We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. This was initiated by Shmueli (2024) where the author introduced the complexity class $\mathbf{BQP^{OI}}$ consisting of promise problems decidable by quantum polynomial time algorithms with access to an oracle for computing order interference. In this work, it was shown that the Graph Isomorphism problem and the Gap Closest Vector Problem (with approximation factor $\mathcal{O}(n^{3/2})$) are in $\mathbf{BQP^{OI}}$. We extend this result by showing that the entire complexity class $\mathbf{SZK}$ (Statistical Zero Knowledge) is contained within $\mathbf{BQP^{OI}}$. This immediately implies that the security of numerous lattice based cryptography schemes will be compromised in a computational model based on superposition of spacetimes, since these often rely on the hardness of the Learning with Errors problem, which is in $\mathbf{SZK}$.
We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. This was initiated by Shmueli (2024) where the author introduced the complexity class BQPOI consisting of promise problems decidable by quantum polynomial time algorithms with access to an oracle for computing order interference. In this work, it was shown that the Graph Isomorphism problem and the Gap Closest Vector Problem (with approximation factor O(n3/2)) are in BQPOI. We extend this result by showing that the entire complexity class SZK (Statistical Zero Knowledge) is contained within BQPOI. This immediately implies that the security of numerous lattice based cryptography schemes will be compromised in a computational model based on superposition of spacetimes, since these often rely on the hardness of the Learning with Errors problem, which is in SZK.
The advent of large-scale quantum computers poses an existential threat to the security of current public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), which rely on the computational intractability of integer factorization and discrete logarithms. This paper explores lattice-based cryptography as a promising candidate for post-quantum security. We delve into the mathematical foundations of lattices, specifically focusing on hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE) problems, which are believed to remain intractable even for quantum computers. We provide a comprehensive review of prominent lattice-based constructions, including NTRU, Kyber, and Dilithium, highlighting their security proofs, efficiency considerations, and suitability for various cryptographic primitives such as public-key encryption, key-exchange mechanisms, and digital signatures. Furthermore, we discuss the challenges and opportunities associated with their deployment, including parameter selection, resistance to side-channel attacks, and integration into existing cryptographic infrastructures. By examining the theoretical underpinnings and practical implications, this work aims to solidify understanding of lattice-based methods as a robust and essential component of the cryptographic landscape in the quantum era, moving beyond the traditional RSA frontier.
The landscape of modern cryptography is fundamentally challenged by the advent of quantum computing. Current public-key cryptosystems, which form the bedrock of secure digital communication, rely on the computational hardness of problems such as integer factorization (RSA) and the discrete logarithm problem (ECC). However, Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these schemes, rendering them insecure. This paper explores the critical transition from these classical number theory problems to lattice-based cryptography, presenting a comprehensive analysis of the motivations, theoretical underpinnings, and practical implications for forging post-quantum cryptographic security. We delve into the mathematical foundations of lattice problems, such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to resist known quantum algorithms. Through a thorough literature review, we trace the evolution of cryptographic thought, detailing the vulnerabilities of pre-quantum schemes and the emergence of lattice-based constructions as leading candidates for post-quantum standardization. The methodology outlines a comparative theoretical analysis of security, efficiency, and robustness against both classical and quantum attacks. Results demonstrate the compelling advantages of lattice-based schemes in terms of provable security reductions and resistance to quantum threats, albeit with considerations for increased key and ciphertext sizes. The discussion highlights the challenges and opportunities in the deployment and standardization of these new cryptographic primitives, emphasizing their critical role in securing future digital infrastructures. This work concludes that lattice-based cryptography offers a robust and theoretically sound pathway to establishing enduring security in a quantum-threatened world.
The advent of quantum computing poses a significant existential threat to the security foundations of current public-key cryptography, most notably the widely adopted RSA algorithm. Shor's algorithm, specifically, can efficiently break the mathematical problems underlying RSA and elliptic curve cryptography, necessitating a paradigm shift towards quantum-resistant solutions. This paper explores lattice-based cryptography as the leading candidate for this post-quantum era. We delve into the fundamental mathematical principles of lattices, including hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Short Integer Solution (SIS), and Learning With Errors (LWE), which form the bedrock of their security. The paper reviews prominent lattice-based schemes such as NTRU, CRYSTALS-Kyber, CRYSTALS-Dilithium, and Falcon, highlighting their construction, security properties, and performance characteristics. We analyze their resilience against both classical and quantum attacks, discuss their progress in the NIST Post-Quantum Cryptography Standardization process, and evaluate their practical implications and implementation challenges. The discussion emphasizes the provable security, versatility, and efficiency of lattice-based methods, positioning them as the robust and essential successor to RSA for safeguarding digital communications and data in a quantum-enabled future.
The security of conventional public-key cryptography schemes is based on mathematical problems which are considered to be infeasible to solve by classical computers in a reasonable amount of time. However, the rise of quantum computers might break this security in the future. Post-quantum cryptography (PQC) aims to achieve secure schemes which are resistant against attacks assisted by quantum computers. Public-key cryptography refers to asymmetric cryptography in which encryption is done with a public key and decryption is done with a private key. Characteristic properties of public-key cryptography include the possibility to derive the private key from public information by solving some presumably hard problem. The hardness of the problem constitutes the basis for the system's security. In the post-quantum era, the security must be based on problems that are too hard to solve even for quantum computers. In this thesis, we will study lattice-based cryptography which is believed to serve as a good candidate for secure post-quantum public-key cryptography. Our main attention is in the problem of learning with errors, which is in the base of some lattice-based schemes and is linked to the hard problem of finding short vectors in a lattice.
In this thesis we will discuss hard computational problems in lattice theory and relate them to cryptographic constructions. Many of these problems enjoy average-case hardness which makes them attractive for cryptography. In addition, lattice based cryptography is a candidate for post-quantum cryptography, as there is no known quantum algorithm which breaks various hardness theorems.\n\nWe build a foundation in algebraic number theory to have the required background to discuss schemes based on discrete algebraic structures. These structures are free Z-modules which permits unique factorization in prime ideals. We relate this algebraic number theory to lattices in R^n so we can use the theory from algebra to our advantage.\n\nWe then define some standard hard computational lattice problems and show how many of these are related to each other. We prove that these problems are at least as hard as finding the shortest vector of a lattice, which we conjecture is computationally infeasible. We then prove a quantum reduction the learning with errors problem, a problem in machine learning . We also show that there is a similar reduction for a variant of this problem over more general rings.
The landscape of modern cryptography, traditionally anchored in the computational hardness of number-theoretic problems such as integer factorization and the discrete logarithm problem, faces an existential threat from the advent of quantum computing. Shor's algorithm, discovered in 1994, demonstrates the theoretical capability of a sufficiently powerful quantum computer to efficiently solve these problems, rendering widely used cryptographic schemes like RSA and Elliptic Curve Cryptography (ECC) insecure. This imminent threat necessitates a paradigm shift towards post-quantum cryptography (PQC), which aims to develop cryptographic algorithms resistant to both classical and quantum attacks. Among the various candidates for PQC, lattice-based cryptography has emerged as a particularly promising area. This paper provides a comprehensive overview of the transition from classical number theory-based cryptography to lattice-based schemes. We explore the fundamental mathematical principles underlying lattices, introduce key lattice problems such as the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE), and discuss their application in constructing quantum-resistant cryptographic primitives. Furthermore, we analyze the security, efficiency, and practical considerations of prominent lattice-based cryptosystems, highlighting their role in the ongoing NIST post-quantum cryptography standardization process. The paper concludes by discussing the challenges and future directions in the deployment and research of lattice-based cryptographic solutions.