# Sender and Receiver FMD

The goal of detection capability is to be able to filter the global stream of state updates into a local stream of state updates that includes all updates related to a particular address, without identifying precisely which updates those are. The rate of updates on this filtered stream should be bounded below, to ensure that there is a large enough anonymity set, and bounded above, so that users processing the stream have a constant and manageable amount of work to process it and catch up with the current chain state. This means that the detection precision must be adaptive to the global message rates: if the false positive rate is too low, the filtered stream will have too few messages, and if it is too high, it will have too many messages.

However, this isn’t possible using the paper’s original definition of fuzzy message detection, because the false positive probability is chosen by the receiver, who has no way to know in advance what probability will produce a correctly sized stream of messages. One way to address this problem is to rename the original FMD definition as Receiver FMD (R-FMD), and tweak it to obtain Sender FMD (S-FMD), in which the sender chooses the detection probability.

The paper’s original definition of fuzzy message detection is as a tuple of algorithms (KeyGen, CreateClue, Extract, Examine).1 The receiver uses KeyGen to generate a root key and a clue key. A sender uses the receiver’s clue key as input to CreateClue to produce a clue. The Extract algorithm takes the root key and a false positive rate (chosen from some set of supported rates), and produces a detection key. The Examine algorithm uses a detection key to examine a clue and produce a detection result.

This scheme should satisfy certain properties, formalizations of which can be found in the paper:

###### Correctness.

Valid matches must always be detected by Examine; i.e., there are no false negatives.

###### Fuzziness.

Invalid matches should produce false positives with probability approximately , as long as the clues and detection keys were honestly generated.

###### Detection Ambiguity.

An adversarial detector must be unable to distinguish between a true positive and a false positive, as long as the clues and detection keys were honestly generated.

In this original definition, the receiver has control over how much detection precision they delegate to a third party, because they choose the false positive probability when they extract a detection key from their root key. This fits with the idea of attenuating credentials, and intuitively, it seems correct that the receiver should control how much information they reveal to their detector. But the information revealed to their detector is determined by both the false positive probability and the amount of other messages that can function as cover traffic. Without knowing the extent of other activity on the system, the receiver has no way to make a principled choice of the detection precision to delegate.

## Sender FMD

To address this problem, we generalize the original definition (now Receiver FMD) to Sender FMD, in which the false positive probability is chosen by the sender.

S-FMD consists of a tuple of algorithms (KeyGen, CreateClue, Examine). Like R-FMD, CreateClue creates a clue and Examine takes a detection key and a clue and produces a detection result. As discussed in the next section, S-FMD can be realized using tweaks to either of the R-FMD constructions in the original paper.

Unlike R-FMD, the false positive rate is set by the sender, so CreateClue takes both the false positive rate and the receiver’s clue key. Because the false postive rate is set by the sender, there is no separation of capability between the root key and a detection key, so KeyGen outputs a clue key and a detection key, and Extract disappears.

In R-FMD, flag ciphertexts are universal with respect to the false positive rate, which is applied to the detection key; in S-FMD, the false positive rate is applied to the flag ciphertext and the detection key is universal.

Unlike R-FMD, S-FMD allows detection precision to be adaptive, by having senders use a (consensus-determined) false positive parameter. This parameter should vary as the global message rates vary, so that filtered message streams have a bounded rate, and it should be the same for all users, so that messages cannot be distinguished by their false positive rate.

1

We change terminology from the FMD paper; the paper calls detection and clue keys the secret and public keys respectively, but we avoid this in favor of capability-based terminology that names keys according to the precise capability they allow. The “clue” terminology is adopted from the Oblivious Message Retrieval paper of Zeyu Liu and Eran Tromer; we CreateClue and Examine clues rather than Flag and Test flag ciphertexts.