Quantum Computing Advancements: Researchers Modify Shor’s Algorithm for Enhanced Efficiency and Security

Quantum computing has seen significant advancements since Peter Shor proposed a polynomial-time quantum algorithm for finding discrete logarithms and factoring integers. However, the current state of quantum computers has limitations, such as high gate error rates and a small number of physical qubits. Researchers Jinyoung Ha, Jonghyun Lee, and Jun Heo have conducted a study to estimate the number of physical qubits and execution time by decomposing an implementation of Shor’s algorithm for elliptic curve discrete logarithms. They also present modified quantum circuits for elliptic curve discrete logarithms and compare their results with those of the original quantum circuit implementations.

What is the Impact of Noisy Qubits on Quantum Computing for Elliptic Curve Discrete Logarithms?

Quantum computing, a field that has seen significant advancements since Peter Shor proposed a polynomial-time quantum algorithm for finding discrete logarithms and factoring integers, is currently being explored by several corporations, including Google and IBM. However, the current state of quantum computers has clear limitations, such as high gate error rates and a small number of physical qubits. This has led to active research on noisy-intermediate-scale quantum operation and quantum error-correcting (QEC) code in a quantum processor.

In addition, research is being conducted on quantum processor architecture to reduce the resources necessary for quantum computers and quantum computing software for efficient operation. Despite these advancements, the quantum resources requirements for Shor’s factoring algorithms have been investigated. Still, there are few studies on the physical resource analysis of Shor’s algorithm for elliptic curve discrete logarithms.

This study by Jinyoung Ha, Jonghyun Lee, and Jun Heo aims to estimate the number of physical qubits and execution time by decomposing an implementation of Shor’s algorithm for elliptic curve discrete logarithms into universal gate units at the logical level when surface codes are used. The researchers also presented modified quantum circuits for elliptic curve discrete logarithms and compared their results with the original quantum circuit implementations at the physical level.

How are Elliptic Curves Utilized in Quantum Computing?

Elliptic curves create public key methods such as key exchange and digital signatures, widely employed in cryptographic systems. Notable curves with widespread use include NIST curves P-256, P-384, and P-521, which are Weierstrass curves over special primes of sizes 256, 384, and 521 bits, respectively.

Elliptic curve cryptography is a public key cryptography approach based on the algebraic structure of elliptic curves over finite fields. The difficulty of computing discrete logarithms in elliptic curve groups, that is, the elliptic curve discrete logarithm problem, is used to secure elliptic curve cryptography.

However, the outcomes of assessments on the physical resource analysis of Shor’s algorithm for elliptic curve discrete logarithms vary greatly depending on the assumptions used. Therefore, examining the resources required under various scenarios is vital.

What are the Modifications Made to the Quantum Circuits for Elliptic Curve Discrete Logarithms?

In addition to the resource analysis, the researchers modified the algorithm to reduce the required resources of the algorithm. The modifications were performed by focusing on the method of performing modular operations in the Roetteler algorithm (RA).

First, the serial constant adder was modified to a parallel constant adder. As the parallel constant adder uses dirty ancilla qubits to reduce the operation depth, the operation speed can be increased without using additional logical qubits for data. Second, the continuous adders to add or subtract p for modular operation were modified into a Takahashi adder. Takahashi adders must use additional logical qubits for data because they must input the number to be added as quantum values. However, as the structure is simple, it is expected to speed up the operation.

What are the Contributions of this Study?

The contributions of this study are significant. First, the researchers express the number of physical qubits (Nphy) and execution time (Tr) required for RA as a closed-form equation for the bit length in elliptic curve cryptography. In addition, they proposed two types of modified algorithms to reduce the resources required for the RA: an algorithm that parallelizes a constant adder and an algorithm that transforms constant adders for modular operation into Takahashi adder.

In the case of the first modified algorithm, the Nphy is increased but the execution time is reduced compared to the original RA. In the case of the second modified algorithm, the Nphy remained almost unchanged. In contrast, the execution time was confirmed to be the shortest among the two modified algorithms and original RA. These results suggest that using the Takahashi adder can be more efficient than using the constant adder even if additional logical qubits are used when performing modular operations.

How does this Study Impact the Field of Quantum Computing?

Finally, the researchers compared the resources required for attacking elliptic curve cryptography with those required for attacking RSA, analyzed in a previous article, and confirmed again that elliptic curve cryptography is more vulnerable to quantum computing attacks at the physical level.

This study provides valuable insights into the resource analysis and modifications of quantum computing with noisy qubits for elliptic curve discrete logarithms. It also highlights the need for further research to overcome the limitations of current quantum computers and enhance the efficiency and security of quantum computing operations.