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Analysis of the Sybil defense of Duniter-based cryptocurrencies

Lucas Isenmann

TL;DR

The paper analyzes the Sybil-defense capabilities of the Duniter Web of Trust via a graph-theoretical model, isolating the certifications network from time-based constraints. It derives a Moore-like upper bound on how many fake vertices a Sybil attack can generate within distance $k$, using a bound $g_k = c(A) \frac{(\Delta-\delta+1)^k-1}{\Delta-\delta}$ and auxiliary definitions $c(X)$ and $\alpha$, and contrasts general bounds with a practical basic attack strategy. The results show that while worst-case bounds can be extremely large, real networks with thousands of users exhibit much slower growth due to distance rules and referent dynamics; social constraints such as Dunbar's number can further limit attack size. Overall, the work supports the resilience of Duniter's Web of Trust against large-scale Sybil inflations under the studied graph-theoretical framework and highlights areas for future modeling, including time constraints and potential vertex/arc deletions.

Abstract

Duniter-based cryptocurrencies, which are providing a kind of universal basic income, are using a system called "Web of Trust" based on a social network whose evolution is subject to graph theoretical rules, time constraints and a licence in order to avoid large Sybil attacks. We investigate in this article the largest size of a Sybil attack that a simplified version of the graph theoretical rules of a Web of Trust can undergo depending on the number of attackers and on the parameters of the system. We show that even if in theory, without considering social and time constraints, this system cannot in general prevent huge attacks, in the real-world case of a Duniter-based cryptocurrency (with thousands of users), the system can prevent attacks of large size with only graph theoretical rules.

Analysis of the Sybil defense of Duniter-based cryptocurrencies

TL;DR

The paper analyzes the Sybil-defense capabilities of the Duniter Web of Trust via a graph-theoretical model, isolating the certifications network from time-based constraints. It derives a Moore-like upper bound on how many fake vertices a Sybil attack can generate within distance , using a bound and auxiliary definitions and , and contrasts general bounds with a practical basic attack strategy. The results show that while worst-case bounds can be extremely large, real networks with thousands of users exhibit much slower growth due to distance rules and referent dynamics; social constraints such as Dunbar's number can further limit attack size. Overall, the work supports the resilience of Duniter's Web of Trust against large-scale Sybil inflations under the studied graph-theoretical framework and highlights areas for future modeling, including time constraints and potential vertex/arc deletions.

Abstract

Duniter-based cryptocurrencies, which are providing a kind of universal basic income, are using a system called "Web of Trust" based on a social network whose evolution is subject to graph theoretical rules, time constraints and a licence in order to avoid large Sybil attacks. We investigate in this article the largest size of a Sybil attack that a simplified version of the graph theoretical rules of a Web of Trust can undergo depending on the number of attackers and on the parameters of the system. We show that even if in theory, without considering social and time constraints, this system cannot in general prevent huge attacks, in the real-world case of a Duniter-based cryptocurrency (with thousands of users), the system can prevent attacks of large size with only graph theoretical rules.

Paper Structure

This paper contains 7 sections, 6 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Web of Trust of a Duniter-based cryptocurrency
  3. Differences between the protocol and this theoretical study.
  4. Maximum size of a Sybil attack
  5. Sybil attacks
  6. Upper bounds
  7. Conclusion

Key Result

Theorem 2.3

Let G be a digraph such that $d$ is the diameter of the graph and such that $\Delta$ is the maximum out-degree. Then $|G| \leq \Delta^d$.

Figures (4)

  • Figure 1: Step 1 of the Sybil optimal strategy (in the case where $\delta = 5$): creation of the first $\delta-1$ fake vertices. The order of creation of the fake vertices is noted on the vertices.
  • Figure 2: Step 2 of the Sybil optimal strategy: creation of a fake vertex with certifications from the $\delta-1$ most recent fake vertices and from one attacker. The order of creation of the fake vertices is noted on the right of the vertices.
  • Figure 3: Example of a Sybil basic attack with $\delta = 2$, $\Delta = 6$ and $|A| = 2$.
  • Figure 4: Statistics about the size of basic Sybil attack in the real world Web of Trust of the Duniter-based cryptocurrency Ğ1 which is of size $2590$ in april 2020 for a sample of 100.000 random sets of $5$ attackers.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: Introducer group
  • Definition 1.4: Vertex creation
  • Definition 1.5: Arc creation
  • Definition 2.1: Sybil attack
  • Definition 2.2
  • Theorem 2.3: Moore Moore
  • Theorem 2.4
  • proof
  • ...and 6 more