Diffie-Hellman

Cryptocurrency forecasting presents a distinctive combination of extreme cross-asset scale heterogeneity, non-stationary dynamics, and structural dependencies among Open, High, Low, and Close (OHLC) variables. We present CryptoL, a unified framework designed to address these challenges within multivariate time-series forecasting. CryptoL evaluates forecasting error in context-normalized coordinates within the RevIN pipeline, preventing inverse normalization from introducing an additional squared-scale weighting into the MSE objective. We formally characterize this effect through the empirical risk and parameter-gradient geometry, establishing the conditions under which large-scale assets can disproportionately influence shared-model optimization. Beyond loss-space normalization, CryptoL examines channel-independent and channel-dependent normalization for OHLC data, showing that a shared channel-dependent affine transformation preserves candle-order relations that independent channel transformations need not preserve. The framework further incorporates scale-adaptive numerical stabilization to reduce distortions caused by a fixed normalization constant across assets spanning many orders of magnitude, together with a soft feasibility loss that penalizes violations of the defining OHLC inequalities. Experiments across heterogeneous cryptocurrency assets evaluate these components through controlled ablations and demonstrate improvements in forecasting accuracy, training stability, and the frequency of financially valid OHLC predictions relative to the considered baselines. CryptoL therefore provides an integrated approach to scale-balanced optimization, structure-preserving normalization, numerical stabilization, and constraint-aware cryptocurrency forecasting.

Robot Operating System 2 (ROS 2) standardizes application code against a middleware interface (RMW) whose reference implementations are built on the Data Distribution Service (DDS). We present AXON, an alternative ROS 2 RMW implementation that separates transport policy by deployment scope. A Rust core and C++ adapter use POSIX shared-memory rings for same-host communication, QUIC for remote communication, and a daemon for discovery and graph synchronization. We then describe two fail-closed TLS 1.3 key-establishment configurations for remote traffic. The classic configuration offers only the hybrid X25519MLKEM768 group, preventing negotiation of a classical-only group. The qkd configuration imports a 256-bit key obtained through the ETSI GS QKD 014 API as a pairwise external PSK and offers no Diffie-Hellman group. Its default messages10 strategy additionally protects remote application messages with AES-256-GCM, rotating KME material after ten outgoing messages and using a fresh nonce per envelope; session relies on QUIC protection alone. The external-PSK path requires a narrow extension to rustls, now bundled with AXON. We define the threat model, distinguish peer authentication in the two configurations, and delimit the implementation-level validation from ROS 2 conformance, comparative performance, and physical-QKD validation.

Cryptographic code is critical infrastructure that must be correct, yet formally verifying production libraries remains difficult. Existing language-model proof systems solve isolated obligations with specifications and premises already given, leaving production-library verification unresolved. We present CryptoProver, an AI-based system that synthesizes internal specifications and Verus-checked proofs from high-level API contracts. Without changing executable code, CryptoProver constructs a new independent proof of curve25519-dalek and verifies RustCrypto's previously unverified chacha20 implementation against an RFC 8439 specification. These cryptographic lineages underpin deployed systems including Signal and Shadowsocks; Signal has an estimated 218M global downloads. The independent, human-led curve25519-dalek verification was developed publicly over eight months by five main contributors. Given the API contracts and a fixed trusted library of field specifications, arithmetic facts, axioms, and vstd, CryptoProver synthesizes the internal specifications and proofs in 11.4 hours with USD 466.99 in recorded API cost. CryptoProver follows a trust-first design principle: mechanical gates reject specification weakening, invented axioms, and cross-module breakage, while isolation blocks reference proof retrieval, including from git history.

The Word Problem has been a subject of intensive mathematical study for over a century, initially driving advances in combinatorial group theory and more recently emerging as a foundational hardness assumption in post-quantum cryptography (PQC). While generally undecidable, several families of infinite non-abelian groups exhibit solvable or algorithmically fast word problems, making them attractive platforms for cryptographic design. This paper introduces WPNet, a novel Graph Neural Network architecture capable of solving the Word Problem heuristically, which is demonstrated on the Baumslag-Solitar group $BS(1,2)$ and on an Artin group. By mapping unreduced words to dynamic graph structures, the model learns to cluster algebraically equivalent elements in a continuous embedding space, effectively identifying the geodesic representative of a word without executing discrete reduction steps. As an application, a model variant is developed that can predict the geodesic length of an unreduced word in both groups. To demonstrate the cryptographic severity of this structural leakage, WPNet is successfully deployed against the Wagner-Magyarik public-key cryptosystem.

We report genuine-un-compiled, textbook-faithful-quantum cryptanalysis of symmetric-cipher structures executed on real IBM quantum hardware (ibm_kingston, Heron generation). Using Simon's algorithm we recover the hidden period of the Even-Mansour cipher up to security parameter N = 10 on real hardware, beyond the largest previously reported real-hardware key recovery of N = 4, and we cleanly recover the periods of a 3-round Feistel (DES-family) construction at block sizes 6 and 8; a 21-qubit block-10 instance is verified in simulation and submitted to hardware. We further provide a breadth-first benchmark of five genuine quantum attacks spanning four symmetric-cipher design paradigms -- Bernstein-Vazirani (linear structure, single query), Grover (SPN key search, quadratic), and Simon (Even-Mansour, CBC-MAC forgery, and Feistel; exponential-to-polynomial in query complexity) -- validated to the classical-simulation ceiling of 25 qubits. We are deliberately explicit about scope: these attacks target reduced or structured constructions in the Q2 (quantum-query) model, asymptotically follow the birthday bound and therefore do not constitute quantum advantage over classical collision-finding, do not break full AES/RSA or 16-round DES, and rely on error mitigation rather than fault-tolerant error correction. Our contribution is the real-hardware demonstration at record structure sizes, the breadth of genuine algorithmic coverage across four paradigms, and an honest, reproducible benchmark with public artifacts.

Autonomous agents are moving from sandboxed text generators to operators of code, data, and physical infrastructure, and they increasingly learn while deployed. This reopens a question that alignment techniques answer only probabilistically: after an agent has adapted in the field, is the running system still confined to what its operator authorised? Here we show that confinement can be guaranteed as an invariant of the agent's execution architecture rather than a probabilistic outcome of its training. Governed individuation binds an agent at boot to a cryptographically frozen identity digest, and routes every action through a gate defined over the semantic effect of the action rather than its name. We prove that no amount of learning, skill acquisition, or self-induced governance abstraction can widen the agent's permitted authority without an operator-signed change to its identity; the guarantee holds even when the agent induces its own safety principle and that principle is wrong. Empirically, in an open-ended tool-use benchmark where a large action space rules out name-based blocking, ungoverned software agents under reward pressure attempt to tamper with their own evaluation at a task-dependent rate that reaches every run on the hardest task, whereas the gate reduces executed forbidden effects to zero as a verified property of the construction while preserving task success. An adversarial evaluation of monitors of increasing semantic depth shows false-allows falling from 75% (name-based gating) to zero (dynamic effect tracing), and refusal history transfers compliance to held-out red-line families. Trust in a deployed learning agent shifts from a wager on its continued alignment to a check anyone can run at boot.

In recent work it has been shown that colluding AI agents can use steganographic methods to exchange malicious information. Whether a transformer can implement steganographic methods depends on what cryptographic functions it can implement, since a transformer that can implement a cryptographic function within its layers has source-free randomness access. Despite existing circuit-complexity results, no prior work maps specific cryptographic constructions to transformer architectures. As Merrill et al. have shown that saturated transformers can be seen as threshold circuits, we first generate threshold circuits for three different cryptographic constructions (Keccak functions, Merkle--Damgard constructions and Merkle Trees) and then map these circuits to different transformer architectures. We derive verified scaling laws for the width and depth of the circuits which implement each cryptographic construction and propose two different mappings: no-attention mapping, tokens-as-gates mapping. Beyond its security implications, this work contributes to by establishing a methodology for deriving structural guarantees on transformer computational capacity. Specifically, we derive constructive upper bounds on what a transformer of a given depth and width could plausibly compute, providing a principled foundation for capability evaluations of transformer-based AI systems.

Fully Homomorphic Encryption (FHE) enables computations to be performed directly on encrypted data while preserving data confidentiality. However, its practical applications remain limited by high computational costs and development complexity. This paper presents ComputeFHE, an open-source C++ library that facilitates the development of privacy-preserving applications based on the TFHE cryptosystem. The library provides encrypted integer and fixed-point data types together with arithmetic, logical, comparison, conditional, and oblivious array-access operations which allow developers to implement algorithms using a familiar imperative programming paradigm. ComputeFHE supports both conventional TFHE arithmetic based on standard two-input logic gates and an optimized Arithmetic Logic Unit (ALU) architecture utilizing FHE-friendly logic primitives. Experimental results demonstrate significant reductions in the number of required bootstrapping operations, achieving performance improvements of up to 3.9x for selected operations. In addition, the library includes a simulation mode that enables testing, debugging, and complexity analysis without performing actual cryptographic computations while providing circuit complexity and bootstrapping costs. Built on top of OpenFHE, ComputeFHE offers a practical and accessible framework for developing and evaluating privacy-preserving algorithms and applications.

Internet of Medical Things (IoMT) devices operate under strict resource constraints while handling highly sensitive health data, making security and privacy critical concerns. Federated learning (FL) further complicates this landscape, as model updates exchanged during training may unintentionally expose private medical information. Emerging quantum computing capabilities threaten the long-term viability of conventional lightweight cryptographic mechanisms, motivating the integration of Post-Quantum Cryptography (PQC) into IoMT systems. This article discusses key enabling technologies for quantum-resilient IoMT, including post-quantum key establishment, lightweight encryption, and edge-native orchestration. We propose a scalable Kubernetes-based framework that integrates PQC into FL-enabled IoMT environments and validate it on a Raspberry Pi testbed. Results demonstrate that distributed cryptographic processing significantly reduces latency compared to sequential designs while maintaining feasible resource overhead. The primary contribution of this work lies in the design and validation of a secure orchestration and communication framework for FL-enabled IoMT systems. We conclude by outlining future directions toward energy-aware architectures, intelligent security optimization, and resilient next-generation Intelligent Internet of Medical Things (IIoMT) ecosystems.

The intersection of crypto x AI is spawning papers, products, online posts, and companies. All the surrounding buzz, though, obscures what exactly has been done, what the opportunities and challenges are, and what open questions deserve attention. This survey paper asks what AI can do for blockchain-based technologies (broadly construed as "crypto") (crypto x AI), and vice versa (AI x crypto). We systematize existing work, summarize key takeaways, highlight open research questions, and offer a perspective on pervasive industry misconceptions, concluding that AI and crypto are still in the very early stages of meaningful integration.

Neural distinguishers are a cryptanalysis method for symmetric-key cryptography that trains machine learning models on pairs of plaintexts and ciphertexts with specific differences in order to recover a secret key. To the best of our knowledge, no existing work has explored the use of large language models (LLMs) for neural distinguishers. In this paper, we propose LLM-based neural distinguishers through a prompt design and conduct extensive experiments with them on SPECK-32/64 to investigate whether LLMs can strengthen neural distinguishers. We then found three key insights. First, by comparing the results of LLM-based neural distinguishers with ResNet in the existing work, we demonstrate that LLMs provide no observable improvement in the performance of neural distinguishers. Second, we confirm that, at high rounds, the choice of differences is no longer effective for LLM-based neural distinguishers as well as ResNet. Third, we show that the performance of LLM-based neural distinguishers can be significantly improved by incorporating only the XOR operation results as a prompt design.

Secure aggregation allows a server to aggregate users' local updates while preserving update privacy. Existing information-theoretic problems typically assume that correlated random keys are provided by a trusted third party (TTP) or generated via prescribed groupwise structures, while the communication cost for establishing such correlated keys is often ignored. Consequently, the fundamental limits under general key-distribution mechanisms remain unknown. In this paper, we study the $T$-colluding information-theoretic secure aggregation problem with $N$ users under a general two-phase framework consisting of a key distribution phase and an update aggregation phase. Unlike prior work, we model key distribution through user-to-user communication and allow arbitrary user-generated key-distribution mechanisms, eliminating TTP or prescribed structures. This enables a joint characterization of three resources: randomness for security, key-distribution communication, and aggregation communication. We completely characterize the capacity region among these three resources by constructing a novel secure aggregation scheme together with a matching information-theoretic converse. In particular, we develop an explicit deterministic capacity-achieving construction over any finite field of size at least $N$, whereas most existing schemes either rely on TTP or employ randomized or existential constructions over sufficiently large finite fields. We further show that the optimal performance can be achieved using only pairwise shared keys, enabling implementation via Diffie--Hellman key exchange. Compared with Google's seminal secure aggregation scheme, the proposed scheme requires fewer random masking keys while preserving the same aggregation communication overhead.

The deployment of embodied artificial intelligence via world-model-based robotics presents a transformative opportunity for blockchain infrastructure, establishing urgent demand for trustworthy data provenance, cross-organizational governance, and incentive-compatible sharing across decentralized ecosystems. Simultaneously, quantum computing advances recognized by the 2025 Nobel Prize in Physics and the Turing Award threaten the cryptographic primitives securing these data economies, creating an interdependent imperative: long-lived verification for embodied AI depends on crypto-agile architectures capable of withstanding quantum adversaries. This tutorial examines blockchain as the coordination layer bridging this dual transition, from financial substrate to foundational Cyber-Physical-Social Systems infrastructure that simultaneously secures against quantum cryptanalysis and enables scalable, trustworthy data economies. The session opens with an immersive AWS Braket demonstration engaging participants with superconducting, trapped-ion, and neutral-atom hardware to assess cryptographic threat timelines and witness ECDSA-to-post-quantum signature transitions. Five integrated modules progress from embodied AI and world-model requirements through quantum hardware reality and evidence-based security migration, to scalable cross-shard architectures via BrokerChain protocols, trustworthy data economies implementing Croissant metadata standards and robotic learning provenance, and industry ecosystem integration for multi-modal cloud deployment. By bridging quantum hardware realities with embodied AI data requirements, this tutorial charts blockchain as unified infrastructure for next-generation decentralized intelligent environments, providing open-source frameworks and roadmaps for architecting quantum-resistant, interoperable, and data-trustworthy systems.

Quantum key distribution (QKD) is increasingly considered for deployment in realistic communication networks, where long distances, heterogeneous fiber infrastructure, and coexistence with classical traffic present substantial challenges. Here, we demonstrate trusted-node QKD between Linköping University and the Stockholm hub of the Swedish national quantum communication infrastructure over 270 km of deployed single-mode fiber, extended by a 33 km multi-core fiber (MCF) segment emulating a metropolitan access link, for a total distance of 303 km. The two sub-links use commercial QKD systems whose receivers are interfaced with external superconducting nanowire single-photon detectors, enabling operation at losses beyond those supported by standard internal gated-mode detectors. We operate the link while actively switching the QKD channel between two MCF cores, with co-propagating Ethernet traffic and injected broadband optical noise in the other cores. The results demonstrate the integration of commercial QKD into demanding, dynamically reconfigurable fiber infrastructure relevant to future hybrid quantum-classical networks. Finally, using the generated secret keys, we illustrate how limited and time-varying QKD throughput affects one-time-pad-protected image transmission: image fidelity depends strongly on the available QKD-generated key budget and the choice of compression algorithm, highlighting application-level challenges for QKD-based encryption in realistic scenarios.

Current AI agent observability is structurally compromised: the entity producing the activity log is the same entity whose activity is being logged. A compromised or buggy agent can omit, alter, or fabricate its own traces, and the operator running the agent has no independent way to detect tampering. We propose a class of protocols that resolves this by inverting the trust boundary: the service that receives an agent's call signs a receipt of what it observed using its own key, encrypts the receipt to the agent's owner, and publishes it to a public transparency log. The owner reconstructs a tamper-evident trail without trusting the agent or its operator. We instantiate the class as Sello, a protocol combining four properties absent in any current system: (P1) receiver-side signing, (P2) HPKE encryption to an owner public key bound to the authorization token via JWS, (P3) publication to a witness-cosigned Merkle log, and (P4) owner-side discovery by token reference. We describe the protocol, analyze its security under an adversary that controls the agent and its operator, present microbenchmarks of the cryptographic operations, and situate Sello among adjacent receipt-protocol work (Signet, AgentROA, Agent Passport System, draft-farley-acta, SCITT). We discuss known limitations including the suppression attack, service collusion, and the adoption-incentive problem.

Traditional cryptography, rooted in problems, e.g., integer factorisation or discrete log, is inevitably vulnerable to a fully operational quantum computer. Although it remains an engineering frontier, the looming threat extends to encrypted data stored today, which could be decrypted in the future with quantum capabilities. To safeguard against this eventuality, the backbone of the modern quantum-safe cryptography is the Shortest Vector Problem (SVP). We enhance Laarhoven's treatment of Ajtai et al.'s sieving as a genetic algorithm (GA) for the SVP by incorporating domain-informed SVP representation and crossover while naturally extending application to the module lattices.

This paper examines the erosion of Public Key Cryptography (PKC) security under adaptive adversarial optimisation driven by artificial intelligence. The problem addressed is the growing mismatch between algorithm-centric cryptographic security models and operational attack realities, where adversaries exploit implementation-level observability rather than breaking cryptographic primitives.

Security systems demand continuous, cryptograph- ically robust identity verification without requiring subjects to carry physical tokens, smart cards, or dedicated hardware authenticators. This paper presents BIDO (Biometric Identity Online), a device-free authentication standard that achieves Au- thenticator Assurance Level 2 (AAL2) per NIST SP 800-63B with- out storing long-lived biometric templates, facial images, or any other form of Personally Identifiable Information (PII). BIDO derives Elliptic Curve Digital Signature Algorithm (ECDSA) key material deterministically from a live biometric measurement salted with a user-defined memorized secret at every authen- tication event, eliminating persistent private-key storage while enabling verification from any commodity sensor terminal. The generated credentials are non-discoverable (non-resident) Web Authentication (WebAuthn) credentials, fully compatible with all FIDO2-enabled websites and services without modification on the server side. A multi-stage pipeline, comprising capture of 200 valid biometric samples, feature extraction using the Dlib 68- point facial landmark predictor, affine face alignment, frontality gating, Euclidean distance computation from the inter-eye mid- point, floor-division quantization with divisor q = 8, inter-session drift stabilization, and majority-voting SHA-256 hash binding, produces a Verification Seed (Vseed) from which the WebAuthn credential is transiently derived and immediately zeroized after signing. Evaluated against three prominent face benchmarks (VGGFace2, LFW, and MegaFace), achieving 99.51% verification accuracy on LFW and 92.14% Rank-1 identification accuracy on MegaFace Challenge 1 at 10^6 distractors, with a cryptographic False Accept Rate (FAR) of 0.03%, a False Reject Rate (FRR) of 0.90%.

Cryptocurrency pump-and-dump schemes coordinated via Telegram threaten market integrity. However, existing research addressing this specific threat has not yet produced solutions that combine reliable results with fast response. This is in part due to the absence of publicly available, message-level labeled data, as well as design choices. In this paper, we address both issues. In particular, we introduce a corpus of over 280,000 Telegram posts from 39 pump-organizing groups, all manually reviewed to identify 2,246 pump announcements and their targeted cryptocurrency and exchange. Leveraging this dataset, we define two tasks: real-time pump-announcement detection and target cryptocurrency/exchange extraction. For detection, we compare two machine-learning models: a lightweight tree-based LightGBM classifier (F1=0.79, latency=9.4 s/sample) and a transformer-based BGE-M3 (F1=0.83, latency=50 ms/sample). With our proposed approach, we show that message analysis can achieve near-instant pump detection at the level of individual Telegram message windows. Unlike prior work that relies purely on market data and typically detects pumps tens of seconds after abnormal trading activity is observed, our method operates directly on the coordination messages themselves and can be evaluated in microseconds per window on commodity hardware. To our knowledge, we also establish the first benchmark for manipulated coin and exchange extraction. We demonstrate that traditional rule-based extraction methods, widely relied upon in prior literature, are ineffective due to ticker ambiguity. In contrast, LLMs achieve the highest accuracy with a score of 0.91.

The increasing deployment of Internet-of-Things (IoT) devices has accelerated the use of distributed learning frameworks, where data remains local while model updates are shared across decentralized systems. Although this reduces centralized data collection, it introduces privacy risks through the exchange of gradients, model parameters, and intermediate representations. A variety of privacy-preserving techniques have been proposed to address these risks, including differential privacy, cryptographic methods, and lightweight system-level approaches. However, existing surveys often evaluate these methods in isolation and lack a unified framework for comparing their effectiveness under realistic attack models and IoT resource constraints. This paper presents a structured analysis of privacy-preserving techniques for distributed learning in IoT environments. A unified threat model is introduced that captures model inversion, membership inference, gradient leakage, and communication-based attacks. Building on this model, an evaluation framework is developed to compare methods in terms of both privacy robustness and system-level efficiency, including computational, memory, and communication overhead. Using this framework, representative approaches including differential privacy, homomorphic encryption, secure multi-party computation, distributed selective stochastic gradient descent, and Bloom Filter-based methods are analyzed. The results highlight a fundamental trade-off between privacy strength and system efficiency. In particular, Bloom Filter-based encodings are shown to provide lightweight privacy through collision-induced ambiguity while maintaining low computational and communication overhead. The paper provides a unified perspective on privacy-preserving design choices for distributed learning in IoT systems.

Our computing ecosystem is being transformed by two emerging paradigms: the increased deployment of agentic AI systems and advancements in quantum computing. With respect to agentic AI systems, one of the most critical problems is creating secure governing architectures that ensure agents follow their owners' communication and interaction policies and can be held accountable for the messages they exchange with other agents. With respect to quantum computing, existing systems must be retrofitted and new cryptographic mechanisms must be designed to ensure long-term security and quantum resistance. In fact, NIST recommends that standard public-key cryptographic algorithms, including RSA, Diffie-Hellman (DH), and elliptic-curve constructions (ECC), be deprecated starting in 2030 and disallowed after 2035. In this paper, we present MAGIQ, a framework for policy definition and enforcement in multi-agent AI systems using novel, highly efficient, quantum-resistant cryptographic protocols with proven security guarantees. MAGIQ (i) allows users to define rich communication and access-control policy budgets for agent-to-agent sessions and tasks, including global budgets for one-to-many agent sessions; (ii) enforces such policies using post-quantum cryptographic primitives; (iii) supports session-based enforcement of policies for agent-to-agent and one-to-many agent sessions; and (iv) provides accountability of agents to their users through message attribution. We formally model and prove the correctness and security of the system using the Universal Composability (UC) framework. We evaluate the computation and communication overhead of our framework and compare it with the state-of-the-art agentic AI framework SAGA. MAGIQ is a first step toward post-quantum-secure solutions for agentic AI systems.

Large language models (LLMs) are increasingly deployed in high-stakes domains, yet a unified treatment of their overlapping safety challenges remains lacking. We present SafeLM, a framework that jointly addresses four pillars of LLM safety: privacy, security, misinformation, and adversarial robustness. SafeLM combines federated training with gradient smartification and Paillier encryption for privacy, integrates defenses against training and inference-time attacks, employs contrastive grounding with calibrated decoding to reduce hallucinations, and introduces alignment-aware binarized aggregation to enhance robustness while maintaining bounded reconstruction quality. Across benchmarks on factuality, toxicity, and membership inference, SafeLM achieves 98.0% harmful content detection accuracy, reduces communication by 96.9%, and lowers gradient inversion PSNR from 31.7 dB to 15.1 dB. Ablations show that each component contributes independently, whereas their integration yields a strong privacy utility efficiency trade-off for deploying trustworthy LLMs.

Large language model (LLM) routing has emerged as a critical strategy to balance model performance and cost-efficiency by dynamically selecting services from various model providers. However, LLM routing adds an intermediate layer between users and LLMs, creating new privacy risks to user data. These privacy risks have not been systematically studied. Although cryptographic techniques such as Secure Multi-Party Computation (MPC) enable privacy-preserving computation, their protocol design and implementation remain under-explored, and naïve implementations typically incur prohibitive computational overhead. To address this, we propose a privacy-preserving LLM routing framework (PPRoute). PPRoute includes multiple strategies to speed up encoder inference and nearest neighbor search under the MPC and maintain the quality of LLM routing. First, PPRoute uses MPC-friendly operations to boost the encoder inference. Second, PPRoute uses a multiple-step model training algorithm to maintain routing quality despite the constraints of the encrypted domain. Third, PPRoute proposes an unsorted Top-k algorithm with $O(1)$ communication complexity for secure sorting in model search, significantly reducing communication latency. Across different datasets, PPRoute achieves the performance of plaintext counterparts, while achieving approximately a 20$\times$ speedup over naïve MPC implementations.

Softmax Mixture Models (SMMs) are discrete $K$-component mixture models for the probabilities of selecting one of $p$ candidate feature vectors $X_1,\ldots,X_p\in\mathbb{R}^L$ in heterogeneous populations and are widely used in econometrics and scientific applications. Related softmax mixture mechanisms also appear in modern LLM architectures. We provide a theoretical and methodological study of SMMs, focusing on the Expectation-Maximization (EM) algorithm and the Method of Moments (MoM). We show that EM recovers the mixture atoms at the parametric rate, up to logarithmic factors, after $\mathcal{O}(\log N)$ iterations, provided atom separation is at least of order $\log K$. This improves on separation conditions in existing analyses of EM for high-dimensional Gaussian mixtures. We also develop MoM procedures for parameter and subspace estimation. Although MoM parameter estimates converge more slowly than EM and can deteriorate with $K$, they provide provable warm starts for EM and are useful for small $K$. For general $K$, we estimate the atom subspace via MoM and recommend running EM from multiple random initializations within this subspace. Finally, as $p\to\infty$, we show that SMMs approximate mixtures of exponential tilts of the feature distribution, yielding asymptotic identifiability.