Introduction

Problems:

1. Multidimensional Byzantine consensus problem. It assumes a sync system: it requires processes to decide on a single d-dimensional vector v (which is decided by the no-faulty party)
2. Multidimensional Byzantine approximate agreement. It assumes an async system, and it requires processes to decide on multiple d-dimensional vectors. All within a fixed distance.

Result:

To tolerate up to f Byzantine failures in a system,

1. For the sync system, it requires n > max(3f, (d+1)f).
2. For the async system, it requires n > (d+2)/f
3. All with full proofs.

Assumes

1. total processes n >=2, and f >=1
2. message is passed in FIFO order.
3. Reliable delivery and sender identification.

Notes

Two important communication primitive

1. Reliable broadcast

   • Objective: The reliable broadcast technique aims to prevent Byzantine processes from sending different messages to different processes in a single round of communication.

   • Conditions: The technique is effective as long as there are more than three times the number of non-faulty processes compared to Byzantine processes, i.e., n > 3f.

   • Communication: The communication is organized in asynchronous rounds, and messages are labeled with sender identification (p), round tag (r), and contents (c), making them appear as M = (p, r, c).

   • Properties: It has four properties: Non-faulty Integrity, Non-faulty Liveness, Global Uniqueness, Global Liveness

2. The witness technique

   • Objective: it is to promote agreement among processes while avoiding indefinite waiting for messages, which is crucial in scenarios where some processes may crash.
   • Steps:
     • Collecting Values: A process p, reliably receives messages from other processes, storing them in Val. It aims to collect at least n - f messages.
     • Transmitting Reports: Once p has collected n - f messages, it transmits its report, containing these collected messages. It also reliably receives reports from other processes and stores them in Rep.
     • Identifying Witnesses: A witness for p is a process whose report contains messages received by p. These witnesses are gathered until there are at least n - f of them in Wit.
     • Common Values: By using this technique, any two non-faulty processes are guaranteed to have at least n - f values in common, ensuring agreement.

Safe Area

In this section, the paper introduces the concept of value message sets, which are sets of messages exchanged among processes, with a specific geometric arrangement. These message sets are characterized as forming (k, f)-simplicial states, where:

1. There are k different standard basic vectors present in the message set’s content.
2. Each of these vectors appears at most f times in the content of the message set.

This concept is essential for the development of the paper’s algorithms, and it helps ensure the agreement and convergence of non-faulty processes in a distributed system.

The section also establishes a key relationship: when a message set forms a (k, f)-simplicial state, the safe area (Safef(X)) is empty. The safe area represents a region in a multidimensional space where the processes can safely operate without violating consensus or agreement properties. This result underscores the importance of (k, f)-simplicial states in the paper’s algorithm design.

Additionally, the section demonstrates that it is possible to construct a valued message set with certain properties. Specifically, this message set contains at most (d + 1) f messages, and it also leads to an empty safe area (Safef(X)). This finding is significant as it provides insights into the bounds and possibilities for message sets in the context of the paper’s algorithms.

Proofs

1. n > max{3 f, (d + 1) f } is not only necessary but also sufficient conditions to solve the multidimensional consensus problem on sync systems.
2. n > (d + 2) f is not only necessary but also sufficient conditions to solve the multidimensional approximate agreement on asynchronous systems. This is illustrated via two distinct algorithms.
   • The Mendes–Herlihy algorithm
     • Advantage: The algorithm for multidimensional approximate agreement over vectors offers flexibility and adaptability for scenarios where processes need to agree on vector values. Its holistic approach ensures coordinated computations across dimensions, and the incorporation of a safe area concept guarantees that updated vectors remain within the convex hull of non-faulty inputs. Additionally, convergence rounds are employed to reach an agreed-upon value, providing a reliable mechanism for achieving consensus.
     • Disadvantage: One notable drawback of this algorithm is its asymptotic dependence on the number of processes (n) for convergence, which can result in slower convergence as the number of processes increases. Furthermore, the algorithm involves complex communication patterns, including multiple rounds of reliable broadcast and witness reports, leading to increased communication overhead, particularly in asynchronous systems. The coordination required for computations across dimensions and message exchange introduces additional complexity, which may affect the algorithm’s practicality in certain scenarios. Additionally, the algorithm’s performance may be influenced by the specific input vectors provided, and achieving agreement may require a sufficient degree of overlap among these input vectors.
   • The Vaidya–Garg Algorithm
     • Advantage: Simpler geometric primitives make the algorithm more versatile for certain applications.
     • Disadvantage: The algorithm has a slower convergence factor that grows with the number of processes, n.

Summarize

Summarize of the paper

This paper addresses the challenging problems of Multidimensional Byzantine Consensus (MBC) and Multidimensional Byzantine Approximate Agreement (MBAA) in the presence of Byzantine failures in distributed systems.

• Multidimensional Byzantine consensus protocol:
  • Agreement (the output vector at all non-faulty processes must be identical), containment (The output vector at all non-faulty processes must be in the convex hull of the non-faulty inputs), and termination (Each non-faulty process must terminate within a finite amount of time.)
• Multidimensional Byzantine approximate agreement protocol:
  • Agreement (The output vectors of non-faulty processes should be within Euclidean distance e > 0, a constant defined a prior), Containment(The output vector at all non-faulty processes must be inside the convex hull of the non-faulty input), and termination (Each non-faulty process must terminate within a finite amount of time)

The authors provide necessary and sufficient conditions for solving these problems on both synchronous and asynchronous systems.

• n > max{3 f, (d + 1) f } is not only necessary but also sufficient conditions to solve the multidimensional consensus problem on sync systems.
• n > (d + 2) f is not only necessary but also sufficient conditions to solve the multidimensional approximate agreement on asynchronous systems. This is illustrated via two distinct algorithms: the Mendes–Herlihy algorithm and the Vaidya–Garg algorithm.
  • The Mendes–Herlihy algorithm
    • Advantage: The algorithm for multidimensional approximate agreement over vectors offers flexibility and adaptability for scenarios where processes need to agree on vector values. Its holistic approach ensures coordinated computations across dimensions, and the incorporation of a safe area concept guarantees that updated vectors remain within the convex hull of non-faulty inputs. Additionally, convergence rounds are employed to reach an agreed-upon value, providing a reliable mechanism for achieving consensus.
    • Disadvantage: One notable drawback of this algorithm is its asymptotic dependence on the number of processes (n) for convergence, which can result in slower convergence as the number of processes increases. Furthermore, the algorithm involves complex communication patterns, including multiple rounds of reliable broadcast and witness reports, leading to increased communication overhead, particularly in asynchronous systems. The coordination required for computations across dimensions and message exchange introduces additional complexity, which may affect the algorithm’s practicality in certain scenarios. Additionally, the algorithm’s performance may be influenced by the specific input vectors provided, and achieving agreement may require a sufficient degree of overlap among these input vectors.
  • The Vaidya–Garg Algorithm
    • Advantage: Simpler geometric primitives make the algorithm more versatile for certain applications.
    • Disadvantage: The algorithm has a slower convergence factor that grows with the number of processes, n.

The introduction of the safe area concept is a novel contribution that aids in achieving convergence despite potential Byzantine influences in multidimensional scenarios.

My recommendation

The paper is solid with all the detailed proofs for the theory and clear analysis.

The paper provides necessary and sufficient conditions for solving these problems on both synchronous and asynchronous systems.

The paper also introduces the concept of value message sets, which are sets of messages exchanged among processes, with a specific geometric arrangement. These message sets are characterized as forming (k, f)-simplicial states, where:

1. There are k different standard basic vectors present in the message set’s content.
2. Each of these vectors appears at most f times in the content of the message set.

This concept is essential for the development of the paper’s algorithms, and it helps ensure the agreement and convergence of non-faulty processes in a distributed system.

Main Strengths:

• The paper clearly illustrates the basic primitives used in the algorithm. For example, it clearly illustrate two communication primitives

  • Reliable broadcast

    • Objective: The reliable broadcast technique aims to prevent Byzantine processes from sending different messages to different processes in a single round of communication.

    • Conditions: The technique is effective as long as there are more than three times the number of non-faulty processes compared to Byzantine processes, i.e., n > 3f.

    • Communication: The communication is organized in asynchronous rounds, and messages are labeled with sender identification (p), round tag (r), and contents (c), making them appear as M = (p, r, c).

    • Properties: It has four properties: Non-faulty Integrity, Non-faulty Liveness, Global Uniqueness, Global Liveness

  • The witness technique

    • Objective: it is to promote agreement among processes while avoiding indefinite waiting for messages, which is crucial in scenarios where some processes may crash.

    • Steps:

      • Collecting Values: A process p, reliably receives messages from other processes, storing them in Val. It aims to collect at least n - f messages.
        • Transmitting Reports: Once p has collected n - f messages, it transmits its report, containing these collected messages. It also reliably receives reports from other processes and stores them in Rep.
        • Identifying Witnesses: A witness for p is a process whose report contains messages received by p. These witnesses are gathered until there are at least n - f of them in Wit.
        • Common Values: By using this technique, any two non-faulty processes are guaranteed to have at least n - f values in common, ensuring agreement.

• The paper makes significant contributions to the field of distributed systems by addressing complex problems related to consensus and approximate agreement in multidimensional spaces, extending beyond the traditional scalar counterparts.

• The establishment of necessary and sufficient conditions for both synchronous and asynchronous systems provides a clear and solid theoretical foundation for addressing these problems, contributing to the understanding of their inherent challenges and limitations.

• The presentation of explicit protocols for MBC and MBAA, including two algorithms for the latter, demonstrates practical applicability and offers a range of options for different scenarios.

• The introduction of the safe area concept is a novel and effective approach to handling multidimensional consensus problems and ensures systematic convergence, which is a notable contribution to the field.

• The paper clearly illustrates how the proposed protocols can be applied in practical scenarios, such as robot convergence, distributed voting, and optimization problems in convex spaces.

Main Weaknesses:

• The paper does not discuss potential limitations or challenges in implementing the proposed algorithms in practical distributed systems.
• The paper assumes point-to-point reliable, complete, and FIFO communication channels. It would be valuable to discuss potential limitations or adaptations of the protocols in scenarios where these assumptions do not hold.

What we learn

The paper has a very good flow, it illustrates

1. The problem definition, clearly illustrates what is Multidimensional Byzantine consensus and Multidimensional Byzantine approximate agreement.

2. Then the paper illustrates the applications to help the reader better understand the context

3. Then the paper provides the preliminaries, illustrating the basic techniques used in the paper, including reliable broadcast and witnessing techniques.

4. Then the paper illustrates the safe area, a foundation for the following algorithm design.

5. Finally, the paper gives proof of the necessary and sufficient conditions to solve multidimensional consensus for both sync and async systems.

The overall flow is well-structured and clear

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