ENGLISH ABSTRACT : Bitcoin is a peer to peer (P2P) electronic payment system proposed by Nakamoto in 2008. Central to the operation of Bitcoin is the blockchain, which is, in essence, a public ledger of all transactions. The blockchain is maintained by a group of volunteers called miners, who are rewarded with bitcoins for successfully mining blocks. Mining a block involves collecting and validating transactions, adding validated transactions to a block, and finally adding the block to the blockchain and broadcasting the block to the peer network. The purpose of this thesis is to examine the profitability of block discarding attacks in the Bitcoin network. To this end, we developed a discrete event simulator to model the dynamics of the blockchain. We use a simplistic network model where all participants are miners. Initially, our investigation focusses on a model where all miners are honest, following the Bitcoin rules, and all block transmissions are subject to delays. We show that the delays cause forks to occur in the blockchain, which are quickly resolved. The simulation results for our network model closely agree with those observed from the real Bitcoin network. We then examine a block discarding attack referred to as selfish-mine [32], where it is claimed that a small group of colluding miners can subvert the Bitcoin protocol. We evaluate this claim using relative revenue (the fraction of the total revenue that is credited to the pool of colluding miners) and confirmed blocks (the absolute number of blocks credited to the pool of colluding miners at the end of a mining period) for the case where there is instantaneous block transmission, and the case where block transmission is subject to delays. We show that this claim is true for the first case. However, for the latter case, selfish-mine is only profitable (in terms of the relative revenue) when the pool commands more than 30% of the total computing power of the network. We further show that a higher relative revenue does not necessarily entail a larger number of confirmed blocks and that the presence of selfish-mine causes both the honest and the dishonest miners to fare worse. Finally we present a generalized form of selfish-mine, representing a greater variety of block discarding attacks, and use a genetic algorithm to search for an optimal configuration of the generalized selfish-mine that yields a better performance for the pool. We attempt to maximize either the relative revenue or the number of confirmed blocks. Our results show that the generalized form of selfish-mine when optimally configured yields better performance than standard selfish-mine, both in terms of the relative revenue and confirmed blocks.
Cryptocurrencies became popular with the emergence of Bitcoin and have shown an unprecedented growth over the last few years. As of November 2016, more than 720 cryptocurrencies exist, with Bitcoin still being the most popular one. We show the statistical properties of the most important cryptocurrencies. We characterize their exchange rates versus the US Dollar by fitting parametric distributions to them, including the Student t distribution, the generalized hyperbolic distribution as well as the asymmetric normal inverse Gaussian and the asymmetric variance gamma distribution. Our findings show that cryptocurrencies exhibit strong non-normal characteristics, with standard heavy-tailed distributions such as the Student t distribution giving good descriptions of the data. This is the first study that looks at the parametric distribution of cryptocurreny returns. The results are important for investment and risk management purposes.
Mathematics has been considered as very important subject since ancient times. We find very elaborate proof of this in Vedas, which were compiled around 6000 BC. The concept of division, addition etc. was used even that time. Concepts of zero and infinite were also there. We also find roots of Beez Ganit in Vedas. When Indian Beez Ganit reached Arab, they called it Algebra. Algebra was name of the Arabic book that described Indian concepts. This knowledge reached Europe from there. This fact was well known to intellectuals of India, which is why they gave special importance to the development of Mathematics, right from the beginning. When this knowledge was negligible in Arab and Europe, India had acquired great achievements. This paper discusses development of mathematics in India from ancient time. Key words: Mathematics, India, Mathematicians, History
The paper contains main concepts of the interactive proof theory and suggests zero-knowledge proof of Diffie–Hellman problem solution with bilinear maps.
It is shown that any free action of a zero-dimensional compact group on the -dimensional Menger compactum is -universal for free actions, and that the orbit space is -classifying. Nonexistence of equivariant mappings between and implies that the orbit space has infinite dimension, where is any compact ANR-space with free action of the group of -adic integers. Knowledge of such nonexistence would then permit proof of the Hilbert-Smith conjecture under the assumption of finite dimensionality for the orbit space.
T. It is shown that any free action of a zero-dimensional compact group G on the О·-dimensional Menger compactum Mn is О·-universal for free actions, and that the orbit space MnjG is В«-classifying. Nonexistence of equivariant mappings between Mn+m and Mn implies that the orbit space R/Ap has infinite dimension, where R is any compact ANR-space with free action of the group Ap of p-adic integers. Knowledge of such nonexistence would then permit proof of the Hilbert-Smith conjecture under the assumption of finite dimensionality for the orbit space.
P. What is Number Theory? 1. The Integers. Numbers and Sequences. Sums and Products. Mathematical Induction. The Fibonacci Numbers. 2. Integer Representations and Operations. Representations of Integers. Computer Operations with Integers. Complexity of Integer Operations. 3. Primes and Greatest Common Divisors. Prime Numbers. The Distribution of Primes. Greatest Common Divisors. The Euclidean Algorithm. The Fundemental Theorem of Arithmetic. Factorization Methods and Fermat Numbers. Linear Diophantine Equations. 4. Congruences. Introduction to Congruences. Linear Congrences. The Chinese Remainder Theorem. Solving Polynomial Congruences. Systems of Linear Congruences. Factoring Using the Pollard Rho Method. 5. Applications of Congruences. Divisibility Tests. The perpetual Calendar. Round Robin Tournaments. Hashing Functions. Check Digits. 6. Some Special Congruences. Wilson's Theorem and Fermat's Little Theorem. Pseudoprimes. Euler's Theorem. 7. Multiplicative Functions. The Euler Phi-Function. The Sum and Number of Divisors. Perfect Numbers and Mersenne Primes. Mobius Inversion. 8. Cryptology. Character Ciphers. Block and Stream Ciphers. Exponentiation Ciphers. Knapsack Ciphers. Cryptographic Protocols and Applications. 9. Primitive Roots. The Order of an Integer and Primitive Roots. Primitive Roots for Primes. The Existence of Primitive Roots. Index Arithmetic. Primality Tests Using Orders of Integers and Primitive Roots. Universal Exponents. 10. Applications of Primitive Roots and the Order of an Integer. Pseudorandom Numbers. The EIGamal Cryptosystem. An Application to the Splicing of Telephone Cables. 11. Quadratic Residues. Quadratic Residues and nonresidues. The Law of Quadratic Reciprocity. The Jacobi Symbol. Euler Pseudoprimes. Zero-Knowledge Proofs. 12. Decimal Fractions and Continued. Decimal Fractions. Finite Continued Fractions. Infinite Continued Fractions. Periodic Continued Fractions. Factoring Using Continued Fractions. 13. Some Nonlinear Diophantine Equations. Pythagorean Triples. Fermat's Last Theorem. Sums of Squares. Pell's Equation. 14. The Gaussian Integers. Gaussian Primes. Unique Factorization of Gaussian Integers. Gaussian Integers and Sums of Squares.