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arXiv:1209.5491 · Quantum Information and Computation

Quantum binary field inversion: improved circuit depth via choice of basis representation

Brittanney Amento, Martin Rötteler, Rainer Steinwandt

Published Jan 1, 2013• 13 citations
arXivHTMLPDF

Abstract

Finite fields of the form ${\mathbb F}_{2^m}$ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth $\bigO(m\log(m))$. To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields ${\mathbb F}_{2^m}$ based on an elementary gate set that is useful for fault-tolerant implementation.

DOI
10.26421/qic13.1-2-7

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Finite fields of the form F2m{\mathbb F}_{2^m}F2m​ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth \bigO(mlog⁡(m))\bigO(m\log(m))\bigO(mlog(m)). To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields F2m{\mathbb F}_{2^m}F2m​ based on an elementary gate set that is useful for fault-tolerant implementation.
Efficient quantum circuits for binary elliptic curve arithmetic: reducing TTT-gate complexity
Elliptic curves over finite fields F2n{\mathbb F}_{2^n}F2n​ play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of TTT-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in F2n\mathbb F_{2^n}F2n​ in depth $\bigO(n\log_2 n)$ using a polynomial basis representation, which may be of independent interest.
Works on quantum computing and cryptanalysis has increased significantly in the past few years. Various constructions of quantum arithmetic circuits, as one of the essential components in the field, has also been proposed. However, there has only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor's algorithm for Elliptic Curve Discrete Logarith Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for binary finite field. In particular, we propose follow a complete waterfall approach to translate the Itoh-Tsujii's variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (CNOT count), which contributes to reduced overall depth and gate count. Furthermore, compare the cost by firstly constructing our method and previous work's in Qiskit quantum computer simulator and perform the resource analysis. Our approach can serve as an alternative for a time-efficient implementation.
Elliptic curves over finite fields GF(2^n) play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of T-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in GF(2^n) in depth O(n log n) using a polynomial basis representation, which may be of independent interest.
Works on quantum computing and cryptanalysis have increased significantly in the past few years. Various constructions of quantum arithmetic circuits as one of the primary elements in the field have also been proposed. However, there have only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor’s algorithm for Elliptic Curve Discrete Logarithm Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat’s Little Theorem (FLT)-based inversion circuit for the binary finite field. In particular, we propose to follow a complete waterfall approach to translate the Itoh-Tsujii’s variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (i.e., CNOT count) as well as slightly reducing theTTTdepth, which contributes to a reduced overall depth and gate count. Furthermore, we concretely verify our method and compare it with the previous work in Qiskit, a quantum computer simulation environment, by constructing both our method and the previous work from scratch and performing the resource analysis. Additionally, we propose employing the relative-phase Toffoli gate by Gidney as opposed to the standard Toffoli implementation, which yields a significantly lowerTTTdepth while further reducing the overall depth. Our approach can serve as an alternative for a time-efficient implementation.
The Shor's algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor's algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh-Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).