Leaderless k-of-n random beacon

#1

TLDR: We suggest a random beacon scheme where committees of size n generate random numbers if k participants participate correctly. It is in a similar vein to Dfinity’s random beacon (without use of BLS) and has the same message complexity of k messages per beacon output.

Part 1: A single random output

For clarity we first show how k-of-n participants generate a single output in three phases. In part 2 we combine the phases for messaging efficiency.

  • Phase 1—ephemeral identities: Every participant is committed to an ephemeral secret key sk_i and shares the corresponding ephemeral public key pk_i.
  • Phase 2—encrypted shares: Every participant uses his secret key sk_i to deterministically build a degree-(k-1) polynomial P_i by interpolating the points (0, sk_i), (1, H(sk_i || 1)), ..., (k-1, H(sk_i || k - 1)). The polynomial P_i is then used to create n shares P_i(1), ..., P_i(n). Finally, the shares P_i(j) are encrypted with respect to the public keys pk_j, and the encrypted shares are committed to and shared publicly. We say that a participant “commits correctly” if both the public key pk_i and the encrypted shares are well-formed relative to the secret key sk_i.
  • Phase 3—reveal: All n participants are invited to reveal their ephemeral secret key sk_i. When a participant reveals his secret key sk_i anyone can check if that participant committed correctly. (Participants that do not commit correctly can be slashed.) Assuming k participants reveal their secret key and committed correctly, if it possible to determine which of the participants who did not reveal their secret key committed properly, and extract the secret key for those who did commit properly. Indeed, k of the n encrypted shares of any participant can be decrypted with the revealed secret keys, which is enough to recover a candidate polynomial P_i and a corresponding secret key sk_i, and check that the participant committed correctly. We now define the random output to be the sum of the secret keys sk_i for which the corresponding participants committed correctly.

Part 2: The random beacon

Similar to RANDAO, every participant is committed to a hash onion and we use the preimages as ephemeral secret keys. In order to combine the above three phases into one message, the reveal phase is used to “refill” ephemeral public keys and encrypted shares for future beacon rounds.

For maximum messaging efficiency and to cater for non-participation in some of the rounds we have a buffer of publicly shared future commitments (of size, say, 10) which is refilled appropriately in the reveal phases.

Discussion

One benefit of the leaderless approach is that there is no leader who has monopoly knowledge over the next beacon output and can singlehandedly abort (a weakness of the RANDAO plus a k-of-n committee approach).

By using a quantum-secure encryption scheme (e.g. supersingular elliptic curve Diffie-Hellman?) we can mimic Dfinity’s random beacon with the same optimal messaging complexity of k messages per round.

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#2

By using a quantum-secure encryption scheme (e.g. supersingular elliptic curve Diffie-Hellman?)

One problem with the quantum-secure public key stuff is that it all has fairly high overhead. From Wikipedia:

In 2014, researchers at the University of Waterloo developed a software implementation of SIDH. They ran their partially optimized code on an x86-64 processor running at 2.4 GHz. For a 768-bit modulus they were able to complete the key exchange computations in 200 milliseconds thus demonstrating that the SIDH is computationally practical.[12]

That’s 200 milliseconds, multiplied by whatever k is, and in any case it’s a non-standardized component that would greatly expand consensus code complexity.

Another issue with using this at consensus layer is that I don’t think it technically adds that much, particularly if large CAS committees are present. An attacker with \alpha < 0.5 stake can reduce the main chain’s growth rate to 1 - \alpha, and the attacker’s chain will grow at exponentially slow speed, so the attacker will not win. Because of this, the attacker cannot manipulate to increase their revenue. Furthermore, in the long run the beacon chain will also be responsible for including cross-links and other consensus-related transactions, and the leader will in any case have a monopoly on those.

For lottery-related use cases, you don’t want 50% resistance, you want 99.9999% resistance, because the RNG is a second-layer construct (caveat: unless we really want to include a consensus-layer RNG) and so it’s much more likely that an attacker could buy up more than a 50% share in it, so I think some VRF construction is optimal.

#3

I think you mean “it is possible” :wink:

#4

The difference between this and a BLS-based common coin is that common coin is fully asynchronous. It looks like this algorithm is synchronous since you assume that if a node does not publish anything for a certain period of time,the node is bad.