujii's variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (CNOT count), which contributes to reduced overall depth and gate count. Furthermore, compare the cost by firstly constructing our method and previous work's in Qiskit quantum computer simulator and perform the resource analysis. Our approach can serve as an alternative for a time-efficient implementation.Comment: version 0.

Efficient Algorithms for Elliptic Curve Cryptosystems

Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF((2n)m). The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide ablolute performance measures for the field operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation

Design of Quantum Circuits for Galois Field Squaring and Exponentiation

This work presents an algorithm to generate depth, quantum gate and qubit optimized circuits for $GF(2^m)$ squaring in the polynomial basis. Further, to the best of our knowledge the proposed quantum squaring circuit algorithm is the only work that considers depth as a metric to be optimized. We compared circuits generated by our proposed algorithm against the state of the art and determine that they require $50 \%$ fewer qubits and offer gates savings that range from $37 \%$ to $68 \%$. Further, existing quantum exponentiation are based on either modular or integer arithmetic. However, Galois arithmetic is a useful tool to design resource efficient quantum exponentiation circuit applicable in quantum cryptanalysis. Therefore, we present the quantum circuit implementation of Galois field exponentiation based on the proposed quantum Galois field squaring circuit. We calculated a qubit savings ranging between $44\%$ to $50\%$ and quantum gate savings ranging between $37 \%$ to $68 \%$ compared to identical quantum exponentiation circuit based on existing squaring circuits.Comment: To appear in conference proceedings of the 2017 IEEE Computer Society Annual Symposium on VLSI (ISVLSI 2017

A comprehensive analysis of constant-time polynomial inversion for post-quantum cryptosystems

Post-quantum cryptosystems have currently seen a surge in interest thanks to the current standardization initiative by the U.S.A. National Institute of Standards and Technology (NIST). A common primitive in post-quantum cryptosystems, in particular in code-based ones, is the computation of the inverse of a binary polynomial in a binary polynomial ring. In this work, we analyze, realize in software, and benchmark a broad spectrum of binary polynomial inversion algorithms, targeting operand sizes which are relevant for the current second round candidates in the NIST standardization process. We evaluate advantages and shortcomings of the different inversion algorithms, including their capability to run in constant-time, thus preventing timing side-channel attacks

Efficient software implementation of elliptic curves and bilinear pairings

Orientador: Júlio César Lopez HernándezTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O advento da criptografia assimétrica ou de chave pública possibilitou a aplicação de criptografia em novos cenários, como assinaturas digitais e comércio eletrônico, tornando-a componente vital para o fornecimento de confidencialidade e autenticação em meios de comunicação. Dentre os métodos mais eficientes de criptografia assimétrica, a criptografia de curvas elípticas destaca-se pelos baixos requisitos de armazenamento para chaves e custo computacional para execução. A descoberta relativamente recente da criptografia baseada em emparelhamentos bilineares sobre curvas elípticas permitiu ainda sua flexibilização e a construção de sistemas criptográficos com propriedades inovadoras, como sistemas baseados em identidades e suas variantes. Porém, o custo computacional de criptossistemas baseados em emparelhamentos ainda permanece significativamente maior do que os assimétricos tradicionais, representando um obstáculo para sua adoção, especialmente em dispositivos com recursos limitados. As contribuições deste trabalho objetivam aprimorar o desempenho de criptossistemas baseados em curvas elípticas e emparelhamentos bilineares e consistem em: (i) implementação eficiente de corpos binários em arquiteturas embutidas de 8 bits (microcontroladores presentes em sensores sem fio); (ii) formulação eficiente de aritmética em corpos binários para conjuntos vetoriais de arquiteturas de 64 bits e famílias mais recentes de processadores desktop dotadas de suporte nativo à multiplicação em corpos binários; (iii) técnicas para implementação serial e paralela de curvas elípticas binárias e emparelhamentos bilineares simétricos e assimétricos definidos sobre corpos primos ou binários. Estas contribuições permitiram obter significativos ganhos de desempenho e, conseqüentemente, uma série de recordes de velocidade para o cálculo de diversos algoritmos criptográficos relevantes em arquiteturas modernas que vão de sistemas embarcados de 8 bits a processadores com 8 coresAbstract: The development of asymmetric or public key cryptography made possible new applications of cryptography such as digital signatures and electronic commerce. Cryptography is now a vital component for providing confidentiality and authentication in communication infra-structures. Elliptic Curve Cryptography is among the most efficient public-key methods because of its low storage and computational requirements. The relatively recent advent of Pairing-Based Cryptography allowed the further construction of flexible and innovative cryptographic solutions like Identity-Based Cryptography and variants. However, the computational cost of pairing-based cryptosystems remains significantly higher than traditional public key cryptosystems and thus an important obstacle for adoption, specially in resource-constrained devices. The main contributions of this work aim to improve the performance of curve-based cryptosystems, consisting of: (i) efficient implementation of binary fields in 8-bit microcontrollers embedded in sensor network nodes; (ii) efficient formulation of binary field arithmetic in terms of vector instructions present in 64-bit architectures, and on the recently-introduced native support for binary field multiplication in the latest Intel microarchitecture families; (iii) techniques for serial and parallel implementation of binary elliptic curves and symmetric and asymmetric pairings defined over prime and binary fields. These contributions produced important performance improvements and, consequently, several speed records for computing relevant cryptographic algorithms in modern computer architectures ranging from embedded 8-bit microcontrollers to 8-core processorsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

Low-Resource and Fast Elliptic Curve Implementations over Binary Edwards Curves

Elliptic curve cryptography (ECC) is an ideal choice for low-resource applications because it provides the same level of security with smaller key sizes than other existing public key encryption schemes. For low-resource applications, designing efficient functional units for elliptic curve computations over binary fields results in an effective platform for an embedded co-processor. This thesis investigates co-processor designs for area-constrained devices. Particularly, we discuss an implementation utilizing state of the art binary Edwards curve equations over mixed point addition and doubling. The binary Edwards curve offers the security advantage that it is complete and is, therefore, immune to the exceptional points attack. In conjunction with Montgomery ladder, such a curve is naturally immune to most types of simple power and timing attacks. Finite field operations were performed in the small and efficient Gaussian normal basis. The recently presented formulas for mixed point addition by K. Kim, C. Lee, and C. Negre at Indocrypt 2014 were found to be invalid, but were corrected such that the speed and register usage were maintained. We utilize corrected mixed point addition and doubling formulas to achieve a secure, but still fast implementation of a point multiplication on binary Edwards curves. Our synthesis results over NIST recommended fields for ECC indicate that the proposed co-processor requires about 50% fewer clock cycles for point multiplication and occupies a similar silicon area when compared to the most recent in literature

High Speed and Low-Complexity Hardware Architectures for Elliptic Curve-Based Crypto-Processors

The elliptic curve cryptography (ECC) has been identified as an efficient scheme for public-key cryptography. This thesis studies efficient implementation of ECC crypto-processors on hardware platforms in a bottom-up approach. We first study efficient and low-complexity architectures for finite field multiplications over Gaussian normal basis (GNB). We propose three new low-complexity digit-level architectures for finite field multiplication. Architectures are modified in order to make them more suitable for hardware implementations specially focusing on reducing the area usage. Then, for the first time, we propose a hybrid digit-level multiplier architecture which performs two multiplications together (double-multiplication) with the same number of clock cycles required as the one for one multiplication. We propose a new hardware architecture for point multiplication on newly introduced binary Edwards and generalized Hessian curves. We investigate higher level parallelization and lower level scheduling for point multiplication on these curves. Also, we propose a highly parallel architecture for point multiplication on Koblitz curves by modifying the addition formulation. Several FPGA implementations exploiting these modifications are presented in this thesis. We employed the proposed hybrid multiplier architecture to reduce the latency of point multiplication in ECC crypto-processors as well as the double-exponentiation. This scheme is the first known method to increase the speed of point multiplication whenever parallelization fails due to the data dependencies amongst lower level arithmetic computations. Our comparison results show that our proposed multiplier architectures outperform the counterparts available in the literature. Furthermore, fast computation of point multiplication on different binary elliptic curves is achieved

Concrete quantum cryptanalysis of binary elliptic curves

This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2 (n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2 (n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography