Performance of algebraic graphs based stream-ciphers using large finite fields

Algebraic graphs D(n, q) and their analog graphs D(n, K), where K is a finite commutative ring were used successfully in Coding Theory (as Tanner graphs for the construction of LDPC codes and turbo-codes) and in Cryptography (stream-ciphers, public-keys and tools for the key-exchange protocols. Many properties of cryptography algorithms largely depend on the choice of finite field Fq or commutative ring K. For practical implementations the most convenient fields are F and rings modulo Z modulo 2m. In this paper the reader can find the first results about the comparison of D(n, 2m) based stream-ciphers for m = 8, 16, 32 implemented in C++. They show that performance (speed) of algorithms gets better when m is increased

Teoría de números en criptografía y su debilidad ante la posible era de las computadoras cuánticas

La principal aplicación de la criptografía es la de proteger información para evitar que sea accesible a observadores no autorizados. Sin embargo, también tiene otras aplicaciones, por ejemplo verificar que un mensaje no haya sido modificado intencionadamente por un tercero, verificar que alguien es quien realmente dice ser, etc. El objetivo del presente trabajo es mostrar cómo la matemática juega un papel importante en la criptografía moderna y como ésta aprovecha los problemas difíciles (en el sentido computacional) que existen en la teoría de números para desarrollar protocolos criptográficos. Asimismo se menciona lo que pasaría con los protocolos criptográficos basados en la teoría de números si existiera una computadora cuántica

Factoring Large Numbers with Continued Fractions

The goal of my project was to gain a better understanding of the CFRAC algorithm and to be able to share my knowledge of factorization of large numbers as it relates to the national security of our country. In order to complete my goal I conducted research of the field of mathematics with a specific exploration of the CFRAC algorithm. With RSA being publicly described in 1977, major breakthroughs were established in message encryption. My goal was to find out if it was possible to crack the RSA code through utilization of CFRAC. In order to do this, I needed to explore the special properties of finite and infinite continued fractions. I also needed to further my knowledge of the program Maple which enabled me to work through the CFRAC algorithm much more quickly

BBB-Voting: 1-out-of-k Blockchain-Based Boardroom Voting

Voting is a means to agree on a collective decision based on available choices (e.g., candidates), where participants (voters) agree to abide by their outcome. To improve some features of e-voting, decentralized solutions based on a blockchain can be employed, where the blockchain represents a public bulletin board that in contrast to a centralized bulletin board provides $100\%$ availability and censorship resistance. A blockchain ensures that all entities in the voting system have the same view of the actions made by others due to its immutable and append-only log. The existing blockchain-based boardroom voting solution called Open Voting Network (OVN) provides the privacy of votes and perfect ballot secrecy, but it supports only two candidates. We present BBB-Voting, an equivalent blockchain-based approach for decentralized voting than OVN, but in contrast to it, BBB-Voting supports 1-out-of-$k$ choices and provides a fault tolerance mechanism that enables recovery from stalling participants. We provide a cost-optimized implementation using Ethereum, which we compare with OVN and show that our work decreases the costs for voters by $13.5\%$ in terms of gas consumption. Next, we outline the extension of our implementation scaling to magnitudes higher number of participants than in a boardroom voting, while preserving the costs paid by the authority and participants -- we made proof-of-concept experiments with up to 1000 participants

Maximality of affine group, and hidden graph cryptosystems

We describe a new algebraic-combinatorial method of public key encryption with a certain similarity to the well known Imai-Matsumoto. We use the general idea to treat vertices of a linguistic graph (see [21] and further references) as messages and use the iterative process to walk on such graph as encryption process. To hide such encryption (graph and walk on it) we will use two affine transformation. Like in Imai - Matsumoto encryption the public rule is just a direct polynomial map from the plaintext to the ciphertext. The knowledge about graph and chosen walk on them (the key) allow to decrypt a ciphertext fast. We hope that the system is secure even in the case when the graph is Public but the walk is hidden. In case of "public" graph we can use same encryption as private key algorithm with the resistance to attacks when adversary knows several pairs:(plaintext, ciphertext). We shall discuss the general idea of combining affine transformations and chosen polynomial map of deg ≥ 2 in case of prime field Fp. As it follows from the maximality of affine group each bijection on Fp n can be obtained by such combining