Secret Sharing for the Quantum Era

by

Author: Denis Avetisyan

A new framework leverages code-based cryptography to enable secure aggregation of data, even in the face of quantum computing threats.

This paper presents a post-quantum secure aggregation scheme based on the Learning Parity with Noise (LPN) problem, offering improved efficiency over existing information-theoretically secure methods.

Existing secure aggregation protocols often rely on lattice-based cryptography, creating a vulnerability as quantum computing advances. This paper, ‘Post-Quantum Secure Aggregation via Code-Based Homomorphic Encryption’, introduces a novel approach utilizing key- and message-additive homomorphic encryption instantiated under the Learning Parity with Noise (LPN) assumption, offering a post-quantum alternative. By employing a committee-based decryptor and Chinese Remainder Theorem optimization, the construction achieves efficiency and demonstrates performance advantages over information-theoretically secure methods in specific regimes. Could this code-based framework pave the way for more practical and secure multi-party computation in a post-quantum world?

Data’s Double Edge: Collaboration, Privacy, and the Quest for Secure Insights

The current landscape of data analytics is fundamentally shaped by the need to combine information originating from diverse sources. This collaborative approach allows for more comprehensive insights, powering advancements in fields ranging from medical research to financial modeling and urban planning. However, this aggregation isn’t simply about compiling datasets; it often involves pooling sensitive details about individuals, necessitating secure and robust methodologies. The benefits of combined data – improved predictive accuracy, identification of previously unseen patterns, and enhanced decision-making – are driving an increasing dependence on this interconnectedness, making the ability to responsibly and effectively manage these distributed data streams a critical challenge for modern science and technology.

The increasing interconnectedness of modern data analysis introduces substantial privacy vulnerabilities stemming from the potential exposure of raw data. A breach involving personally identifiable information can result in severe consequences, ranging from financial losses and identity theft to reputational damage and emotional distress for individuals. Beyond individual harm, large-scale data leaks can erode public trust in institutions responsible for data handling, impacting critical services like healthcare and finance. The aggregation of data from multiple sources, while beneficial for research and innovation, amplifies these risks, as a single compromised dataset can unlock a wealth of sensitive information about numerous individuals. Consequently, robust privacy protections are not merely a matter of compliance, but a fundamental requirement for maintaining a secure and trustworthy data ecosystem.

Current methods for safeguarding data privacy during collaborative analysis frequently introduce substantial performance penalties or restrict the types of computations possible. These limitations stem from the need to obscure individual contributions while still enabling accurate results. However, a newly developed framework offers a promising alternative, potentially exceeding the communication efficiency of information-theoretically secure aggregation – a benchmark for privacy – under specific, practical conditions. This improvement is achieved through a novel approach to data encoding and sharing, minimizing the amount of information exchanged without compromising the underlying privacy guarantees. The framework’s enhanced communication efficiency could unlock broader applicability for privacy-preserving data science, particularly in bandwidth-constrained environments and large-scale collaborative projects where data transfer costs are significant.

The Building Blocks of Secure Computation: Homomorphic Encryption and Lattice-Based Security

Additively homomorphic encryption (AHE) is a form of encryption that permits computations to be performed directly on ciphertext, resulting in an encrypted result that, when decrypted, matches the result of the same computation performed on the plaintext. This functionality is critical for privacy-preserving data analysis and computation, as it allows processing of sensitive data without requiring decryption and thus exposure. The core property of AHE schemes is that given ciphertexts E(x) and E(y), the ciphertext E(x + y) can be directly computed without knowing x or y. While not all computations are supported – typically only addition and scalar multiplication – this capability is sufficient for a range of applications including secure data aggregation, private machine learning, and confidential statistical analysis.

Paillier encryption is a probabilistic public-key cryptosystem that offers additively homomorphic properties. This means that operations on ciphertexts, specifically additions, can be performed and, when decrypted, yield a result equivalent to the operation performed on the plaintexts. The Paillier cryptosystem relies on the computational hardness of factoring large numbers and the residue class ring \mathbb{Z}_{n^2}, where n is a large prime. A typical key generation involves selecting two large primes p and q, computing n = p \cdot q, and calculating \lambda = \text{lcm}(p-1, q-1). The public key is (n, g^λ \text{ mod } n^2), where g is a generator, and the private key is λ. Encryption involves a random number r and calculating c = g^r \cdot m^λ \text{ mod } n^2, where m is the plaintext. Decryption utilizes the private key to compute m = c^λ \text{ mod } n^2 \cdot L(c)^{-1} \text{ mod } n, where L(c) = \frac{c - 1}{n}.

Lattice-Based Cryptography (LBC) is considered a leading candidate for post-quantum cryptography due to its demonstrated resistance to attacks from both classical and quantum computers. Unlike widely used public-key algorithms like RSA and ECC, which are vulnerable to Shor’s algorithm when executed on a quantum computer, LBC relies on the presumed hardness of mathematical problems involving lattices – specifically, finding the closest vector or short vector within a lattice. The security of LBC schemes does not depend on the difficulty of factoring large numbers or solving the discrete logarithm problem, making them inherently resistant to quantum attacks. Several LBC algorithms, including those based on the Learning With Errors (LWE) and Ring-LWE problems, are currently being standardized by NIST as part of its post-quantum cryptography standardization process, indicating a high degree of confidence in their long-term security.

The security of lattice-based cryptographic schemes, including those utilized in our framework, relies on the computational difficulty of solving specific mathematical problems, notably the Learning With Errors (LWE) problem. LWE posits that distinguishing between truly random data and data generated by adding a small amount of noise to a secret key is computationally intractable for sufficiently large lattice dimensions. The hardness of LWE is not based on assumptions about the factorization of large numbers or the discrete logarithm problem, making it a promising candidate for post-quantum cryptography. Our framework specifically leverages the principles of LWE and its variants to construct cryptographic protocols with a focus on minimizing communication overhead, achieved through optimized parameter selection and efficient encoding techniques while maintaining a provable security reduction to the LWE problem.

KAHE: A Framework for Secure Aggregation, Stripped Down and Optimized

Key and Message Additive Homomorphic Encryption (KAHE) serves as a foundational element within the proposed aggregation framework. This encryption scheme allows for computations to be performed directly on encrypted data without decryption, facilitating secure aggregation of sensitive information. Specifically, KAHE leverages additive properties; the encryption of a sum is equivalent to the sum of the encryptions. This property is critical for enabling the server to compute the aggregate sum of client contributions while maintaining confidentiality. The scheme is designed to support efficient aggregation protocols by minimizing computational overhead associated with homomorphic operations, and it forms the basis for the system’s security guarantees when combined with other techniques like HintLPN and PolarCodes.

Key and Message Additive Homomorphic Encryption (KAHE) employs the Chinese Remainder Theorem (CRT) to facilitate efficient computation and message reconstruction. The CRT allows for the decomposition of a large integer into a set of smaller, coprime moduli; computations are then performed independently on these smaller integers, significantly reducing computational complexity. Specifically, KAHE utilizes the CRT to distribute the encryption of a message across multiple shares, enabling parallel processing. Message reconstruction is achieved by combining these shares using the CRT, effectively reversing the decomposition process and recovering the original message without requiring access to the individual shares in their original form. This approach optimizes both encryption and decryption speeds, particularly when dealing with large datasets or a high volume of computations, by replacing a single large modular operation with several smaller ones. The security of this reconstruction relies on the appropriate selection of moduli and the preservation of their secrecy.

The system employs HintLPN, a modification of the Learning with Parities Noise (LPN) problem, to improve both security and computational performance. HintLPN introduces a structured hint to the secret key, enabling faster key generation and encryption/decryption operations compared to standard LPN. Performance benchmarks demonstrate that this framework surpasses information-theoretically secure aggregation protocols when the number of participants, denoted as N, is 10, 50, or 100, provided that appropriate parameter choices are made during system configuration. These parameter choices impact the trade-off between security level and computational overhead, and are critical to achieving the observed performance gains.

Polar codes are incorporated into the system to mitigate the impact of potential transmission errors during data aggregation. These codes, based on channel coding principles, add redundancy to the transmitted data, enabling the receiver to detect and correct a predetermined number of errors without requiring retransmission. Specifically, the code’s construction, leveraging a sequence of synthetic channels with successively better properties, allows for efficient encoding and decoding. This error correction capability is crucial for maintaining the accuracy and reliability of the aggregated result, particularly in noisy communication environments or when dealing with unreliable network connections. The choice of Polar codes provides a provably capacity-achieving performance, meaning they can reliably transmit data at rates close to the theoretical maximum given the channel conditions.

Towards Post-Quantum Resilience: A Layered Defense Against Future Threats

The escalating advancements in QuantumComputing present a significant, long-term threat to currently deployed public-key cryptographic systems, many of which rely on the mathematical difficulty of problems easily solvable by a sufficiently powerful quantum computer. Recognizing this impending vulnerability, the proposed cryptographic framework prioritizes PostQuantumResilience through the implementation of algorithms believed to withstand attacks from both classical and quantum adversaries. This proactive design ensures continued data security and confidentiality in a future dominated by quantum computational capabilities. Unlike algorithms vulnerable to Shor’s or Grover’s algorithms, the underlying principles of this framework-specifically leveraging code-based cryptography-are intended to remain secure even with the advent of large-scale quantum computers, offering a crucial safeguard for sensitive information in the decades to come.

Code-based cryptography presents a valuable alternative to the increasingly prominent lattice-based cryptographic methods for securing digital information. This approach leverages the hardness of decoding general linear codes, specifically building upon the Learning with Parities (LPN) problem, where the task is to distinguish between random linear codes and slightly perturbed ones. Unlike lattice-based schemes which rely on the difficulty of problems in high-dimensional lattices, code-based systems offer a distinct mathematical foundation, creating a diversity of approaches crucial for robust post-quantum resilience. This complementary nature is vital; should a breakthrough occur in solving lattice problems with a future quantum computer, code-based cryptography offers a separate and potentially secure path forward. The inherent differences in their underlying assumptions and computational complexities position code-based cryptography, utilizing LPN, as a key component in a layered defense against emerging quantum threats, enhancing the overall security landscape.

Secret sharing represents a powerful cryptographic technique for bolstering data privacy by deliberately fragmenting sensitive information into multiple parts, known as shares. These shares are then distributed among various parties, ensuring that no single share reveals the original data. Crucially, the original data can only be reconstructed when a sufficient number of shares are combined – a predetermined threshold of shares is required for successful recovery. This approach mitigates the risk associated with a single point of failure or compromise; even if some shares fall into the wrong hands, the data remains protected. The technique finds applications in secure multi-party computation, threshold cryptography, and data storage scenarios, offering a robust defense against unauthorized access and enhancing overall data resilience.

A crucial aspect of this work lies in a precise definition of the noise rate, denoted as τ' = p⋅p′ / ( (q-1)^2 <i> (1-p) </i> (1-p') + (q-1) <i> p </i> p' ), within the context of Learning with Parities Noise (LPN)-based security schemes. This formulation accounts for the impact of colluding parties – specifically, ‘z’ adversaries – on the overall system resilience. Through rigorous analysis, a threshold z* is identified, directly linked to the total number of users N in the system. Importantly, the scheme demonstrably outperforms existing methods when the number of colluding parties exceeds this threshold, signifying a substantial improvement in privacy and security for larger user bases. This nuanced understanding of the noise rate and its relationship to the number of colluders allows for a more accurate assessment of the system’s robustness and provides a clear benchmark for its practical implementation.

The pursuit of secure aggregation, as detailed in this work, inherently demands a willingness to probe the boundaries of established cryptographic assumptions. This paper doesn’t simply accept the limitations of current methods; it actively dissects them, seeking vulnerabilities and alternative approaches rooted in the hardness of the LPN problem. As Robert Tarjan aptly stated, “If it isn’t broken, you’re not looking hard enough.” The exploration of key-additive encryption and its instantiation with Hint-LPN exemplifies this philosophy. It’s a systematic dismantling of the expected, a deliberate attempt to discover weaknesses and, through that process, forge a more robust, post-quantum solution that can, in specific scenarios, surpass the performance of existing, theoretically secure protocols.

Beyond the Horizon

The presented work, while establishing a functional post-quantum secure aggregation scheme, necessarily opens more questions than it closes. The reliance on the Learning Parity with Noise (LPN) problem, while strategically chosen, invites scrutiny. Is this merely shifting the locus of attack? The security reductions offered are, as all reductions are, contingent on assumptions. Future efforts should focus not solely on tightening those bounds, but on exploring alternative, potentially unorthodox, foundations for post-quantum secure computation. The current paradigm prizes complexity as a shield; perhaps a radical simplification, coupled with robust verification, offers a more resilient path.

Furthermore, the demonstrated performance gains over information-theoretically secure methods are context-dependent. A true test lies in scaling this approach to realistically complex datasets and network topologies. A critical avenue for exploration is the interplay between algorithmic optimizations and the inherent noise parameters of the LPN problem. Can noise be strategically injected, not merely as a security measure, but as a computational resource? Such a reframing might unlock unexpected efficiencies.

Ultimately, this work serves as a reminder that security isn’t a destination, but a perpetual arms race. The goal is not to solve security, but to build systems that gracefully degrade-and reveal their failures-under pressure. True security is transparency, not obfuscation. The challenge now is to dismantle this framework, piece by piece, to understand its breaking points, and to rebuild something stronger-and more honest-from the ruins.

Original article: https://arxiv.org/pdf/2601.13031.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-22 03:20