2. Quantum Byzantine Agreement without Entanglement
- Agreement: Either all honest parties abort the protocol or all honest parties decide on the same output value .
- Validity: If all parties are honest, then they decide on the same output value .
2.1. Round 1: List Distribution
- For all , for some integer m which is a multiple of 6.
- . numbers on are 0. numbers on are 1. numbers on are 2.
- For all , .
- For all , if , then .
- For all , if , then .
- For all , if , then for all the probability that and that are equal (i.e., the numbers of occurences of 0 and 1 are equal in the list).
- For all , .
2.2. Rounds 2 and 3: Reaching Agreement
- Round 2
- sends a binary number to all , . Together with , sends to the list of numbers , which indicate all positions of on the list . The length of is to be , where is the length of . uses as the final value it outputs.
- Round 3
- checks the obtained message against his own reference list . If the analysis of shows that is consistent with , then he sets and sends to all other receivers . Here, is consistent with means that for all index , . However, if is not consistent with , then immediately ascertains that is dishonest and sends to other receivers message: ⊥, meaning: “I have received an inconsistent message”. To acknowledge the fact that every receiver knows his own output, we formally assume that each of them receives a message from himself.
- Time 3
- After all messages have been exchanged between the receivers, every analyzes the data received from and acts according to the following criteria:
- If there is a set of receivers H with such that, for all , is consistent with , and for some , , then sets his output value to be ⊥.
- If there is a set of receivers H with such that for all , is consistent with and all are the same, and for all , is not consistent with , then sets his output value to be .
- If there is a set of receivers H with such that for all , is consistent with and all are the same, and for all , the message sent by is ⊥, then sets his output value to be .
- In all other cases, sets his value to be ⊥.
- If has sent a message consistent with , then is honest.
- If has sent a message inconsistent with , then is dishonest.
- If has sent ⊥, then may be honest or dishonest. However, if in this case is honest, then must be dishonest.
3. Analysis of the Protocol
- The adversary can control a fixed set of participants and let those participants send arbitrary messages at his will. A participant is dishonest if and only if he is controlled by the adversary. The amount of honest participants is ≥3.
- The adversary can bribe the list distributors to disclose certain information. When being bribed, a list distributor will disclose information with probability p.
- The adversary has unlimited computing power.
- All honest receivers receive consistent data. In this case, there are two sub-cases:
- All honest receivers receive the same data. In this case, according to Criterion (b), all honest participants will output the same value.
- Not all honest receivers receive the same data. Then, according to Criterion (a), all honest receivers will abort the protocol (output ⊥).
- Not all honest receivers receive consistent data. In this case, if there are still two receivers that receive the same and consistent data and all other receivers output ⊥, then, according to Criterion (c), all honest receivers will output the same value. Otherwise, according to Criterion (a) or Criterion (d), all honest receivers will output ⊥.
4. Conclusions and Future Work
Conflicts of Interest
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