Abstract:Quantum modular multiplication circuit is one of the basic quantum computation circuits which are basic functions in quantum algorithms. However, since quantum-quantum modular multipliers require a high cost reversible modular inversion routine for modular multiplication, researchers have been unable to propose a feasible quantum-quantum modular multiplier. In this paper, we proposed efficient quantum-classical modular multipliers and the first quantum-quantum modular multipliers that do not require a reductio… Show more
“…In 2020, Cho et al proposed efficient quantum multipliers over and , as shown in Fig. 7 24 . They reduced the number of quantum gates and depth by half.…”
Section: Preliminariesmentioning
confidence: 99%
“…A quantum-classical version of the multiplier of Jang et al 25 has depth one of the gate. Table 2 shows the quantum resources for two quantum multipliers 24 , 25 . Figure 7 The quantum multiplier over ( gate) proposed by Cho et al 24 .…”
Recent rank-based attacks have reduced the security of Rainbow, which is one of the multi-layer UOV signatures, below the NIST security requirements by speeding up iterative kernel-finding operations using classical mathematics techniques. If quantum algorithms are applied to perform these iterative operations, the rank-based attacks may be more threatening to multi-layer UOV, including Rainbow. In this paper, we propose a quantum rectangular MinRank attack called the Q-rMinRank attack, the first quantum approach to key recovery attacks on multi-layer UOV signatures. Our attack is a general model applicable to multi-layer UOV signature schemes, and in this paper, we provide examples of its application to Rainbow and the Korean TTA standard, HiMQ. We design two quantum oracle circuits to find the kernel in consideration of the depth-width trade-off of quantum circuits. One is to reduce the width of the quantum circuits using qubits as a minimum, and the other is to reduce the depth using parallelization instead of using a lot of qubits. By designing quantum circuits to find kernels with fewer quantum resources and complexity by adding mathematical techniques, we achieve quadratic speedup for the MinRank attack to recover the private keys of multi-layer UOV signatures. We also estimate quantum resources for the designed quantum circuits and analyze quantum complexity based on them. The width-optimized circuit recovers the private keys of Rainbow parameter set V with only 1089 logical qubits. The depth-optimized circuit recovers the private keys of Rainbow parameter set V with a quantum complexity of $$2^{174}$$
2
174
, which is lower than the complexity of $$2^{221}$$
2
221
recovering the secret key of AES-192, which provides the same security level as parameter set III.
“…In 2020, Cho et al proposed efficient quantum multipliers over and , as shown in Fig. 7 24 . They reduced the number of quantum gates and depth by half.…”
Section: Preliminariesmentioning
confidence: 99%
“…A quantum-classical version of the multiplier of Jang et al 25 has depth one of the gate. Table 2 shows the quantum resources for two quantum multipliers 24 , 25 . Figure 7 The quantum multiplier over ( gate) proposed by Cho et al 24 .…”
Recent rank-based attacks have reduced the security of Rainbow, which is one of the multi-layer UOV signatures, below the NIST security requirements by speeding up iterative kernel-finding operations using classical mathematics techniques. If quantum algorithms are applied to perform these iterative operations, the rank-based attacks may be more threatening to multi-layer UOV, including Rainbow. In this paper, we propose a quantum rectangular MinRank attack called the Q-rMinRank attack, the first quantum approach to key recovery attacks on multi-layer UOV signatures. Our attack is a general model applicable to multi-layer UOV signature schemes, and in this paper, we provide examples of its application to Rainbow and the Korean TTA standard, HiMQ. We design two quantum oracle circuits to find the kernel in consideration of the depth-width trade-off of quantum circuits. One is to reduce the width of the quantum circuits using qubits as a minimum, and the other is to reduce the depth using parallelization instead of using a lot of qubits. By designing quantum circuits to find kernels with fewer quantum resources and complexity by adding mathematical techniques, we achieve quadratic speedup for the MinRank attack to recover the private keys of multi-layer UOV signatures. We also estimate quantum resources for the designed quantum circuits and analyze quantum complexity based on them. The width-optimized circuit recovers the private keys of Rainbow parameter set V with only 1089 logical qubits. The depth-optimized circuit recovers the private keys of Rainbow parameter set V with a quantum complexity of $$2^{174}$$
2
174
, which is lower than the complexity of $$2^{221}$$
2
221
recovering the secret key of AES-192, which provides the same security level as parameter set III.
“…In the quantum setting, the problem of reversible quantum modular multiplier has been well-studied [50,51]. It has been proved that the quantum operator…”
Section: Phase Search Through Interval Amplificationmentioning
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop a new algorithmic framework "Quantum phase processing" that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled-unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigen-information of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase-related. As a notable application, we propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the least ancilla qubits and matches the best performance so far. We further exploit the power of our QPP framework by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.
“…This optimized quantum modular adder will be very useful for quantum operations that require a full adder over GF(2 n − 1). For example, Cho et al proposed an efficient classical quantum and quantum-quantum modular multiplication circuit over GF(2 n ) and GF(2 n − 1) [34]. Their multiplication circuit can be applied to any full adder, and they used QCLA focused on speed in their simulation.…”
Section: Quantum Modular Adder Over Gf(2 N − 1)mentioning
Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field GF(2n−1), which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2n−1 adders. Our proposed adder can produce modulo sum within the range 0,2n−2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over GF(2n−1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM’s simulator ProjectQ.
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