Comprehensive Guide to Post-Quantum Cryptography: Types and Applications
The digital landscape faces an escalating change: quantum computing. Traditional encryption methods are vulnerable to the immense computational power of quantum computers. Abelian is at the forefront of safeguarding digital assets by developing cutting-edge Post-Quantum Cryptography (PQC) solutions.
PQC represents a paradigm shift in cybersecurity. This article explores various types of PQC, including lattice-based, code-based, multivariate, and hash-based cryptography - highlighting their strengths, challenges, and the ongoing efforts to integrate these advanced cryptographic methods into practical applications. As quantum computing advances, proactive measures are essential to educate ourselves and develop solutions that protect sensitive information and assets. Abelian is committed to ensuring that all members of the Web3 community stay protected against quantum attacks.
Lattice-Based Cryptography
A leading candidate for post-quantum cryptography due to its resilience against classical and quantum attacks.
Lattice-based constructions, supporting important standards of post-quantum cryptography, are resistant to attacks by both classical and quantum computers. This resistance is based on hard mathematical problems like the Shortest Vector Problem (SVP) and Learning With Errors (LWE). Lattice-based cryptographic algorithms are known for their high-speed computation and lower energy consumption compared to other post-quantum cryptographic methods, making them suitable for applications requiring efficient real-time processing, such as online gaming and streaming (Geekflare). Additionally, these cryptographic methods can be applied to various security challenges, including but not limited to digital signatures, key exchange, identity-based encryption, zero-knowledge proof systems, and encryption in general. This versatility is a significant advantage, allowing for broad implementation across different domains.
While lattice-based cryptographic schemes offer robust security, their implementation can be complex. The security of lattice-based cryptographic schemes often relies on worst-case hardness assumptions, the premise that certain mathematical problems are incredibly difficult to solve — even in the worst-case scenarios. These problems are assumed to be computationally infeasible to solve efficiently by any current algorithm, including potential future advancements. Theoretical guarantees provide the mathematical proofs that assure the security of these cryptographic schemes under these assumptions. However, translating these theoretical models into practical, real-world applications can introduce challenges.
Practical implementations must ensure that they adhere to these theoretical guarantees without introducing vulnerabilities. This involves careful selection of parameters, rigorous testing, validation, and potential modifications to the algorithms to suit practical constraints. For example, when implementing schemes like Learning With Errors (LWE) or NTRU, developers must optimize algorithms for performance while preventing side-channel attacks (attacks based on extra information that can be gathered) ensuring secure handling of all cryptographic operations in both hardware and software environments.
Lattice-based cryptography encompasses several subtypes, each based on different hard lattice problems:
- NTRUEncrypt and NTRUSign: These are based on the hardness of certain lattice problems and offer efficient encryption and signature schemes suitable for constrained environments.
- Ring-Learning With Errors (RLWE): This includes schemes like qTESLA, which are built on the RLWE problem, providing strong security guarantees against quantum attacks.
- CRYSTALS-Dilithium: Selected by NIST for standardization. A digital signature scheme based on module-LWE and module-SIS.
- CRYSTALS-Kyber: Selected by NIST for standardization. A public-key encryption and key encapsulation mechanism based on the hardness of the Learning With Errors (LWE) problem. It offers strong security against quantum attacks and is noted for its efficiency in terms of computational resources, making it suitable for a wide range of applications. Selected by NIST for standardization.
- FALCON: Selected by NIST for standardization. FALCON is based on the NTRU lattice, The algorithm uses FFT techniques to optimize the polynomial multiplications required in lattice-based cryptography. This results in efficient computation generating more compact key sizes than those provided by Dilithium.
Abelian employs PQC cryptography inspired by Crystals-Kyber and Crystals-Dilithium, implemented in C and Golang.
2. Code-Based Cryptography
Dating back to the 1970s with the McEliece cryptosystem, this approach relies on the challenge of decoding a general linear code. Think of it as deciphering a secret language. The complexity of these problems makes it tough for quantum computers to crack.
Key concepts in code-based cryptography revolve around error-correcting codes and the decoding problem. Error-correcting codes are fundamental to code-based cryptography, as they are used to detect and correct errors in data transmission. These cryptographic systems leverage the mathematical structures of these codes to establish secure encryption methods. Imagine sending a picture with extra parts added so that if any part gets smudged, the receiver can still figure out what it is supposed to be. This allows the receiver to detect and correct a certain number of errors without needing to re-transmit the entire message.
The central challenge, known as the decoding problem, provides the security backbone for code-based cryptography. It involves the formidable task of decoding a general linear code by finding the original message from an encoded message that has been altered by errors, a computationally hard problem. The difficulty of this problem forms the basis of the security for code-based cryptographic schemes.
Code-based cryptography primarily revolves around the difficulty of decoding general linear codes:
- McEliece Cryptosystem: Uses the hardness of decoding random linear codes for encryption. It’s known for its long public keys but remains secure against quantum attacks. One notable disadvantage is the relatively large size of the public key compared to other cryptographic systems.
- Niederreiter Cryptosystem: Similar to McEliece, it is based on the decoding problem but uses different code structures to offer a compact and efficient alternative. A variant of the McEliece system, it employs different code structures, such as generalized Reed-Solomon codes, to achieve similar security properties.
Challenges remain in integrating code-based cryptography into practical applications, notably the large public key sizes. Optimizing key sizes and enhancing efficiency is a work in progress.
3. Multivariate Cryptography
Multivariate Cryptography’s security relies on solving systems of multivariate polynomials. Imagine recreating a dish from taste alone without knowing the ingredients. This cryptographic approach uses complex equations with many variables (multivariate polynomials) ensuring security through their difficulty.
Multivariate cryptography has received mixed reviews in the cryptographic community. Its primary strength lies in the inherent difficulty of solving systems of multivariate polynomial equations over finite fields, a problem known to be NP-hard. NP-hard (Non-deterministic Polynomial-time hard) refers to a class of problems for which no efficient solution algorithm is known. Solving an NP-hard problem means that, in the worst case, the time required grows exponentially with the size of the input. This makes these problems computationally infeasible to solve as they scale.
However, multivariate cryptosystems also face several challenges. These include vulnerabilities to specific structural attacks such as the MinRank attack which exploits weaknesses in the rank structure of matrices for attackers to analyze and reduce the problem to a simpler form that enables efficient solving for the secret key. In addition, the process of finding Gröbner bases for direct attacks remains a significant concern.
Multivariate cryptography relies on solving systems of multivariate polynomial equations:
- Rainbow Signature Scheme: Utilizes multiple layers of polynomial equations to enhance security. It’s particularly notable for its efficiency and has been considered as one of the finalists during the standardisation process of NIST.
- HFE (Hidden Field Equations): Employs hidden field equations to provide robust encryption and signature schemes, though it’s complex and susceptible to specific algebraic attacks.
Despite some challenges, ongoing research aims to improve the security and efficiency of multivariate schemes that leverages Multivariate cryptography’s inherent difficulty.
4. Hash-Based Cryptography
Hash-based cryptography, particularly hash-based signatures, was among the earliest practical methods proposed for post-quantum cryptography. These signatures depend on cryptographic hash functions, which is a function that transforms an input (or ‘message’) into a fixed-size string of bytes. While the hashing process is straightforward, reversing it to find the original message is computationally difficult. This principle underlies the security of Bitcoin mining, where miners solve complex hash puzzles to add transactions to the blockchain.
Hash-based cryptographic methods are rooted in the difficulty of reversing cryptographic hash functions:
- Lamport Signatures: One-time signatures that use hash functions for secure digital signatures. Lamport signatures are one-time signatures, meaning each key pair can only sign a single message securely. This limitation makes them impractical for applications requiring the signing of multiple messages for each key pair.
- Merkle Signature Scheme: Extends Lamport signatures to support multiple signatures, making it more practical for real-world applications by allowing multiple transactions or messages to be securely signed using a single key pair. While more efficient than Lamport signatures, the Merkle Signature Scheme still involves relatively large keys and signatures which can be computationally intensive
- SPHINCS+: Selected by NIST for standardization. A stateless hash-based signature scheme that combines many-time signature schemes to provide efficient and secure digital signatures. The design of SPHINCS+ involves multiple layers of hash functions and tree structures, making it complex to implement correctly. Ensuring that the implementation is both secure and efficient can be challenging.
Hash-based cryptographic methods like Lamport signatures, Merkle signature schemes, and SPHINCS+ are considered post-quantum because their security relies on the hardness of reversing cryptographic hash functions or finding hash collisions, tasks which quantum computers do not efficiently solve. However, Bitcoin’s primary vulnerability lies in its use of ECC for key generation and digital signatures, which is not resistant to quantum attacks.
Why Hash-Based Cryptography is Considered Post-Quantum but Bitcoin is Not Quantum-Resistant
The primary cryptographic technique used in Bitcoin is the SHA-256 hash function, which is employed in the mining process and for creating addresses. However, this is distinct from hash-based cryptographic signature schemes. Bitcoin’s primary vulnerability lies in its use of elliptic curve cryptography (ECC) for public key generation and digital signatures, which is susceptible to quantum attacks. While Bitcoin mining and transaction integrity rely on cryptographic hash functions (SHA-256), which quantum computers can only marginally improve in efficiency using Grover’s algorithm, the significant threat comes from Shor’s algorithm, which can efficiently solve the elliptic curve discrete logarithm problem (ECDLP) underlying ECC. This allows a quantum computer to derive private keys from public keys, potentially enabling the theft of bitcoins. Therefore, for Bitcoin to become fully quantum-resistant, it would need to replace its ECC-based signature scheme with a post-quantum alternative, such as a hash-based signature scheme or another quantum-resistant algorithm.
Having various types of PQC, such as lattice-based, code-based, multivariate, and hash-based cryptography, provides a robust and diversified approach to securing digital communications. The ongoing progress in PQC development, with research continually improving the security, efficiency, and practicality of these cryptographic schemes, highlights the dynamic and evolving nature of the field.
Post-Quantum Cryptography (PQC) is a field of cryptography designed to secure data against the potential threats posed by quantum computers. To add to that, the benefits of PQC far exceed quantum-resistance. Next time, we will explore more about the benefits of employing PQC and how it can enhance security in various applications.
About Abelian
Abelian is a quantum-resistant blockchain infrastructure which enables digital gold 2.0 and empowers the post-quantum crypto ecosystem. Learn more about the quantum-safe Abelian blockchain & $ABEL Tokenomics at our documentation page.
The Abelian Foundation welcomes all feedback regarding tech developments and upcoming changes. To join the conversation, please visit us on our various social media and community channels linked on our linktree👇
