TLDR:
Privacy is impossible with most usual implementations of CFMMs under reasonable conditions.
There are some strategies that can be used to mitigate this deanonymization and preserve privacy, but they have costs.
Core Research Question
 In what ways are CFMMs privacypreserving or can users’ transactions be deanonymized?
Citation
Background
Constant Function Market Maker (CFMM)  A smart contract that allows for the exchange of coins and derivative assets. It employs a convex function which maps asset quantities to implied prices abiding by the principle of supply and demand. Uniswap and Celo utilize similar CFMMs.
Decentralized Exchange (DEX)  A type of cryptocurrency exchange where traders can exchange assets with each other or with the protocol without the need for a trusted third party.
1homogeneous functions  A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by t^k.
Hyperplane  A hyperplane is a set of potential solution spaces that are subject to a single equality constraint.
εclose  In mathematics, ε represents an infinitesimally small quantity, whose limit is usually taken as ε>0. εcloseness is the property of being a distance from a value that is infinitesimally small.
Constant Sum Market Makers  A market maker that satisfies the equation where the sum of the reserves are constant.
Summary
Transactions in DEXs are public and decentralized. This means it is not necessary to trust a centralized entity to transact. All details of any trade are transparent, meaning they can be directly attributed to a user’s address. This transparency has costs, e.g. frontrunning and deanonymization. CFMMs like Uniswap are subject to deanonymization via statistical analysis. Zeroknowledge systems, however, might be able to privately execute CFMM transactions. Purportedly privacypreserving modifications to Uniswap are unlikely to preserve privacy as identity is leaked from the timing of trades.
The authors of this paper formalize the intuition that CFMM trades can be deanonymized. Combining information about a CFMM’s function and timeordered trade data allows an attacker to reversecalculate trade size. It has been shown in other works that CFMMs benefit from many useful properties of convexity. A downside to this is that trade sizes can be easily recovered.
Method & Results
The researchers 1) use basic convex analysis tools to show how to calculate the uniqueness of transactions and 2) use logical proofs to show how this applies to more general CFMMs and weaker attack models. They also 3) provide some mitigation mechanisms to improve privacy.

Impossibility of Privacy
 The researchers define CFMMs using reserve quantities and a trading function.
 They assume that trading functions are strictly concave and increasing (this does not apply to constant sum market makers).

Adversary definition and attack
 The researchers assume a simple and general adversary model (Eve) attempting to see the quantity traded by an agent (Alice).
 Action space: They assume Eve can query the marginal price before and after trades, whether a trade is valid, that Alice’s transaction took place. As such, this would not apply to protocols where users do not know when transactions take place.
 Attack description: The researchers outline that Eve can always reconstruct the reserves from a marginal price and a single valid trade using basic nonlinear equations. This means that the adversary can compute the reserves before and after Alice’s trade, and thus the trade amounts. Now, Eve only needs to compute the uniqueness of reserve amounts that satisfy these equations. A Newtontype method can be used to get good results, but there are important special cases wthatare much simpler.
 Reserve discovery in Uniswap: In the case of a CPM like Uniswap, this uniqueness problem reduces to a linear system of equations which simplifies to the following equation.
 This produces the unique solution (R1, R2) for the reserves for all pricemoving trades. This means that using marginal price only, Uniswap reserves can be recovered.

Uniqueness of solution
 The researchers show that the reserve discovery equation is unique when the trade function is an increasing, nonnegative, strictly concave 1homogeneous function. This includes CPM and CMM (eg. Uniswap and Balancer). The researchers suspect that this applies when the function is not necessarily 1homogeneous but leave that for future work.
 Reserves at a fixed price: The researchers define a set of reserves consistent with the above constraint. I.e they use a series of proofs to define the set of reserves that map to the marginal prices of trades.
 Reserves consistent with a trade: they then use another series of proofs to show that the intersection of this set and the set of reserves for which a given trade is feasible is a singleton (i.e. a unique solution exists). The researchers also note that Curve is a counterexample since scaling reserves will change the marginal price.

Extensions
 Nonzero fees: Where a function has nonzero fees, there is a different equation that the CFMM must satisfy. The feasibility condition is changed but the proof of uniqueness does not change.
 General homogeneity: This paper assumes 1homogeneity to make the proofs cleaner, but the proof should be similar for more general phomogeneity (for p>0).
 Unknown marginal price: When the marginal price is not known (Eve cannot access), it is not difficult for Eve to compute an arbitrarily good approximation by performing a linear amount of queries.
 Nonstrict concavity: The case where the trading function is not strictly concave is more difficult because some reserve amounts may map to the same marginal price. In some special cases (eg. constant sum market makers), Eve can keep increasing her trade size until a trade becomes feasible and then perform a binary search to get εclose to true reserves. More general CFMMs whose concave functions are not strictly concave are harder but similar. In those cases, Eve can again query progressively larger trades until they become feasible. This requires a more involved argument that is not made by the researchers in this paper as most CFMMs are strictly concave.

Mitigating strategies
 There are a number of conditions that might be used to modify CFMMs to guarantee privacy. The researchers provide examples, but do not prove these conditions in this paper.
 Randomness in price: The CFMM could add some amount of randomness to the marginal price (as is done in differential privacy) to prevent reconstruction of reserve values. This randomness must be fixed at each block or else it would be too easy to approximate the true price. The researchers note that this can quickly become complex and might cause losses for liquidity providers.
 Batching orders: Another option is to batch orders together. This does not work if all the other trades in a batch surrounding Alice’s belong to Eve. Also, the delay in batching orders can lead to a bad user experience and could negatively impact solvency with large price movements.
Discussion and Key Takeaways
More complex mechanisms are needed to preserve privacy than would be initially assumed.
Impossibility of privacy: The researchers primarily make use of two tricks to show their results.
First, because the set of possible trade functions is strictly convex (mathematically simple), any unique hyperplane (i.e. any crosssection of variables) maps to a unique reserve value. This provides a way of identifying reserves at a fixed liquidity. They note Curve as a potential counterexample that all convex trading functions fit the conditions outlined.
The second trick used by the researchers is that due to the monotonicity and strict concavity of the trading function, a given trade will either be too expensive for a given liquidity amount (a better trade exists) or cannot be executed (not enough liquidity).
Further thoughts: There is a tradeoff between privacy and the cost traders or liquidity providers will pay for it. This can be direct (eg. higher prices) or an indirect cost (e.g. risk of failure). It is not currently clear what price users are willing to pay for this. This raises the question of whether it is worth spending resources to achieve such privacy solutions.
Conclusion: As shown by the researchers, privacy in DeFi is difficult to achieve. The researchers suspect there are reasonable variations of CFMM which are privacypreserving. However, solutions to this problem are either hard to implement in practice or suffer from worse user experience.
Implications and Followups
The researchers identify two open questions: 1) does the attack outlined work for all strictly convex CFMMs (not just 1homogenous) and 2) what does a privacypreserving CFMM look like and what guarantees can be made.
Applicability
The work in this paper applies to all constant function DEXs, for example Uniswap. It also potentially applies to more generalizable automated market makers such as Curve. This information “leak” has broad implications for the DeFi design space.