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Quantum Cryptography Through Some Algorithm for Quantum Computers

Koji Nagata, Do Ngoc Diep, Tadao Nakamura

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

We propose quantum cryptography based on an algorithm of determining a function. The security of our cryptography is based on the Ekert 1991 protocol, that is, we use an entangled state. Eve must destroy the entangled state. Consider a function. Alice knows all the mappings concerning the function. Bob knows none of them. His aim is of obtaining all of them without Eve's attack. In classical case, Bob needs some queries. In quantum case, Bob needs just a query. By measuring the single entangled state, which is sent by Alice, Bob can obtain all the mappings concerning the function, simultaneously. This is faster than classical cryptography.

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We propose quantum cryptography based on an algorithm of determining a function. The security of our cryptography is based on the Ekert 1991 protocol, that is, we use an entangled state. Eve must destroy the entangled state. Consider a function. Alice knows all the mappings concerning the function. Bob knows none of them. His aim is of obtaining all of them without Eve's attack. In classical case, Bob needs some queries. In quantum case, Bob needs just a query. By measuring the single entangled state, which is sent by Alice, Bob can obtain all the mappings concerning the function, simultaneously. This is faster than classical cryptography.
In this paper, we propose a method of enciphering quantum states of two-state systems (qubits) for sending them in secrecy without entangled qubits shared by two legitimate users (Alice and Bob). This method has the following two properties. First, even if an eavesdropper (Eve) steals qubits, she can extract information from them with only a certain probability at most. Second, Alice and Bob can confirm that the qubits are transmitted between them correctly by measuring a signature. If Eve measures m qubits one by one from n enciphered qubits and sends alternative ones (the intercept/resend attack), the probability that Alice and Bob do not notice Eve's action is equal to (3/4) m or less. Passwords for decryption and the signature are given by classical binary strings and they are disclosed through a public channel. Enciphering classical information by this method is equivalent to the one-time pad method with distributing a classical key (random binary string) by the BB84 protocol. If Eve takes away qubits, Alice and Bob lose the original quantum information. If we apply our method to a state in iteration, Eve's success probability decreases exponentially. We cannot examine security against the case that Eve makes an attack using entanglement. This remains to be solved in the future.
We define a keyless communication as a message sent from one party (Alice) to another (Bob), such that the message is totally secure. This means that if a third party (Eve) is trying to eavesdrop, she will receive only random bits. She cannot understand the content of the message, even if she is able to intercept it in such away that Alice and Bob do not know it has been intercepted. For such a totally secure message, Alice and Bob do not need to set up any key beforehand, as it can add no further security. Starting with a system that we felt sure met such criteria, we were shocked to find out that it could be broken in at least two ways, by a semi-classical method and by a quantum method, such that Eve could read a message and Alice and Bob would not know it. Alice and Bob can counter such eavesdropping, but only by sending out test messages. This will enable them to ascertain that Eve is on the line, but then she can still occasionally pick up snippets of their conversation. In that case, in order to be totally secure, it is better to establish an unbreakable key beforehand. We postulate and give plausibility arguments that in general it is impossible to establish keyless communication between two parties.
Present cryptography algorithms are based on mathematical problem such as factoring two large prime numbers and N vs NP problem. But computing power and evolution in mathematics increases day by day so it’s easy to reverse one way function to quickly such as that of factoring large integers. So the solution is to build a cryptosystem which secure our communication even the computing power is to high so we introduce quantum physics in cryptography which lead to evaluation of quantum cryptography. Quantum cryptography one of the emerging topics in field of computer industry. This paper is an attempt to review fundamentals of quantum cryptography and also analyses few application areas of quantum cryptography and it’s limitation.
Quantum cryptography uses physics to develop a cryptosystem completely secure against being compromised without knowledge of the sender or the receiver of the messages. Regular, non-quantum encryption can work in a variety of ways, but, generally, a message is scrambled and can only be unscrambled using a secret key. Cracking the private key in a modern cryptosystem would generally require figuring out the factors of a number that is the product of two insanely huge prime numbers. The numbers are chosen to be so large that, with the given processing power of computers. But, in practice, quantum cryptography comes with its own load of weaknesses. It was recognized in 2010, for instance, that a hacker could blind a detector with a strong pulse, rendering it unable to see the secret-keeping photons.
Cryptography is processing for securing communication between two people. The opponent wants to know the message that is encrypted using a secret key. Although the opponent can eavesdrop the message sent between the sender and the receiver, the opponent is unable to decrypt to read the message. Therefore, the secret key is very important. The sender and the receiver agree with the secret key in an insecure channel by using key distribution protocol such as the Diffie–Hellman protocol. Since quantum computer is coming soon, Diffie–Hellman protocol is not secure. We will develop a quantum key distribution protocol. The benefit of the quantum system is the quantum state that cannot copy by no‐cloning theorem. Thus, the opponent does not copy and keeps the message that is quantum. In this paper, a novel quantum key distribution protocol between two people (Alice and Bob) based on quantum operators is developed. The opponent (Eve) wants to know the secret key. Although Eve knows this quantum key distribution protocol, Eve does not behave similarly to Alice and Bob. For example, Eve eavesdrops Alice's quantum state that was sent to Bob, and Eve sends another quantum state. Therefore, we cannot control Eve's behavior. So we give the upper bound of mutual information between the user and opponent by using Holevo's bound. We verify the usual security definition for quantum key distribution that is equality‐and‐uniformity and privacy in the mutual information sense.
So far, the knowledge of Eve has been modeled by a classical random variable. The aim of this chapter is first to discuss how the previous results apply to QKD, where Eve's action may not necessarily be classically described by the random variable Z . Then, we explain the equivalence between BB84 and so-called entanglement purification protocols as a tool to encompass general eavesdropping strategies. Finally, we apply this equivalence to the case of the coherent-state protocol GG02. Eavesdropping strategies and secret-key distillation In Chapters 10 and 11, I analyzed the security of the protocols with regard to individual eavesdropping strategies only. In this particular case, the result of Alice's, Bob's and Eve's measurements can be described classically and the results of Section 6.4 apply. Let us review other kinds of eavesdropping strategies and discuss how the concepts of Chapter 6 can be applied. Note that the individual eavesdropping strategy, the simplest class of strategies, is still technologically very challenging to achieve today in an optimal way. Nevertheless, we do not want the security of quantum cryptography to rely only on technological barriers. By considering more general eavesdropping strategies, we make sure quantum cryptography lies on strong grounds. Following Gisin et al. [64], we divide the possible eavesdropping strategies into three categories: individual attacks, collective attacks and joint attacks .