Breakthrough in Quantum Algorithms for Lattice Problems: Implications for Abelian’s Security
Quantum Algorithms for Lattice Problems
A groundbreaking research paper by Prof. Yilei Chen from Tsinghua University has sent shockwaves through the cryptography community. The paper, titled “Quantum Algorithms for Lattice Problems,” describes an efficient quantum algorithm for solving the Learning With Errors (LWE) problem, which is the foundation of lattice-based cryptosystems like the ones used by Abelian.
Methodology: Prof. Chen’s novel approach leverages —
- Gaussian functions with complex variances in the quantum algorithm design, exploiting the Karst wave feature in the discrete Fourier transform of these functions.
- Windowed quantum Fourier transforms with complex Gaussian windows, merging information from time and frequency domains. The technique is described in the “Overview of our algorithm for solving LWE” section on pages 6–7.
- Transformation of LWE instances into quantum states with purely imaginary Gaussian amplitudes, converting these into classical linear equations tied to LWE’s secret and error terms.
- Use of Gaussian elimination to solve these equations and recover the LWE secret.
Results: Prof. Chen’s findings include —
- A polynomial-time quantum algorithm that resolves the LWE problem for certain polynomial modulus-noise ratios.
Let n, m, q ∈ N, α ∈ (0, 1) be such that m ≥ Ω(n log q), q ∈ ˜Ω((αq)⁴ m²). There is a quantum algorithm that solves LWE_n,m,q,U(Z_q),D_Z,αq in time poly(m, log q, αq).
- Polynomial-time quantum algorithms that address GapSVP and SIVP for all n-dimensional lattices within approximation factors of Õ(n⁴.5).
There exist poly(n) time quantum algorithms that solve SIVP_γ and GapSVP_γ for all n-dimensional lattices for γ ∈ ˜O(n⁴.5).” This corollary presents the establishment of polynomial-time quantum algorithms for solving GapSVP and SIVP for all n-dimensional lattices within approximation factors of Õ(n⁴.5), which is a direct consequence of the main result in Theorem 1.6.
Implications for Abelian’s Security
Lattice-based cryptography has been considered a promising candidate for post-quantum security, as it is believed to be resistant to attacks by both classical and quantum computers. The LWE problem, in particular, has been widely used in the construction of various cryptographic primitives, including the ones selected by NIST for standardization in their Post-Quantum Cryptography (PQC) project. NIST currently states lattice-based Crystals-Kyber and Crystals-Dilithium as the standard for “quantum-safe” security.
Prof. Chen’s algorithm, if proven correct, would imply that lattice-based cryptosystems are not as quantum-resistant as previously thought. However, it’s important to note that the existence of an efficient quantum algorithm does not necessarily mean that these cryptosystems can be broken in practice.
First, although tech powerhouses such as Apple has already implemented PQ3 Protocol using quantum-safe mechanisms to protect iMessages and the NSA reports that practical quantum computing tools are only about 3 to 5 years out from workforce use, the universal quantum computer that can break elliptic curves, widely used in blockchain technology but known to be vulnerable to quantum attacks, is still 5–10 years away from being applied to real life. but the algorithm assumes the existence of a universal quantum computer, which is still years away from being realized.
Second, even if Prof. Chen’s algorithm is theoretically correct, it will take time for the proofs to be thoroughly verified by the cryptography community. If the algorithm holds up to scrutiny, it may prompt NIST to reevaluate their PQC standardization process and consider alternative cryptosystems.
Despite these developments, it’s crucial to understand that the lattice-based cryptosystems used by Abelian are still significantly more quantum-resistant than their elliptic curve counterparts. Prof. Chen himself acknowledges that his algorithm does not directly break the parameters used in schemes like Kyber and Dilithium, which are among the finalists in the NIST PQC competition.
To put things in perspective, if it takes a quantum computer with 100,000 qubits to crack elliptic curve cryptosystems, it would require a machine with over 10,000,000,000 qubits to break the lattice-based schemes used by Abelian. This difference in scale highlights the enduring security advantages of lattice-based cryptography, even in the face of theoretical advances. Currently, IBM’s Quantum Eagle Processor can deploy 127 qubits and Japanese tech centers RIKEN and Fujitsu have collaborated to develop a new 64-qubit superconducting quantum computer .
The Abelian research team is actively monitoring developments related to Prof. Chen’s work and will provide updates and insights as the situation evolves. We remain committed to ensuring the security and reliability of our platform, and we will adapt our cryptographic primitives as necessary to maintain the highest standards of protection for our users.
In conclusion, while Prof. Chen’s research is a significant theoretical breakthrough, it does not pose an immediate practical threat to the security of Abelian or other lattice-based cryptosystems. As the quantum computing landscape continues to evolve in dynamic directions, we will work diligently to stay ahead of the curve and provide our users with the most secure and advanced cryptographic solutions available.
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.
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