Cryptography Series Ten: Quantum Cryptography

Quantum cryptography is an emerging science combining quantum physics and cryptography. In 1994, Shor proposed a probabilistic polynomial time algorithm for integer decomposition and discrete logarithm solution on a quantum computer; in 2003, Shor's quantum algorithm was extended to elliptic curves. These advances have theoretically questioned the security of traditional public-key cryptosystems, and cryptographers have begun to study cryptosystems in the era of quantum computers.

1. The physical basis of quantum cryptography

The security of quantum cryptography is based on the Uncertainty Principle of Heisenberg in quantum mechanics, so breaking the quantum cryptography protocol means negating the laws of quantum mechanics, so quantum cryptography is a theoretically safe cryptographic technology .

(1) Quantum non-cloning theorem

Wootters and Zurek wrote in "Nature" magazine in 1982 and raised the following question: Is there a physical process to realize the exact replication of an unknown quantum state, so that each replica state is exactly the same as the initial quantum state? Wootters and Zurck It is proved that the linear property of quantum mechanics prohibits such duplication, which is the original expression of the quantum non-duplication theorem.

(2) Heisenberg uncertainty principle

The uncertainty principle is a fundamental principle of quantum mechanics. Particles in the microscopic world have many conjugate quantities, such as position and velocity, time and energy are two pairs of conjugate quantities, and when any one of them is measured, the other physical quantity will inevitably be disturbed .

Quantum cryptography uses quantum uncertainty to construct a secure communication channel so that both parties in communication can detect whether the information has been eavesdropped. This property provides absolute security for both parties in key negotiation or key exchange. Quantum cryptography does not rely on the computational difficulty of the problem, but provides provable unconditional security using the fundamental laws of physics. And unlike a one-time pad, it is impossible for anyone eavesdropping on quantum key exchanges and replicating keys to go undetected.

According to the uncertainty principle, the essence of quantum cryptography can be understood as the following problem: Given 4 possible polarization states ( $\leftrightarrow ,\updownarrow$ , ⤢, ⤡) to describe a single photon, can one determine its polarization with certainty? The answer is no. straight line basis ($\leftrightarrow$ and$\updownarrow$ ) is incompatible with the diagonal basis (⤢ and ⤡), so the uncertainty principle prohibits measuring both simultaneously, and more generally, even if only partially reliable, the process of discriminating non-orthogonal states will disturb their states .

2. Quantum key distribution

Quantum key distribution system is the most extensive and in-depth direction of quantum cryptography research. Quantum cryptography communication is not used to transmit ciphertext or plaintext, but to establish and transmit a key, which is absolutely safe. Quantum cryptography communication is currently the only communication method recognized by science that can achieve absolute security. It can ensure that both parties in legal communication can detect potential eavesdroppers and take corresponding measures to prevent eavesdroppers from cracking quantum cryptography, no matter how powerful the cracker is. .

2.1 Basic Principles of Quantum Key Distribution

Quantum cryptography uses the polarization of protons to program codes . Protons can be polarized in four ways, horizontal and vertical, and two diagonal lines.

The most essential difference between quantum key distribution and classical key distribution is that the former uses quantum states to represent random numbers 0 and 1 , while the existing key distribution uses physical quantities to represent bits 0 and 1.

If optical pulses are used to transmit bits, in classical information, a photon in the optical pulse represents 1, and no photon represents 0; in quantum information, the quantum state of a single photon, such as the polarization state, is used to represent the bit information, such as circularly polarized light represents 1. Linearly polarized light represents 0, that is, each light pulse can only have at most one photon. The different quantum states of this photon indicate that it carries different bits of information. Since a single photon is indivisible as a whole, eavesdroppers cannot pass the wave splitting method. to get information.

There is an important theorem in quantum information, that is, the quantum non-cloning theorem , which points out that there is no real physical system that can accurately copy (clone) the unknown quantum state, so eavesdropping cannot obtain information by copying the quantum state of photons , which is just It is the basic principle of quantum mechanics that ensures the security of quantum key distribution, and any eavesdropping process will inevitably leave traces and be discovered by legitimate users . Only when Alice and Bob are sure that the key they share has not been eavesdropped can it be used for secure communication.

2.2 Quantum key distribution system

The quantum key distribution system is a cryptographic system that enables the sender to use quantum channels to share secret information with the receiver, and unauthorized third parties cannot steal information. The system consists of quantum information sources, quantum channels, classical channels, senders, receivers Square and other parts.

The quantum information source is the source of the quantum channel, which mainly provides the required quantum state in the channel. After a quantum code group is selected in the quantum information source, the probability of occurrence of each quantum state in the quantum code group can be in various situations, but the selected quantum states in the quantum cryptography system all appear with equal probability. It should be pointed out that the quantum information source is completely different from the classical information source, because in the quantum information source, when the quantum code group is determined, the quantum channel is also determined.

Quantum channels are used to transmit secret information. The so-called quantum channel is a quantum information channel that transmits qubits from one end to the other. Quantum channel can have many forms, such as free space, optical fiber and so on. The channel in classical information theory only transmits information from the sender to the receiver. However, in quantum information theory, the way information is transmitted in the channel is very important, and the transmission of information is limited by the information carrier.

Quantum key distribution does not rely on the computational difficulty of the problem, but uses fundamental physical laws to provide provably unconditional security for cryptosystems. Quantum key distribution based on the uncertainty principle uses quantum uncertainty to construct a secure communication channel, and enables both parties to detect whether the information has been eavesdropped, providing absolute security for both parties in key agreement or key exchange. .

So far, there are three main categories of quantum cryptography implementations:

• Based on the Heisenberg Uncertainty Principle in Single Photon Quantum Channel
• Based on the Bell principle in the quantum correlation channel
• Based on the properties of two non-orthogonal quantum states

2.3 BB84 protocol

The BB84 protocol is the first quantum cryptography communication protocol. It was proposed by Benntt and Brassard in 1984. The protocol uses four quantum states (such as right-handed, left-handed, horizontal and vertical polarization states) to realize quantum key distribution . It is agreed in advance: left-handed and the horizontal polarization state represent a bit "0", and the right-handed and vertical polarization states represent a bit "1".

The operation steps of quantum key distribution are as follows.

• (1) User A sends multiple photons to user B, and each photon randomly selects any one of the four polarization states of right-handed, left-handed, horizontal or vertical.
• (2) User B randomly chooses a linear polarization base or a circular polarization base to measure the polarization state of photons, and records his measurement results.
• (3) User B tells user A on the public channel which measurement base he chooses each time, but does not publish the measurement results.
• (4) After user A knows the measurement base of user B, he can determine which of user B's measurement bases is correct and which is wrong; he tells user B to leave the measurement of the selected base through an open channel As a result, user A and user B can be $50$ success rate to create exactly the same sequence of random numbers.
• (5) User B samples some bits from the established random bit sequence, generally $1/3$ , and sent to user A.
• (6) User A checks whether the bits sent by user B are consistent with the bits sent by himself. If there is no eavesdropping, they should be consistent; otherwise, eavesdropping must have occurred.
• (7) If no eavesdropping occurs, the two parties can agree to use the remaining $2/3$ bits are used as a shared session key, thereby realizing key distribution. User A and user B can obtain enough bits in this way.