Decoding Generalized Reed-Solomon Codes and Its Application to RLCE Encryption Schemes

This paper compares the efficiency of various algorithms for implementing quantum resistant public key encryption scheme RLCE on 64-bit CPUs. By optimizing various algorithms for polynomial and matrix operations over finite fields, we obtained several interesting (or even surprising) results. For example, it is well known (e.g., Moenck 1976 \cite{moenck1976practical}) that Karatsuba's algorithm outperforms classical polynomial multiplication algorithm from the degree 15 and above (practically, Karatsuba's algorithm only outperforms classical polynomial multiplication algorithm from the degree 35 and above ). Our experiments show that 64-bit optimized Karatsuba's algorithm will only outperform 64-bit optimized classical polynomial multiplication algorithm for polynomials of degree 115 and above over finite field $GF(2^{10})$. The second interesting (surprising) result shows that 64-bit optimized Chien's search algorithm ourperforms all other 64-bit optimized polynomial root finding algorithms such as BTA and FFT for polynomials of all degrees over finite field $GF(2^{10})$. The third interesting (surprising) result shows that 64-bit optimized Strassen matrix multiplication algorithm only outperforms 64-bit optimized classical matrix multiplication algorithm for matrices of dimension 750 and above over finite field $GF(2^{10})$. It should be noted that existing literatures and practices recommend Strassen matrix multiplication algorithm for matrices of dimension 40 and above. All our experiments are done on a 64-bit MacBook Pro with i7 CPU and single thread C codes. It should be noted that the reported results should be appliable to 64 or larger bits CPU architectures. For 32 or smaller bits CPUs, these results may not be applicable. The source code and library for the algorithms covered in this paper are available at http://quantumca.org/

Chaves mais pequenas para criptossistemas de McEliece usando codificadores convolucionais

The arrival of the quantum computing era is a real threat to the confidentiality and integrity of digital communications. So, it is urgent to develop alternative cryptographic techniques that are resilient to quantum computing. This is the goal of pos-quantum cryptography. The code-based cryptosystem called Classical McEliece Cryptosystem remains one of the most promising postquantum alternatives. However, the main drawback of this system is that the public key is much larger than in the other alternatives. In this thesis we study the algebraic properties of this type of cryptosystems and present a new variant that uses a convolutional encoder to mask the so-called Generalized Reed- Solomon code. We conduct a cryptanalysis of this new variant to show that high levels of security can be achieved using significant smaller keys than in the existing variants of the McEliece scheme. We illustrate the advantages of the proposed cryptosystem by presenting several practical examples.A chegada da era da computação quântica é uma ameaça real à confidencialidade e integridade das comunicações digitais. É, por isso, urgente desenvolver técnicas criptográficas alternativas que sejam resilientes à computação quântica. Este é o objetivo da criptografia pós-quântica. O Criptossistema de McEliece continua a ser uma das alternativas pós-quânticas mais promissora, contudo, a sua principal desvantagem é o tamanho da chave pública, uma vez que é muito maior do que o das outras alternativas. Nesta tese estudamos as propriedades algébricas deste tipo de criptossistemas e apresentamos uma nova variante que usa um codificador convolucional para mascarar o código de Generalized Reed-Solomon. Conduzimos uma criptoanálise dessa nova variante para mostrar que altos níveis de segurança podem ser alcançados usando uma chave significativamente menor do que as variantes existentes do esquema de McEliece. Ilustramos, assim, as vantagens do criptossistema proposto apresentando vários exemplos práticos.Programa Doutoral em Matemátic

Secure Key Encapsulation Mechanism with Compact Ciphertext and Public Key from Generalized Srivastava code

Code-based public key cryptosystems have been found to be an interesting option in the area of Post-Quantum Cryptography. In this work, we present a key encapsulation mechanism (KEM) using a parity check matrix of the Generalized Srivastava code as the public key matrix. Generalized Srivastava codes are privileged with the decoding technique of Alternant codes as they belong to the family of Alternant codes. We exploit the dyadic structure of the parity check matrix to reduce the storage of the public key. Our encapsulation leads to a shorter ciphertext as compared to DAGS proposed by Banegas et al. in Journal of Mathematical Cryptology which also uses Generalized Srivastava code. Our KEM provides IND-CCA security in the random oracle model. Also, our scheme can be shown to achieve post-quantum security in the quantum random oracle model

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2

QR codes

In this thesis QR codes are considered. A QR code is a two-dimensional bar code for machine-readable data. Data are first encoded using Reed-Solomon error correction codes to ensure readability even if the QR code is damaged. In the first part of the thesis we first review finite fiels that are needed in the sequel. Next we introduce error correction codes and define Reed-Solomon codes. Reed-Solomon codes attain the Singleton bound which means that they are capable of correcting the maximum possible number of errors with respect to to the number of symbols that are added to the message. We also present algorithms for coding and decoding Reed-Solomon codes, which are used for QR codes. In the second part of the thesis we describe the structure of the QR code symbol in detail. We explain how the message is encoded in the QR code and how a QR code is decoded. We illustrate these procedures with examples. Finally, we discuss the security of QR codes and possibilities of their abuse