Birthday Problem Research Articles

The birthday problem: repeated sampling of animal populations and ethics of experimental design

Researchers who use animals in science must balance statistical power with the need to satisfy the three Rs, whereby researchers are required to reduce numbers of animals, refine what they experience, and use alternatives to (replace) higher animals where possible. In repeated sampling over time-series studies, there is potential loss of power as well as ethics implications posed by repeated sampling of individual animals, where this cannot be managed or avoided. Here, we consider the mathematics of repeated sampling from three perspectives: that of the population at large, from the experience of the individual, and the conditional probability of sampled individuals being sampled again. The calculations are illustrated using four theoretical case studies across veterinary epidemiology with different practical implications and a provided R Shiny tool for researchers. Despite the availability of exact calculations, it is necessary to also consider the biological factors which may affect capture and recapture rates in sampling studies such as animal personality and response to capture. Researchers must also choose their question carefully to avoid inappropriate framing of ethical concerns around repeated sampling.

Partial pre-image attack on Proof-of-Work based blockchains

Blockchain is a type of distributed ledger technology that consists of a growing list of records, called blocks, that are securely linked together using cryptography. Each blockchain-based solution deploys a specific consensus algorithm that guarantees the consistency of the ledger over time. The most famous, and yet claimed to be the most secure, is the Proof-of-Work (PoW) consensus algorithm. In this paper, we revisit the fundamental calculations and assumptions of this algorithm, originally presented in the Bitcoin white paper. We break down its claimed calculations in order to better understand the underlying assumptions of the proposal. We also propose a novel formalization model of the PoW mining problem using the Birthday paradox. We utilize this model to formalize and analyze partial pre-image attacks on PoW-based blockchains, with formal analysis that confirms the experimental results and the previously proposed implications. We build on those analyses and propose new concepts for benchmarking the security of PoW-based systems, including Critical Difficulty and Critical Difficulty per given portion. Our calculations result in several important findings, including the profitability of launching partial pre-image attacks on PoW-based blockchains, once the mining puzzle difficulty reaches a given threshold. Specifically, for any compromised portion of the network (q<0.5; honest majority assumption still holds), the attack is formally proven profitable once the PoW mining puzzle difficulty reaches 56 leading zeros.

Understanding Biometric Entropy and Iris Capacity: Avoiding Identity Collisions on National Scales

The numbers of persons who can be enrolled by their iris patterns with no identity collisions is studied in relation to the biometric entropy extracted, and the decision operating threshold. The population size at which identity collision becomes likelier than not, given those variables, defines iris “capacity.” The general solution to this combinatorial problem is derived, in analogy with the well-known “birthday problem.” Its application to unique biometric identification on national population scales is shown, referencing empirical data from US NIST (National Institute of Standards and Technology) trials involving 1.2 trillion (1.2 × 1012) iris comparisons. The entropy of a given person’s two iris patterns suffices for global identity uniqueness.

Card guessing and the birthday problem for sampling without replacement

Card guessing and the birthday problem for sampling without replacement

Simple versus nonsimple loops on random regular graphs

AbstractIn this note, we solve the “birthday problem” for loops on random regular graphs. Namely, for fixed , we prove that on a random ‐regular graph with vertices, as approaches infinity, with high probability: (i) almost all primitive nonbacktracking loops of length are simple, that is, do not self‐intersect, and (ii) almost all primitive nonbacktracking loops of length self‐intersect.

An Inquiry-Based Learning Approach to Teaching Undergraduate Cryptology

This article describes the author's experience designing and implementing an inquiry-based learning (IBL) pedagogical approach to an upper-division undergraduate cryptology course. The author shares the course goals and how the IBL style supports their achievement. The article concludes with sample activities – in-class exercises that touch on one-time pad, the Birthday Problem, the Meet-in-the-Middle attack, and Diffie–Hellman key exchange protocol, and a programing assignment on the Vigenère cipher.

A note on the exponentiation approximation of the birthday paradox

This note sheds new light on the exponentiation approximation of the probability that all K individuals have distinct birthdays across N calendar days. The exponentiation approximation imposes a pairwise independence assumption, which does not hold in general. We sidestep this assumption by deriving the conditional probability for each pair of individuals to have distinct birthdays given that previous pairs do. An interesting implication is that the conditional probability decreases in a step-function form—not in a strictly monotonical form—as more pairs are restricted to have distinct birthdays. The source of the step-function structure is identified and illustrated. We also establish the equivalence between the pairwise approach and another common approach based on permutations of all individuals.

The Kolmogorov birthday paradox

We prove a Kolmogorov complexity variant of the birthday paradox. Sufficiently large random subsets of strings are guaranteed to have two members x and y with low K(x/y). To prove this, we first show that the minimum conditional Kolmogorov complexity between members of finite sets is very low if they are not exotic. Exotic sets have high mutual information with the halting sequence.

When are there too many collisions? Variants of the birthday problem

Due to restrictions on the use of unique identifiers of individuals in data sets, there may be instances in which two or more data sets have some of the individuals in common, with no direct way to detect such occurrences. More generally, a collision occurs when two or more observations are in agreement with respect to variables associated with the observations. This article discusses several possible statistical/probabilistic approaches to determining when the number of collisions (or near-collisions) exceeds what would be expected by chance if in fact the observations are all distinct. The methods and results are related to the Birthday Problem and to Occupancy Problems.

Towards More Efficient Rényi Entropy Estimation

Estimation of Rényi entropy is of fundamental importance to many applications in cryptography, statistical inference, and machine learning. This paper aims to improve the existing estimators with regard to: (a) the sample size, (b) the estimator adaptiveness, and (c) the simplicity of the analyses. The contribution is a novel analysis of the generalized "birthday paradox" collision estimator. The analysis is simpler than in prior works, gives clear formulas, and strengthens existing bounds. The improved bounds are used to develop an adaptive estimation technique that outperforms previous methods, particularly in regimes of low or moderate entropy. Last but not least, to demonstrate that the developed techniques are of broader interest, a number of applications concerning theoretical and practical properties of "birthday estimators" are discussed.

A groupwise approach to the birthday paradox

A groupwise approach to the birthday paradox

A Lower Bound on Cycle-Finding in Sparse Digraphs

We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set , i.e., a subset of edges whose deletion results in an acyclic graph, in G is large. We prove an information-theoretic lower bound, showing that for N -vertex graphs with constant outdegree, any algorithm for this problem must make Ω̄(N 5/9 ) queries to an adjacency list representation of G . In the language of property testing, our result is an Ω̄(N 5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω (√ N ) lower bound, implicit in the work of Bender and Ron, which follows from a simple birthday paradox argument.

106.30 Threshold functions and the birthday paradox

Our goal is to illustrate the idea of a threshold function in the context of the birthday paradox. We do this by exploring the asymptotics of binomial coefficients.

First Person: The birthday problem

A 2nd-grade teacher who never cared much about birthdays made them part of her classroom routine once she saw how important they were to her students. But forgetting a student’s birthday on a chaotic Friday afternoon led her to question whether these kinds of celebrations should be her responsibility.

Pollard's Ro-Method for Finding a Discrete Logarithm in the Case of its Low Weight

n this paper, we present a modification of Pollard's ro-method for searching for a discrete logarithm in the case when it is expressed by a binary vector of relatively small weight. In this case, it is necessary to change the procedure for iterating through random elements of the group under consideration in order to remain in the set of discrete logarithms expressed by binary vectors of low weight. We assume that this weight does not exceed half the length of the input word in the algorithm under consideration. To the other hand the case, when some close approximation of discrete logarithm is known considered. The paper uses estimates for binomial coefficients, the birthday paradox theorem, and the Berry-Esseen theorem for the Bernoulli scheme.

A Fresh Shot at Statistics in the Classroom: Three Perspectives Using World Cup Soccer Player Data

In this article, we outline several activities revolving around soccer players who participated in the 2018 FIFA World Cup and 2019 FIFA Women’s World Cup. Classroom activities are described from different perspectives, useful for a range of different statistics courses. In a first semester probability theory course, students investigate the counter-intuitive birthday paradox empirically and theoretically. For an introductory data science course, students practice their data wrangling skills using the statistical software R. Additional activities are shared for those instructors interested in emphasizing multivariable thinking in a second semester applied statistical modeling course. The activities shared will provide a range of opportunities for instructors to incorporate the FIFA World Cup data in their own courses. Supplementary files for this article are available online.

Implementation of point-counting algorithms on genus 2 hyperelliptic curves based on the birthday paradox

Our main contribution is an efficient implementation of the Gaudry - Schost and Galbraith - Ruprai point-counting algorithms on Jacobians of hyperelliptic curves. Both of them are low memory variants of Matsuo - Chao - Tsujii (MCT) Baby-Step Giant-Step-like algorithm. We present an optimal memory restriction (a time-memory tradeoff) that minimizes the runtime of the algorithms. This tradeoff allows us to get closer in practical computations to theoretical bounds of expected runtime at 2.45√N and 2.38a√N for the Gaudry - Schost and Galbraith - Ruprai algorithms, respectively. Here N is the size of the 2-dimensional searching space, which is as large as the Jacobian group order, divided by small modulus m, precomputed by using other techniques. Our implementation profits from the multithreaded regime and we provide some performance statistics of operation on different size inputs. This is the first open-source parallel implementation of 2-dimensional Galbraith - Ruprai algorithm.

Asymptotically Guaranteed Anti-Jamming Spread Spectrum Random Access Without Pre-Shared Secret

Our main contribution is the design of a spread spectrum random access scheme without pre-shared secret with asymptotically guaranteed availability against electronic/cyber-based jamming attacks. In order to establish asymptotic anti-jamming guarantees, first we develop the system model and show that for reactive jamming (only limited by the laws of physical propagation), the required location for a successful attack can be controlled by design. Then, for non-reactive protocol-aware jamming, we map a random selection of (publicly known) spreading codes to the computational birthday problem. Differently to the related literature, we formulate the birthday problem using entropic measures (R&#x00E9;nyi entropy), which allows to obtain the optimal spreading code set sizes that guarantee asymptotically zero collision probability. Accordingly, the birthday attack is the best strategy to optimize the (trial and error) blind acquisition at the legitimate detector. We obtain numerical results that illustrate how to use our entropic method as a system design degree of freedom to identify realistic spreading code set sizes. We also derive an upper bound of the detection latency. Finally, we show that the resulting anti-jamming throughput is higher than the throughput under conventional operation (i.e. without anti-jamming guarantees) due to the control of collision probability by-design.

Sequential metric dimension for random graphs

AbstractIn the localization game on a graph, the goal is to find a fixed but unknown target node$v^\star$with the least number of distance queries possible. In thejth step of the game, the player queries a single node$v_j$and receives, as an answer to their query, the distance between the nodes$v_j$and$v^\star$. The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is.The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD except that the player’s queries are non-adaptive. In this work we extend the results of Bollobás, Mitsche, and Prałat [4] on the MD of Erdős–Rényi graphs to the SMD. We find that, in connected Erdős–Rényi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős–Rényi graphs. Connections with source localization, binary search on graphs, and the birthday problem are discussed.

Non-uniform birthday problem revisited: Refined analysis and applications to discrete logarithms

The result of birthday problem is of fundamental importance and has many applications in diverse areas, analyzing the discrete logarithm problem in particular. The birthday problem can be modeled as sampling balls with replacement until one is sampled twice. In the classical scenario, balls are sampled uniformly. Inspired by the analysis of algorithms solving certain discrete logarithms, Galbraith and Holmes considered the non-uniform birthday problem and gave an estimate on the number of sampling before having a collision up to the dominating term. In this paper, we give a refined analysis to the non-uniform birthday problem, and apply it to the analysis of an algorithm solving certain discrete logarithm problem. Experiment results suggest that parameter selection following this refined analysis performs better than related work.