Circuit for Shor's algorithm using 2n+3 qubits

Beauregard, Stéphane

doi:10.26421/qic3.2-8

Cited by 162 publications

(225 citation statements)

References 11 publications

(16 reference statements)

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“…The reference implementation works the way most implementations of Shor's algorithm do, by decomposing exponentiation into iterative controlled modular multiplication [6,31,42,87,90,91]. A register x is initialized to the |1 state, then a controlled modular multiplication of the classical constant g 2 j (mod N ) into x is performed, controlled by the qubit e j from the exponent e, for each integer j from n e − 1 down to 0.…”

Section: Reference Implementationmentioning

confidence: 99%

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

Gidney

Ekerå

2021

Quantum

386

175

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We significantly reduce the cost of factoring integers and computing discrete logarithms in finite fields on a quantum computer by combining techniques from Shor 1994, Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Ekerå-Håstad 2017, Ekerå 2017, Ekerå 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction using plausible physical assumptions for large-scale superconducting qubit platforms: a planar grid of qubits with nearest-neighbor connectivity, a characteristic physical gate error rate of 10−3, a surface code cycle time of 1 microsecond, and a reaction time of 10 microseconds. We account for factors that are normally ignored such as noise, the need to make repeated attempts, and the spacetime layout of the computation. When factoring 2048 bit RSA integers, our construction's spacetime volume is a hundredfold less than comparable estimates from earlier works (Van Meter et al. 2009, Jones et al. 2010, Fowler et al. 2012, Gheorghiu et al. 2019). In the abstract circuit model (which ignores overheads from distillation, routing, and error correction) our construction uses 3n+0.002nlg⁡n logical qubits, 0.3n3+0.0005n3lg⁡n Toffolis, and 500n2+n2lg⁡n measurement depth to factor n-bit RSA integers. We quantify the cryptographic implications of our work, both for RSA and for schemes based on the DLP in finite fields.

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Section: Reference Implementationmentioning

confidence: 99%

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

Gidney

Ekerå

2021

Quantum

386

175

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“…In this work, the Shor Factorizing algorithm has been executed on IBM's quantum computer, as shown in Figure 5. In earlier research, the theoretical circuits of Shor's algorithm use 2n+3 qubits for factoring [2], yet the circuits consume 4n+2 qubits for factoring with IBM Q Experience in practice, according to our observation. We infer that the overhead comes from auxiliary quantum registers used in addition and multiplication of algorithm implementation.…”

Section: Factorization Attack Test On Ibm Quantum Cloud Computersmentioning

confidence: 66%

Quantum-Resistant Network for Classical Client Compatibility

Lin

Fuh

2021

ITC

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Quantum computing is no longer a thing of the future. Shor’s algorithm proved that a quantum computer couldtraverse key of factoring problems in polynomial time. Because the time-complexity of the exhaustive keysearch for quantum computing has not reliably exceeded the reasonable expiry of crypto key validity, it is believedthat current cryptography systems built on top of computational security are not quantum-safe. Quantumkey distribution fundamentally solves the problem of eavesdropping; nevertheless, it requires quantumpreparatory work and quantum-network infrastructure, and these remain unrealistic with classical computers.In transitioning to a mature quantum world, developing a quantum-resistant mechanism becomes a stringentproblem. In this research, we innovatively tackled this challenge using a non-computational difficulty schemewith zero-knowledge proof in order to achieve repellency against quantum computing cryptanalysis attacks foruniversal classical clients.

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“…The first one is to double the size of the private keys of existing cryptosystems. Some authors speculate that doubling the size of current secure instances of RSA and AES should be enough to resist attacks by quantum computers, but if the development of quantum-based hardware has a behavior similar to Moore's law, doubling the size of the private keys will prove to be only a temporary solution [2].…”

Section: A Replacing the Integer Factorization Problemmentioning

confidence: 99%

“…Beauregard's variation uses 2n + 3 qubits to perform O(n 3 log n) operations to factorize N = pq with p and q primes, and it can also be modified to perform n 2+o (1) if N fits into n bits. Also, by a simple algebraic transformation, Shor's algorithm can be used to solve the discrete logarithm problem [2]. Besides Shor's algorithm, Grover's algorithm has also been studied.…”

Section: Introductionmentioning

confidence: 99%

O2MD²: A New Post-Quantum Cryptosystem With One-to-Many Distributed Key Management Based on Prime Modulo Double Encapsulation

Rodas

Lin

Lu³

et al. 2021

IEEE Access

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Polynomial-time attacks designed to run on quantum computers and capable of breaking RSA and AES are already known. It is imperative to develop quantum-resistant algorithms before quantum computers become available. Computationally hard problems defined on lattices have been proposed as the fundamental security bases for a new type of cryptography. The National Institute of Standards and Technology (NIST) recently hosted the Post-Quantum Cryptography Standardization project, aiming to create a roster of innovative post-quantum cryptosystems. These candidates have been publicly available for testing since early 2017. As they are currently under analysis, new proposals are still desirable. As such, we use the ring learning with errors (RLWE) problem combined with arithmetic functions to propose the O2MD 2 cryptosystem, which provides a one-to-many private/public key architecture having a distributed key refresh for a network of users while working on multiple polynomial rings over different prime order fields. Our solution has three different frameworks that reach AES-256 equivalent security, and provides message integrity and message authenticity verifications. We compare our solution's speed against the speed of the twenty-six different implementations from seven popular candidates in the NIST project, and our cryptosystem performs from 2 to 4 orders of magnitude faster than them. We also propose six different implementations that reach the security levels 1, 3 and 5 proposed in the NIST competition. Finally, we used the NIST Statistical Test Suite to verify the indistinguishability of our produced ciphertexts against randomly generated noise.

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Circuit for Shor's algorithm using 2n+3 qubits

Cited by 162 publications

References 11 publications

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

Quantum-Resistant Network for Classical Client Compatibility

O2MD²: A New Post-Quantum Cryptosystem With One-to-Many Distributed Key Management Based on Prime Modulo Double Encapsulation

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