SEMINÁRIO DE AVALIAÇÃO - SÉRIE: A Cryptographic Algorithms based on Ramanujan Graphs

Orientadores:
Fábio Borges de Oliveira - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Fábio Borges de Oliveira - Laboratório Nacional de Computação Científica - LNCC (presidente)
Renato Portugal - Laboratório Nacional de Computação Científica - LNCC
Bruno Richard Schulze - Laboratório Nacional de Computação Científica - LNCC

Suplentes:
Jack Baczynski - Laboratório Nacional de Computação Científica - LNCC

Resumo:

The Discrete Logarithm Problem (DLP) has been applied in cryptography since 1976, when Diffie and Hellman proposed the DH key-agreement protocol. In 1985, ElGamal presented an encryption scheme based on DLP. Even though it is applied in several cryptographic algorithms, DLP is vulnerable to quantum attacks based on Shor's algorithm. In 2011, Jao and De Feo proposed a key-agreement isogeny-based algorithm resistant to quantum attacks. It is based on supersingular isogeny (Ramanujan) graph walks. Ramanujan graphs are also applied to block ciphers, such as Advanced Encryption Standard (AES). Much of the security of AES is present in its S-Box, in the same way that other block ciphers. In 2019, we identified the relationship between the Ramanujan graph and the affine transformation matrix in the construction of AES-like S-Boxes. Recently, an algorithm based on supersingular isogeny graphs reached the second round of the NIST's standardization process on post-quantum cryptography, the Supersingular Isogeny Key Encapsulation (SIKE). In this work, we present the results based on the Ramanujan graph in two parts, construction of S-Boxes for block ciphers and isogeny-based post-quantum cryptography. First, we present four evolving versions of ElGamal-like cryptography schemes b ased on supersingular isogeny. We show why its first two versions can be reduced to the DLP and are, therefore, vulnerable to quantum attacks, and how they were modified to produce two quantum attacks resistant algorithms in its last two versions. These last two algorithms are based on isogenies composition presented in SIKE. In addition, SIKE is not homomorphic. To our knowledge, our latest two versions are the first ElGamal-like algorithms resistant to quantum attacks and additive homomorphic. Finally, we define B-Ramanujan matrix as a {0,1}-circulant adjacency matrix of a Ramanujan graph and propose a theorem that uses B-Ramanujan to optimize the search for binary matrices for the construction of secure S-Boxes. For the case of AES-256, we should choose a matrix in a set with approximately 1018 nonsingular binary matrices. However, our result reduces the search to a set of 247 B-Ramanujan matrices.