arXiv subjects

Christopher Blake

Publications and source records attributed to Christopher Blake.

5 recordsLinked to original sources

arXiv2026

Rigorous and Generalized Proof of Security of Bitcoin Protocol with Bounded Network Delay

A proof of the security of the Bitcoin protocol is made rigorous, and simplified in certain parts. A computational model in which an adversary can delay transmission of blocks by time $\Delta$ is considered. The protocol is generalized to allow blocks of different scores and a proof within this more general model is presented. An approach used in a previous paper that used random walk theory is shown through a counterexample to be incorrect; an approach involving a punctured block arrival process is shown to remedy this error. Thus, it is proven that with probability one, the Bitcoin protocol will have infinitely many honest blocks so long as the fully-delayed honest mining rate exceeds the adversary mining rate. This means that an adversary cannot censor future transactions of a user in perpetuity, which would render the protocol useless.

cs.CR↗

arXiv2026

Merged Bitcoin: Proof of Work Blockchains with Multiple Hash Types

Proof of work blockchain protocols using multiple hash types are considered. It is proven that the security region of such a protocol cannot be the AND of a 51\% attack on all the hash types. Nevertheless, a protocol called Merged Bitcoin is introduced, which is the Bitcoin protocol where links between blocks can be formed using multiple different hash types. Closed form bounds on its security region in the $\Delta$-bounded delay network model are proven, and these bounds are compared to simulation results. This protocol is proven to maximize cost of attack in the linear cost-per-hash model. A difficulty adjustment method is introduced, and it is argued that this can partly remedy asymmetric advantages an adversary may gain in hashing power for some hash types, including from algorithmic advances, quantum attacks like Grover's algorithm, or hardware backdoor attacks.

cs.CR↗

arXiv2025

The Finite Element Neural Network Method: One Dimensional Study

The potential of neural networks (NN) in engineering is rooted in their capacity to understand intricate patterns and complex systems, leveraging their universal nonlinear approximation capabilities and high expressivity. Meanwhile, conventional numerical methods, backed by years of meticulous refinement, continue to be the standard for accuracy and dependability. Bridging these paradigms, this research introduces the finite element neural network method (FENNM) within the framework of the Petrov-Galerkin method using convolution operations to approximate the weighted residual of the differential equations. The NN generates the global trial solution, while the test functions belong to the Lagrange test function space. FENNM introduces several key advantages. Notably, the weak-form of the differential equations introduces flux terms that contribute information to the loss function compared to VPINN, hp-VPINN, and cv-PINN. This enables the integration of forcing terms and natural boundary conditions into the loss function similar to conventional finite element method (FEM) solvers, facilitating its optimization, and extending its applicability to more complex problems, which will ease industrial adoption. This study will elaborate on the derivation of FENNM, highlighting its similarities with FEM. Additionally, it will provide insights into optimal utilization strategies and user guidelines to ensure cost-efficiency. Finally, the study illustrates the robustness and accuracy of FENNM by presenting multiple numerical case studies and applying adaptive mesh refinement techniques.

cs.CE↗

arXiv2015

On the Energy Complexity of LDPC Decoder Circuits

It is shown that in a sequence of randomly generated bipartite configurations with number of left nodes approaching infinity, the probability that a particular configuration in the sequence has a minimum bisection width proportional to the number of vertices in the configuration approaches $1$ so long as a sufficient condition on the node degree distribution is satisfied. This graph theory result implies an almost sure $\Omega\left(n^{2}\right)$ scaling rule for the energy of capacity-approaching LDPC decoder circuits that directly instantiate their Tanner Graphs and are generated according to a uniform configuration model, where $n$ is the block length of the code. For a sequence of circuits that have a full set of check nodes but do not necessarily directly instantiate a Tanner graph, this implies an $\Omega\left(n^{1.5}\right)$ scaling rule. In another theorem, it is shown that all (as opposed to almost all) capacity-approaching LDPC decoding circuits that directly implement their Tanner graphs must have energy that scales as $\Omega\left(n\left(\log n\right)^{2}\right)$. These results further imply scaling rules for the energy of LDPC decoder circuits as a function of gap to capacity.

cs.IT↗

arXiv2014

Energy Consumption of VLSI Decoders

Thompson's model of VLSI computation relates the energy of a computation to the product of the circuit area and the number of clock cycles needed to carry out the computation. It is shown that for any family of circuits implemented according to this model, using any algorithm that performs decoding of a codeword passed through a binary erasure channel, as the block length approaches infinity either (a) the probability of block error is asymptotically lower bounded by 1/2 or (b) the energy of the computation scales at least as Omega(n(log n)^(1/2)), and so the energy of successful decoding, per decoded bit, must scale at least as Omega((log n)^(1/2)). This implies that the average energy per decoded bit must approach infinity for any sequence of codes that approaches capacity. The analysis techniques used are then extended to the case of serial computation, showing that if a circuit is restricted to serial computation, then as block length approaches infinity, either the block error probability is lower bounded by 1/2 or the energy scales at least as fast as Omega(n log(n)). In a very general case that allows for the number of output pins to vary with block length, it is shown that the average energy per decoded bit must scale as Omega(n(log n)^(1/5)). A simple example is provided of a class of circuits performing low-density parity-check decoding whose energy complexity scales as O(n^2 log log n).

cs.IT↗