International Association
for Cryptologic Research

In this short note, we show that two identity-based encryption schemes, introduced by Joye and Zhao et al., which appear different, are essentially the same scheme. The only difference between them is that one prioritizes speed (Zhao et al.), while the other prioritizes bandwidth (Joye). We also show how to speed-up Joye's scheme at the cost of adding at most one integer to the public key, thereby achieving a better encryption complexity, while having the same bandwidth requirements.

While deep learning is ubiquitous, centralized pro- cessing exposes sensitive sequential data—such as natural lan- guage—to untrusted servers, forcing an unacceptable privacy- utility trade-off. Fully Homomorphic Encryption (FHE) re- solves this by computing directly on encrypted data. However, standard neural networks ported to FHE suffer from severe latency bottlenecks, particularly because continuous non-linear activations dominate the computational budget. To overcome this, we introduce the Blind Spiking LSTM (BSLSTM), a TFHE-optimized recurrent architecture for privacy-preserving sequential inference. By co-designing the network with the cryptographic framework, we replace expen- sive continuous non-linearities with an efficient multi-threshold programmable bootstrapping paradigm. Evaluated on stan- dard NLP tasks, BSLSTM achieves an inference latency of 5.2 seconds for a 128-token sequence, significantly outperform- ing traditional homomorphic approaches while maintaining competitive accuracy. Operating at an amortized cost of 211 microseconds per bootstrapping operation, our work demon- strates the practical viability of low-latency, fully homomorphic inference for real-world applications.

While quantum position verification aims to certify a prover's location using quantum information, existing security definitions only guarantee that part of the successful adversarial party is in the claimed location. This leaves open the possibility that a distributed team of adversaries can jointly simulate a prover in a way that defeats the intended meaning of ``being at a location'' in position-based cryptography.

We introduce stronger notions of position verification that we call quantum localization, which requires that there is a specified, unclonable state at the verified spacetime point -- and that this state can be found nowhere else. We show that quantum localization leads naturally to a meaningful notion of trajectory verification, in which quantum information is verifiably tracked through space and time. We construct quantum localization and trajectory verification protocols using quantum anchor states, which generalize coset states from unclonable cryptography. The security of our schemes is proven in the classical oracle (i.e. ideal obfuscation) model, which can be heuristically instantiated in the plain model using post-quantum indistinguishability obfuscation.

We also introduce and instantiate the concept of functionality localization, which guarantees that the adversary has the ability to compute a secret function at the verified spacetime point, and this function cannot be computed anywhere else. This raises the intriguing possibility of localizing computational capabilities in space and time.

More broadly, we believe our notions of quantum localization and our feasibility results provide stronger foundations for position-based cryptography.

The cost of homomorphic multiplications for existing FHEs to evaluate a degree-$D$ polynomial $f(x)$ at some point $x$ is very expensive. When $x$ is encoded in a plaintext slot having a power-of-two degree $d = 2^\ell$ and $D \leq d$, one can efficiently evaluate $f(x)$ with \(O(\log d)\) multiplications using the heuristic algorithms of Okada et al. (ASIACRYPT 2023). However, neither $d = 2^\ell$ nor $D\leq d$ is satisfied for most practical FHE parameters, and the Paterson–Stockmeyer (P-S) method with \(O(\sqrt{D})\) multiplications remains the state-of-the-art for $d \neq 2^\ell$ or $D>d$. In this paper, we first present a polynomial evaluation algorithm with \(O(\log d)\) multiplications for any non-power-of-two $d$ and $D\leq d$, which achieves the same asymptotic complexity as that of Okada et al. Then, we gave a polynomial evaluation algorithm with $O(\sqrt{D/d})$ multiplications for plaintext modulus $p>2$ and $d < D\leq d\log p$, which beats the P-S method by a factor of $\sqrt{d}$ and essentially achieves logarithmic multiplication complexity when $D \leq d \cdot \min(\log^2 D, \log p)$. As a major application, we implement our algorithms in experiment to evaluate the digit extraction polynomials of the BGV bootstrapping with parameter $d$ ranging from $14$ to $45$ in HElib, and obtain a \(1.22-2.16\times\) speedup over the state-of-the-art work of Ma et al. (EUROCRYPT 2024).

$1+1 \equiv 0$---while looking strange at first glance, this is certainly true in GF(2). In this paper we propose the two field-like structures \FlipInts\ and \FlipPolys\ as potential alternative building block for crypographic schemes---in particular in the post-quantum setting. Both structures have $2^d$ elements for some positive integer $d$ and are derived from the natural numbers $N$ in the first case and the univariate polynomial ring GF(2)$[t]$ in the second case. We call the generalization of these two structures FlipFields. In addition, we see examples how they can be used for post-quantum cryptography, in particular Unbalanced Oil and Vinegar, Learning with Errors and Saber. There is also a discussion of cryptographic primitives that are most likely not suitable for FlipFields or at least not easy to tweak. As the structures are very new, this paper also includes a list of OpenProblems.

The recent work [Devadas-Hopkins-Kalai-Kothari-Lombardi-Mathialagan, STOC 2026] proposed a low-norm Nullstellensatz hypothesis for the "AND code": every polynomial $f$ vanishing on the "AND-code ideal'' should admit a Nullstellensatz decomposition over the local AND constraints whose total coefficient \(\ell_1\)-norm is only polynomially larger than the \(\ell_1\)-norm of $f$.

We give a counterexample to this conjecture by proving an exponential lower bound on the total coefficient \(\ell_1\)-norm. The core idea of the proof was discovered by ChatGPT 5.5 Pro, and we verified and reorganized the proof to improve its exposition. The proof constructs a dual linear functional, whose analysis leverages the rank of the quadratic forms to bound Fourier correlations.

The counterexample can also be extended to give the first \(\ell_1\)-norm lower bound for Nullstellensatz refutations over the \(\{\pm1\}\)-basis. Previously, \(\ell_1\)-norm lower bounds for Nullstellensatz refutations were known only over the \(\{0,1\}\)-basis, due to Potechin and Zhang [ICALP 2024]. We believe this is of independent interest to proof complexity.

The FS-FS hybrid signature scheme of Bindel and Hale [12] couples two independent Fiat-Shamir components through a single shared challenge c = H(w1, w2, D(m)), achieving one of the strongest known proof composability and simultaneous verification properties among hybrid designs, but its EUF-CMA security was stated without proof. We present the first machine-checked EUF-CMA security proof of the FS-FS hybrid, formalised in EasyCrypt in the Random Oracle Model and parametrised over abstract sigma-protocol interfaces; the bound applies to any heterogeneous FS-based pair, classical or post-quantum. We prove two symmetric security bounds, one reducing to each component independently, so that security holds whenever either component is EUF-CMA secure; the FS-FS-Schnorr corollary confirms the result is non-vacuous. We further show that the second-preimage-resistance assumption of [12] is subsumed by the ROM guessing term 1/|R|, reducing the effective assumptions from three to two: EUF-CMA of either component under the shared hybrid-hash challenge, and collision resistance of the digest. The mechanisation uncovers two proof obligations invisible at the theorem level—a logging invariant over the shared lazy oracle and a module-restriction framing argument for the abstract digest—which we isolate as reusable EasyCrypt proof patterns.

Unrolled cryptographic hardware architectures are increasingly deployed to maximize throughput, inherently intro- ducing massive algorithmic noise floors that frequently thwart traditional temporal Side-Channel Analysis (SCA). However, the reliance on structural combinational noise as a standalone coun- termeasure remains underexplored against adaptive, AI-driven profiling. This work presents a novel autonomous framework uti- lizing a Large Language Model (LLM) agent to orchestrate and execute differential power evaluations against a 161,000-gate fully unrolled AES-128 core on a target CW305 FPGA. We first estab- lish a baseline, demonstrating that standard Correlation Power Analysis (CPA) systematically fails to penetrate the unrolled noise floor, yielding statistically insignificant correlations (r ≈ 0.11). In response to this heuristic failure, the autonomous agent dynamically pivots to a Zero-State Differential Power Isolation methodology. By leveraging single-channel baseline subtraction, the agent mathematically cancels multi-round algorithmic noise from the global power trace, successfully isolating the target combinational leakage and achieving peak correlations exceeding r = 0.318 across all 16 state bytes. Furthermore, by comparing the extracted physical signatures to the logical target state, the framework autonomously extracts 16 unique physical-to-logical routing maps. This demonstrates that while automated Electronic Design Automation (EDA) synthesis inadvertently introduces physical bit-level obfuscation, these synthesis optimizations can be systematically reverse-engineered by agentic profiling. Ultimately, this work proves that unrolled combinational architectures can- not serve as a robust defense against adaptive, autonomous side- channel characterization.

Blind signatures are a central primitive for privacy-preserving applications such as e-cash, anonymous credentials, and e-voting. In the post-quantum setting, existing constructions typically follow one of two paradigms: either signatures are realized as non-interactive zero-knowledge ($\mathsf{NIZK}$) proofs of valid underlying signatures, or they are obtained from identification schemes via the Fiat--Shamir transform. In both approaches, the resulting signatures deviate syntactically from standard signatures, incurring additional verification overhead and limiting compatibility with existing infrastructures. In contrast, classical constructions such as blind Schnorr yield signatures that are indistinguishable from ordinary ones. Achieving this property in the lattice setting has remained an open problem.

We present $\mathsf{BRaccoon}$, the first lattice-based blind signature scheme that achieves concurrent security while producing signatures that are syntactically identical to those of a standard signature scheme. Our construction builds on the rejection-free lattice signature scheme $\mathsf{Raccoon}$, and extends the ``blind signatures from a signature assumption'' paradigm of Fuchsbauer and Wolf (EUROCRYPT~2024) to lattices. At a high level, we introduce blinding at the commitment stage and enforce correct challenge and response generation via linearly homomorphic encryption combined with $\mathsf{NIZK}$ proofs. As a result, $\mathsf{BRaccoon}$ signatures preserve the algebraic structure of $\mathsf{Raccoon}$ signatures while remaining compact: in an optimized instantiation, signatures are $32$ KB, public keys are $10$ KB, and total communication is $847$ KB for up to $2^{32}$ signatures.

A central technical challenge stems from discrete Gaussian sampling, where blinding induces a non-trivial distributional shift that precludes direct security reductions. To overcome this, we introduce a modified scheme $\mathsf{Raccoon}^\star$ that explicitly captures this shift. We prove that one-more unforgeability of $\mathsf{BRaccoon}$ tightly reduces to the unforgeability of $\mathsf{Raccoon}^\star$, which in turn reduces to that of $\mathsf{Raccoon}$.

For a concrete instantiation, we develop a hybrid proof framework that combines lattice-based zero-knowledge arguments for linear relations with arithmetic zk-SNARKs for hash computations, linked via structured commitments. Our work demonstrates that concurrently secure blind signatures with standard-signature syntax can be achieved in the lattice setting, providing a viable path toward practical and interoperable post-quantum privacy-preserving systems.

Functional encryption (FE) which covers the notion of attribute-based encryption (ABE), is the cryptographic tool to realize fine-grained control on the accessibility of encrypted data. The traditional FE requires a central trusted authority to issue secret keys. It depends on the full-trust model, and is vulnerable to the security issue caused by key-escrow. While the registered FE (Reg-FE) achieves the zero-trust model and addresses the security issue by removing the use of central authority. It allows users to generate secret keys themselves and join the system by registering corresponding public keys to a curator.

This work introduces delegated Reg-FE, which is a primitive with a new registration paradigm. It allows the registration of certain authorities that can issue secret keys for their respective classical FE sub-systems, beyond the prior work of registering plain users. Delegated Reg-FE implements a hybrid trust model within a two-level hierarchy. By redefining key escrow as a functional mechanism rather than a security concern, this model employs a zero-trust upper level which removes key-escrow, while the subsystem of each authority is locally full-trust and retains key-escrow mechanism.

We construct four delegated Reg-FE schemes for functionalities that can be described as the $2\times 2$ combinations of linear function and policy check. Namely, Delegated Reg-IPFE, Delegated Reg-ABE, Reg-IPFE with delegated ABE, and Reg-ABE with delegated IPFE. All concrete schemes support bounded registrations and delegations, and achieve standard adaptive security under MDDH assumption on prime-order bilinear group. Furthermore, these schemes only rely on black-box techniques. Technically, these schemes rely on dual-system techniques as prior registration-based works. And we devise a new "hierarchically invoked dual-system" technique on schemes which have sub-ABE delegation systems.

Furthermore, we present a generic construction of Delegated Reg-FE from the combination of Reg-FE and FE. The instantiations of this generic construction demonstrate the feasibility of delegated Reg-FE, supporting arbitrary functions as well as unbounded numbers of registrations and delegations. However, this approach requires non-black-box techniques and achieves weaker semi-adaptive security without malicious registration, where the semi-adaptive means the adversary claims the challenge after seeing common reference string but before making any query. Its security relies solely on the underlying assumptions of the Reg-FE and FE components.

In this work, we present a generic side-channel protected design of Ascon that achieves high efficiency by dynamically reconfiguring the hardware countermeasures during message processing. The resultant implementation is protected and capable of meeting stringent performance requirements whilst minimising resource overhead. The experimental results obtained demonstrate that the implementation meets the required security and achieves superior throughput-to-area ratio across all protection orders. Ascon, recently selected by NIST as the lightweight cryptography standard, is widely deployed in resource-constrained devices that demand both high performance and resistance against threats such as side-channel analysis (SCA). Exploiting Ascon's mode-level structure, which does not require protection against differential power analysis during bulk operations, we introduce a modified masking gadget with dual functionality: serving as a countermeasure during critical operations, and processing multiple data paths in parallel to accelerate bulk computation. Our architecture supports any configurable security order and instantiates only the minimum hardware resources needed to maximize throughput per round. We also evaluate an enhanced Ascon architecture based on the Changing of the Guards technique, which eliminates the need for fresh randomness. Security validation is performed using fixed-vs-random t-tests on both first- and second-order masked implementations. Finally, we compare our masked design against state-of-the-art solutions.

The Learning With Errors (LWE) problem is a cornerstone of lattice-based cryptography and underpins the security of numerous cryptographic schemes. To enhance efficiency, practitioners often employ sparse secrets in LWE, where the secret vector $\mathbf{s}$ has a significantly lower Hamming weight than its dimension $n$. While this approach improves performance, it raises security concerns, particularly against side-channel attacks that can leak partial information, or “hints,” about the secret key.

In this paper, we revisit the LWE with side information framework on sparse ternary secrets, focusing on approximate/perfect hints of the form $(\mathbf{v}, l)$ satisfying $l = \langle \mathbf{v}, \mathbf{s} \rangle + e$, where $e$ is a small error term, or $l = \langle \mathbf{v}, \mathbf{s} \rangle$. While previous results needed about $n/2$ perfect or modular hints to break LWE in polynomial time, we show empirically, supported by a conservative lower-bound analysis under the Gaussian Approximation Assumption (GAA), that the task can be accomplished with only $O(h \log_2 h)$ hints, where $h$ denotes the Hamming weight of $\mathbf{s}$.

We demonstrate the effectiveness of our algorithm on practical parameter sets used in Fully Homomorphic Encryption (FHE) schemes. For instance, for a sparse-secret FHE bootstrapping regime with $(n, h) = (2^{15}, 32)$, our method requires only 320 approximate/perfect hints to recover the secret key, compared to the $2^{14}$ perfect/modular hints required by previous methods. For the OpenFHE library with $(n, h) = (2^{15}, 192)$, we heuristically confirm secret-key recovery via $O(h \log_2 h)$ perfect hints; approximate hints have not yet been validated in this setting. After collecting the necessary hints, our algorithm recovers the secret key in polynomial time in dimension $n$.

The SHA-2 family hash is standardized by NIST and mainly includes two variants, SHA-256 and SHA-512. Due to its widespread deployment, its security has attracted continuous attention from various parties. Although Li et al. have developed open-source SAT/SMT-based tools and proposed new memory-efficient collision attack frameworks for SHA-2 in recent two years, practical collision attacks are only achieved for 31-step SHA-256 and 29-step SHA-512, respectively. To push the limit of such an attack framework for SHA-2, we carefully investigate existing strategies to choose message differences used in 38/39-step semi-free-start collision attacks. We found that by selecting message words $(W_{4+i}, \ldots, W_{8+i}, W_{12+i}, W_{13+i}, W_{20+i}, W_{22+i})_{0\leq i \leq 3}$ to inject differences, and employing the open-source SAT/SMT-based automated tools to search for the corresponding differential characteristics, notable improvement can be achieved for practical and theoretical collision attacks. Specifically, the first practical collision attacks on 35-step SHA-256 and SHA-512 can be achieved for $i=0$, improving the best practical collision attacks on SHA-256 and SHA-512 by 4 and 6 steps, respectively. When $i\in\{1,2\}$, theoretical collision attacks on both SHA-256 and SHA-512 can reach up to 36/37 steps. We have also tried collision attack up to 38 steps by setting $i=3$, but the uncontrolled differential probability is too low to be used for effective attacks.

A witness pseudorandom function (WPRF) is a PRF which has an additional mode of public evaluation. Given the public key, it can be evaluated publicly if one provides a valid NP witness for that input, while the output remains pseudorandom to anyone without such a witness. WPRFs are powerful objects, and general-purpose constructions are currently only known from assumptions that imply indistinguishability obfuscation.

In this work, we construct a WPRF for a specific language related to the Libert-Yung vector commitment (TCC 2010). More specifically, public evaluation of the WPRF on an input is possible if a valid local opening for that input is provided. Our construction relies only on standard assumptions on pairing groups and is fully black-box.

We further show that this primitive enables us to solve several open problems in the study of communication-efficient secure computation: - Rate-1 Laconic Oblivious Transfer. We construct a laconic oblivious transfer protocol with total communication complexity $2k+\mathsf{poly}(\lambda)$ for $k$ executions. Previously, achieving this efficiency required non-falsifiable assumptions such as evasive LWE (Wee, CRYPTO'24). - Near-optimal Laconic Private Set Intersection. We construct laconic private set intersection for which the amortized communication complexity approaches $\lambda$ bits per element in the sender's set (assuming each set element is represented by $\lambda$ bits) and is independent of the receiver's set. This is within a constant factor of the information-theoretic lower bound. - Rate-1 Batch Registration-Based Encryption. We construct a registration-based encryption scheme in which, when the encryptor sends multiple messages to multiple receivers, the ciphertext overhead is only two group elements.

Advances in quantum computing technology have motivated the development of post-quantum cryptography (PQC) algorithms. HAWK is a new post-quantum digital signature scheme and the only lattice-based candidate selected for Round 3 of the "Additional Digital Signatures" phase of the NIST PQC Standardization process. HAWK offers compact key and signature sizes compared to NIST standard ML-DSA (Dilithium), and its simple design avoids the use of floating-point arithmetic unlike NIST standard FN-DSA (FALCON). This makes HAWK very well suited for resource-constrained applications. We perform software runtime profiling of HAWK signature computation and verification on a resource-efficient Vex RISC-V processor core, and identify Keccak permutations and polynomial transformations as the most computationally expensive functions. In this work, we demonstrate light-weight hardware-software co-design of HAWK with these operations accelerated using Vex RISC-V Custom Function Units and accompanying custom instructions. We present multiple design variants with different degrees of acceleration, and our best design achieves $\approx 3 \times$ speedup and $\approx 40\%$ reduction in area-time-product compared to the baseline when implemented on a Xilinx Artix-7 FPGA.

Authenticated Private Information Retrieval (Authenticated PIR) allows the client to retrieve the desired database entry without revealing any information about the query, while safely aborting if malicious behavior by the server is detected (presented in USENIX '23). However, two key challenges remain: existing single-server authenticated PIR schemes with sublinear online communication have not yet been clearly and fully implemented; incremental updates to the digest introduce unnecessary overhead. In this paper, we implement the previously outlined idea and present two complete and concrete single-server authenticated PIR schemes with $O(\sqrt{N})$ online communication, namely LWE-AuthPIR and DDH-AuthPIR, along with detailed security proofs. Furthermore, we introduce the notion of single-server authenticated and incremental PIR and propose a corresponding concrete construction, LWE-AuthIncPIR. LWE-AuthIncPIR supports immediate updates to individual entries and integrates a communication-efficient row aggregation for periodic update scenarios. When 1\%-8\% of the entries in a 1GB database are modified, LWE-AuthIncPIR reduces offline preprocessing computation by 19-88$\times$ compared to previous incremental update method. In the password breach detection, LWE-AuthIncPIR achieves a 67$\times$ reduction in preprocessing time and a 2.9$\times$ reduction in communication overhead.

Traitor tracing [Chor-Fiat-Naor; CRYPTO'94] has historically been formalized through the lens of a $\textit{single}$ trusted authority that samples the master keys and that, therefore, can read every ciphertext on its own. This $\textit{key escrow}$ problem makes traditional traitor tracing fundamentally incompatible with end-to-end encrypted broadcast networks. In this work, we study introduce $\textit{multi-authority traitor tracing}$ (MA-TT) [Goyal-Yadugiri; ePrint] a new decentralized model for traitor tracing in which the setup is split across $K$ asynchronous and non-interacting authorities, and a user can decrypt only by combining partial keys from $\textit{every}$ authority. We require that semantic security holds even when an arbitrary set of authorities is corrupted (as long as some honest partial key remains hidden for every user), and that traceability never accuses a user for whom an honest partial key remains hidden.

We design MA-TT by combining any multi-authority attribute-based encryption (MA-ABE) scheme with a new primitive that we introduce, distributed mixed functional encryption (DMFE), a careful decentralization of the mixed functional encryption notion [Goyal-Koppula-Waters; STOC'18]. We construct DMFE from LWE via single-key key-homomorphic private constrained PRFs. Plugging in known MA-ABE schemes, we obtain MA-TT from LWE plus pairings with ciphertext size $K \cdot \mathsf{poly}(\lambda, \log N)$. As feasibility, we also give MA-TT from any PKE (with ciphertexts of size $K \cdot N \cdot \mathsf{poly}(\lambda)$) and from any multi-authority functional encryption (with fully succinct parameters).

Repeat-accumulate-accumulate (RAA) codes combine a very simple linear-time encoding procedure with strong distance behavior, making them attractive both in classical coding theory and in recent cryptographic applications such as code-based polynomial commitments, zkSNARKs, and pseudorandom correlation generators. Existing concrete analyses of RAA codes are strongest over the binary field, while large-field cryptographic applications require distance guarantees over fields whose size grows with the block length. In this regime, the usual binary-field weight-enumerator and union-bound arguments lose the large-field cancellation gains needed to obtain sharp tails.

We give a gap-covering proof of an optimal-tail distance bound for the large-field RAA ensemble \(G=RP_1AP_2A\). For every fixed repetition factor $r\ge9$ and every field size satisfying $q-1\ge(eN)^2$, we prove \[ Pr[d_{\min}(G)\le \delta N]\le \widetilde O_r(N^{1-r}) \] for every fixed $0<\delta<1/2$, where $N$ denotes the code length. We also prove the matching large-field lower bound $\Omega_r(N^{1-r})$, showing that the upper bound is optimal up to polylogarithmic factors.

In addition, we prove a binary-field companion lower bound. In particular, for every fixed $0<\delta<1/2$ and even $r$, \[ Pr[d_{\min}(G)\le \delta N] \ge c_{r,\delta}N^{1-r/2}. \] For even $r$, together with the improved binary upper bound in the literature, this identifies the binary tail up to polylogarithmic factors, while our large-field result gives the tight tail $\widetilde\Theta_r(N^{1-r})$. Thus the binary and large-field RAA ensembles have genuinely different low-distance tail exponents: the large-field improvement is an actual polynomial separation, not merely a separation between available proof techniques.

Public blockchains like Ethereum deliver transparency, but adding anonymity remains a fundamental challenge. Existing proposals either offer limited anonymity guarantees or rely on heavy cryptographic machinery, e.g., zero-knowledge proofs.

We introduce \emph{Blockchain Anonymous Tokens} (BAT), a system for efficient \emph{sender}-anonymous transactions on transparent blockchains. Building on the observation that \emph{one-time-spendable tokens suffice for many applications}, BAT has a lightweight design using classic anonymous tokens due to Chaum (1983). Unlike such tokens, though, BAT is designed to work in a transparent decentralized setting, where issuers are untrusted (i.e., any single potentially malicious entity can be the issuer) and spends happen publicly. BAT issuance is compact: A client can receive $\ell$ tokens with just $\mathcal{O}(1)$ on-chain communication and computation.

We formalize the notion for BAT, and provide a concretely efficient construction. Our BAT construction requires no on-chain verification of expensive zero-knowledge proofs; just a signature verification during spends and a single exponentiation on-chain during issuance. We present several applications of BAT in various blockchain contexts.

We prove the security of our BAT scheme assuming hardness of the one-more computational Diffie-Hellman assumption in a bilinear pairing group in the random oracle model and with any secure digital signature scheme. We implement and evaluate BAT and show that compared to the closest baseline, Zcash transactions, BAT tokens are more than 50$\times$ shorter, 9$\times$ faster to verify, and 7,000$\times$ faster to generate.

Decentralized exchanges built on Automated Market Maker (AMM) protocols have become a cornerstone of Decentralized Finance (DeFi), offering token swaps without conventional order-book matching. However, supplying liquidity to these AMM pools exposes participants to distinctive market risks, most notably impermanent loss (IL) - the potential underperformance of a liquidity-provider portfolio relative to simply holding the underlying tokens. This paper presents a comprehensive overview of IL, unifying its main theoretical models, empirical evidence, and mitigation strategies. Our survey spans constant-function and concentrated liquidity market makers, synthesizes findings from leading DeFi protocols, and reviews mitigation methods that include both protocol-level adaptations and financial-engineering approaches. Across these perspectives, we highlight recurring trade-offs, cost, complexity, and security. Finally, we outline open questions for managing IL's systemic effects, keeping decentralized liquidity provision both profitable for participants and sustainable for the broader ecosystem.