Congratulations to Dr. Diksha Gupta who defended her dissertation last week and is now locked in quarantine in a hotel room in Singapore. More proof that education really opens doors – or closes and locks them as the case may be.

Here’s our main research paper. To our knowledge, this is the first theoretical result on general defense against Sybil attacks.

Filed under: Uncategorized | Tags: Byzantine agreement, resource competitive, security, theory

I recently gave a talk at DISC on our paper Secure and Scalable Computation Among Strangers. Short abstract below:

“The last decade has seen substantial progress on designing Byzantine agreement algorithms that do not require all-to-all communication. However, these protocols do require each node to play a particular role determined by its ID. Motivated by the rise of permissionless systems such as Bitcoin, where nodes can join and leave at will, we extend this research to a more practical model, where initially, each node does not know the identity of its neighbors. In particular, a node can send to new destinations only by sending to random (or arbitrary) nodes, or responding to messages received from those destinations.

We assume a synchronous and fully-connected network, with a full-information, but static Byzantine adversary. A major drawback of existing Byzantine protocols in this setting is that they have at least n^2 message complexity, where n is the total number of nodes. In particular, the communication cost incurred by the honest nodes n^2, even when Byzantine node send no messages. In this paper, we design protocols for fundamental problems which are message-competitive, i.e., the total number of bits sent by honest nodes is not significantly more than the total sent by Byzantine nodes. We describe a message-competitive algorithm to solve Byzantine agreement, leader election, and committee election. Our algorithm sends an expected O((T+n) log n) bits and has latency O(polylog(n))(even in the CONGEST model), where T=O(n^2) is the number of bits sent by Byzantine nodes.”

Filed under: Uncategorized | Tags: distributed computing, game theory, proof-of-work, security, Sybil attack, theory

Above is a link to the keynote talk I gave at SIROCCO 2020. Short abstract is below:

How can we defend Blockchains and peer-to-peer systems, when no central authority provides admission control? Resource-burning (proof-of-work, proof-of-state, CAPTCHAs) is one of the most used tools to defend such systems, but it is currently poorly understood mathematically. In this talk, I survey recent research to better understanding resource burning, in order to reduce its cost, and thereby improve system security.

Keywords: Distributed algorithms, game theory, costly signaling, money burning, Sybil attack, blockchains, cryptocurrencies, peer-to-peer.

Filed under: Uncategorized | Tags: distributed algorithms, golden ratio, security, theory

I’ll be giving two talks at the upcoming SIROCCO. The first is a keynote on how to ensure security in systems where nodes can come and leave at will, with no admission control. The second is a paper that won the best paper award; it is on robust, distributed search, and makes use of the golden ratio in an interesting way.

Anyone who registers can participate in the livestream of the talk, and the organizers of SIROCCO have made registration **free** this year. Below are links to talk times and registration information .

Talk abstracts are below:

**Resource Burning for Permissionless Systems** (Monday, June 29th, 6-7pm CEST)

Proof-of-work puzzles and CAPTCHAS consume enormous amounts of energy and time. These techniques are examples of resource burning: verifiable consumption of resources solely to convey information.

Can these costs be eliminated? It seems unlikely, since resource burning shares similarities with “money burning`"`

and “costly signaling”, which are concepts foundational to game theory, biology, and economics. Can these costs be reduced? Yes, research shows we can significantly reduce asymptotic costs of resource burning in disparate domains.

In this talk, we survey the literature on resource burning; take positions based on predictions of how the technique is likely to evolve; and propose several open problems targeted at the theoretical distributed-computing research community.

**ANTS on a Plane** (Wednesday, July 1st, 7:00-7:30 pm CEST)

In the ANTS (Ants Nearby Treasure Search) problem, multiple searchers, starting from a central location, search for a target. The searchers cannot communicate and have few bits of initial knowledge, called advice, when they begin the search. In this paper, we initiate the study of ANTS in the geometric plane.

Our main result is an algorithm, GoldenFA, that tolerates arbitrarily many crash failures caused by an adaptive adversary, and requires no bits of advice. GoldenFA takes expected time to find the shape, for a shape of diameter D, at distance L from the central location, with N searchers, t<N of which suffer adversarial crash-failures.

We complement our algorithm with a lower bound, showing that it is within logarithmic factors of optimal. Additionally, we empirically test GoldenFA and a related heuristic, and find that the heuristic is consistently faster than the state-of-the-art. Our algorithms and analysis make critical use of the golden ratio.

My PhD student Abhinav Aggarwal wrote a really nice blog post on a problem I assigned on a final exam. This was a kind of interesting dynamic programming problem about playing a simple game against an opponent (who is not always playing optimally). I don’t usually like to give out solutions to problems that I write, but in this case, Abhinav’s writeup is so nice that I’ll point to it here.

I’ve been asked to post about travel support available for this upcoming FOCS for US-based students and postdocs. The site with information is: http://dimacs.rutgers.edu/FOCS16/travel-support.html, and the deadline is September 12th.

Just a quick link to a new CS blog: Hunt for the Towel. The blog is by my former student Amitabh Trehan (now at Queens). To quote the Hitchiker’s Guide on Amitabh: *“There’s a frood who really knows where his towel is.”*

My student Abhinav Aggarwal has just written the inaugural post on his new blog (the name of which, I hope, is not a comment on the soporific qualities of my advisement 🙂 His first post is on the Clever algorithm by Jon Kleinberg. This is inarguably, the paper that started the modern, spectral-based approach to web search that Google has built on so successfully.

Some questions worth pondering while reading through Abhinav’s summary (and hopefully also the seminal paper itself): 1) Why did PageRank “beat out” the Clever algorithm in real-world web search? 2) Are there domains besides web search where one might use a “top left and right eigenvector” approach like the Clever algorithm? 3) What about the “soft” results in the paper like the use of the second eigenvector for clustering? Can these be formalized in an interesting way?

*[The following report on the Counting Complexity and the Phase Transitions Boot Camp at the Simons Institute was written by my student Abhinav Aggarwal]*

### 2016 Boot Camp on Counting Complexity and Phase Transitions

### Jan 25-28, Simons Institute for the Theory of Computing, University of California, Berkeley

Recently I visited the Simons Institute at UC Berkeley to attend a boot camp on counting complexity and phase transitions. The program for the same is available here. There were a total of 16 talks spread over 4 days, with various speakers covering topics from basics of counting complexity, dichotomy theorems, Markov chain mixing times and random instances. All the videos from the lectures are available here. The following is a brief summary of talks in two areas that seemed interesting to me and are related to my research at UNM.

Nayantara Bhatnagar from University of Delaware and Ivona Bezakova from Rochester Institute of Technology presented 3 talks about properties of Markov chains and their mixing times, along with a couple of applications of the same. The talks started with basics of Markov chain properties and conditions under which a given chain is ergodic (irreducible, aperiodic and finite). Once an ergodic Markov chain is obtained, a metric called total variation distance is defined between the initial distribution on the states and the stationary distribution of this chain (which is unique because of ergodicity). This distance measures how close the former is to latter, purely on the basis of the difference in probability mass allocated by the former to the events in the sample space.

A well-known result is that for stochastic matrices (like the transition probability matrix for Markov chains), the decrease in the distance from the current distribution to the stationary distribution over time depends solely on the spectral gap. The smaller this gap, the larger the distance, and consequently, the more time taken by the Markov chain to approach its stationary distribution. This property is exploited by techniques called Markov chain Monte Carlo (MCMC), which are used heavily for sampling from non-trivial distributions. Some examples of this technique that were presented in the talks include sampling a uniformly random coloring of a given undirected graph and approximate counting of the number of matchings in a given graph. The details of the construction and proof can be found here.

The talks continued with a discussion of various techniques that are popular while sampling and counting using Markov chains. Four of these techniques are:

**Almost uniform sampler**– Given a tolerance parameter*d >*0, produce a sample from the distribution that is within*d*total variation distance from the uniform distribution.**Fully polynomial almost uniform sampler (FPAUS) –**An almost uniform sampler that runs in time polynomial in the input size and log 1/*d***Randomized approximation scheme –**Given a counting problem and a tolerance parameter ε>0, produce a count which is within ±ε of the actual count with probability at least 3/4.**Fully polynomial randomized approximation scheme (FPRAS) –**A randomized approximation scheme that runs in time polynomial in the input size and (1/ε).

Upon defining these techniques, our goal then becomes to design a Markov chain with small mixing time so that the runtime of FPRAS can be minimized. The talks further discussed details about coupling theory, both between Markov chains and probability distributions. Nayantara presented these concepts using coupling between two sequences of coin tosses and the famous card shuffling example, which is used to prove that the top-to-random shuffle takes O(n log n) steps to produce a perfect shuffle of a deck of n cards. Within these many steps, the total variation distance of the Markov chain underlying this shuffling reaches sufficiently close to the uniform distribution.

Another session that I thoroughly enjoyed was the one on approximate counting by Leslie Ann Goldberg. The main topic covered in her talk was relative complexity and its relation to the #BIS problem (Bipartite Independent Sets). She started with the discussion of the Potts model and the partition function in the context. This was in relation to the number of proper q-colorings in a given graph. The aim of the model was to approximate the count of these colorings. She discussed an FRPAS algorithm for the same, which outputs a rational number z such that for a given tolerance ε > 0 and the actual count C, we have Pr (Ce^-ε <= z <= Ce^ε) >= 3/4. The counting problem using FRPAS can be NP Hard in general., however Leslie showed that the problem cannot be much harder than that.

An outline of the proof uses the bisection technique by Vazirani and Valiant, which shows that #SAT can be approximated by a probabilistic polynomial time Turing machine using an oracle for SAT. Leslie then defined the relative complexity for approximate counting using AP-reductions from a function f to a function g. Concretely, a function f is AP-reducible to a function g if 1) there exists a randomized algorithm A to compute f using an oracle for g, and 2) A is a randomized approximation scheme for f whenever the oracle is a randomized approximation scheme for g. This makes the class of functions with an FPRAS closed under AP-reductions.

An immediate dichotomy result that comes out of this formulation is that if NP != RP, then within #P, all FPRASable problems form a class and the rest remain unFPRASable. All problems in a given class are AP-inter-reducible, but an FPRAS doesn’t necessarily exist unless NP = RP. However, within the class of unFPRASable problems, a dichotomy further exists. This class is partitioned into two subclasses, one of which consists of problems that are AP-reducible to #SAT and the other consists of problems that are AP-reducible to #BIS. Leslie concluded her talk by giving an example of this trichotomy in the context of graph-homomorphisms.

Filed under: Uncategorized | Tags: distributed computing, students, theory

Congratulations to Mahnush Movahedi who today just passed with distinction her dissertation defense on Efficient and Robust Distributed Computing. Serving on her dissertation committee were Sean Luan (UNM), David Evans (UVA) and Maxwell Young (MSU). After graduation, Mahnush will be will be moving to the University of Virginia as a postdoc to work with David Evans on Secure Computation.

It’s a great pleasure to have worked with Mahnush! She and Mahdi will both be greatly missed in our research group.