Dining cryptographers problem

Dining cryptographers problem illustration

Three cryptographers gather around a table for dinner. The waiter informs them that the meal has been paid for by someone, who could be one of the cryptographers or the National Security Agency (NSA). The cryptographers respect each other's right to make an anonymous payment, but want to find out whether the NSA paid. So they decide to execute a two-stage protocol.

In the first stage, every two cryptographers establish a shared one-bit secret, say by tossing a coin behind a menu so that only two cryptographers see the outcome in turn for each two cryptographers. Suppose, for example, that after the coin tossing, cryptographer A and B share a secret bit ${\displaystyle 1}$, A and C share ${\displaystyle 0}$, and B and C share ${\displaystyle 1}$.

In the second stage, each cryptographer publicly announces a bit, which is:

• if they didn't pay for the meal, the exclusive OR (XOR) of the two shared bits they hold with their two neighbours,
• if they did pay for the meal, the opposite of that XOR.

Supposing none of the cryptographers paid, then A announces ${\displaystyle 1\oplus 0=1}$, B announces ${\displaystyle 1\oplus 1=0}$, and C announces ${\displaystyle 0\oplus 1=1}$. On the other hand, if A paid, she announces ${\displaystyle \lnot (1\oplus 0)=0}$.

The three public announcements combined reveal the answer to their question. One simply computes the XOR of the three bits announced. If the result is 0, it implies that none of the cryptographers paid (so the NSA must have paid the bill). Otherwise, one of the cryptographers paid, but their identity remains unknown to the other cryptographers.

David Chaum coined the term dining cryptographers network, or DC-net, for this protocol.

The DC-net protocol is simple and elegant. It has several limitations, however, some solutions to which have been explored in follow-up research (see the References section below).

Collision

If two cryptographers paid for the dinner, their messages will cancel each other out, and the final XOR result will be ${\displaystyle 0}$. This is called a collision and allows only one participant to transmit at a time using this protocol. In a more general case, a collision happens as long as any even number of participants send messages.

Disruption

Any malicious cryptographer who does not want the group to communicate successfully can jam the protocol so that the final XOR result is useless, simply by sending random bits instead of the correct result of the XOR. This problem occurs because the original protocol was designed without using any public key technology and lacks reliable mechanisms to check whether participants honestly follow the protocol.[2]

Complexity

The protocol requires pairwise shared secret keys between the participants, which may be problematic if there are many participants. Also, though the DC-net protocol is "unconditionally secure", it actually depends on the assumption that "unconditionally secure" channels already exist between pairs of the participants, which is not easy to achieve in practice.

A related anonymous veto network algorithm computes the logical OR of several users' inputs, rather than a logical XOR as in DC-nets, which may be useful in applications to which a logical OR combining operation is naturally suited.

David Chaum first thought about this problem in the early 1980s. The first publication that outlines the basic underlying ideas is his.[3] The journal version appeared in the very first issue of the Journal of Cryptology.[4]

DC-nets are readily generalized to allow for transmissions of more than one bit per round, for groups larger than three participants, and for arbitrary "alphabets" other than the binary digits 0 and 1, as described below.

The measure originally suggested by David Chaum to avoid collisions is to retransmit the message once a collision is detected, but the paper does not explain exactly how to arrange the retransmission.

Dissent avoids the possibility of unintentional collisions by using a verifiable shuffle to establish a DC-nets transmission schedule, such that each participant knows exactly which bits in the schedule correspond to his own transmission slot, but does not know who owns other transmission slots.[9]

Herbivore divides a large anonymity network into smaller DC-net groups, enabling participants to evade disruption attempts by leaving a disrupted group and joining another group, until the participant finds a group free of disruptors.[10] This evasion approach introduces the risk that an adversary who owns many nodes could selectively disrupt only groups the adversary has not completely compromised, thereby "herding" participants toward groups that may be functional precisely because they are completely compromised.[11]

Dissent implements several schemes to counter disruption. The original protocol[9] used a verifiable cryptographic shuffle to form a DC-net transmission schedule and distribute "transmission assignments", allowing the correctness of subsequent DC-nets ciphertexts to be verified with a simple cryptographic hash check. This technique required a fresh verifiable before every DC-nets round, however, leading to high latencies. A later, more efficient scheme allows a series of DC-net rounds to proceed without intervening shuffles in the absence of disruption, but in response to a disruption event uses a shuffle to distribute anonymous accusations enabling a disruption victim to expose and prove the identity of the perpetrator.[5] Finally, more recent versions support fully verifiable DC-nets - at substantial cost in computation efficiency due to the use of public-key cryptography in the DC-net - as well as a hybrid mode that uses efficient XOR-based DC-nets in the normal case and verifiable DC-nets only upon disruption, to distribute accusations more quickly than is feasible using verifiable shuffles.[8]