We correct the double spend race analysis given in Nakamoto's foundational Bitcoin article and give a closed-form formula for the probability of success of a double spend attack using the Regularized Incomplete Beta Function. We give a proof of the exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are necessary compared to those given by Nakamoto. We also compute the probability conditional to the known validation time of the blocks. This provides a finer risk analysis than the classical one.To the memory of our beloved teacher André Warusfel who taught us how to have fun with the applications of mathematics.Date: February 9th 2017. 2010 Mathematics Subject Classification. 68M01, 60G40, 91A60, 33B20.

We review the so called selfish mining strategy in the Bitcoin network and compare its profitability to honest mining. We build a rigorous profitability model for repetition games. The time analysis of the attack has been ignored in the previous literature based on a Markov model, but is critical. Using martingale's techniques and Doob Stopping Time Theorem we compute the expected duration of attack cycles. We discover a remarkable property of the bitcoin network: no strategy is more profitable than the honest strategy before a difficulty adjustment. So selfish mining can only become profitable afterwards, thus it is an attack on the difficulty adjustment algorithm. We propose an improvement of Bitcoin protocol making it immune to selfish mining attacks. We also study miner's attraction to selfish mining pools. We calculate the expected duration time before profit for the selfish miner, a computation that is out of reach by the previous Markov models.

The following text is a short version of a forthcoming preprint about torsors. The adopted viewpoint is an old reformulation of torsors recalled recently by Kontsevich [Kon]. We propose a unification of the definitions of torsors in algebraic geometry and in Poisson manifolds (Example 2 and section 2.2). We introduce the notion of a quantum torsor (Definition 2.1). Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other (Theorem 3.1.) In the finite dimensional case, these two Hopf algebras share the same finite dimension (Proposition 3.1). We show that any Galois extension of a field is a torsor (Example 4) and that any torsor is a Hopf-Galois extension (section 3.2). We give also examples of non-commutative torsors without character (Example 5). Torsors can be composed (Theorem 3.2). This leads us to define for any Hopf algebra, a new group-invariant, its torsors invariant (Theorem 3.3). We show how Parmentier's quantization formalism of "affine Poisson groups" is part of our theory of torsors (Theorem 3.4).I am very grateful to B. Enriquez for his help and his support. I am also grateful to P. Cartier, A. Chambert-Loir, S. Natale and S. Parmentier for comments and remarks.

We explicitly quantize the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p − 1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in characteristic p (of Cartan type). In particular, this yields new families of noncommutative and noncocommutative Hopf algebras of dimension p p in char p.  2004 Published by Elsevier Inc.

No abstract