Quantum cryptography in a Nutshell
By: Mohamed Yassine Ferjani
Tags: Quantum Computing, Quantum Cryptography, Quantum Physics, Cryptography, Crypto

Quantum cryptography in a Nutshell

Cryptography is the art of rendering information exchanged between two parties unintelligible to any unauthorized person. Although it is an old science, its scope of applications remained mainly restricted to military and diplomatic purposes until the development of electronic and optical telecommunications.

Classical Cryptography:

Introduction to classical cryptography:

Classical cryptography can be divided into two major branches; secret or symmetric key cryptography and public key cryptography, which is also known as asymmetric cryptography.

Secret key cryptography represents the most traditional form of cryp-tography in which two parties both encrypt and decrypt their messages using the same shared secret key. While some secret key schemes, such as one-time pads, are perfectly secure against an attacker with arbitrary computational power , they have the major practical disadvantage that before two parties can communicate securely they must somehow establish a secret key.

In order to establish a secret key over an insecure channel, key distribution schemes based on public key cryptography, such as Diffie-Hellman, are typically employed.

One-time-pad and key distribution problem:

In conventional cryptography, an unbreakable code does exist. It is called the one-time-pad and was invented by Gilbert Vernam in 1918.

In the one-time-pad method, a message (traditionally called the plain text) is first converted by Alice into a binary form (a string consisting of “0”s and “1”s) by a publicly known method. A key is a binary string of the same length as the message.

By combining each bit of the message with the respective bit of the key using XOR (i.e. addition modulo two), Alice converts the plain text into an encrypted form (called the cipher text) .Alice then transmits the cipher text to Bob via a broadcast channel. Anyone including an eavesdropper can get a copy of the cipher text. However, without the knowledge of the key, the cipher text is totally random and gives no information whatsoever about the plain text. For decryption, Bob, who shares the same key with Alice, can perform another XOR (i.e. addition modulo two) between each bit of the cipher text with the respective bit of the key to recover the plain text .

The one-time-pad method is unbreakable, but it has a serious drawback: it supposes that Alice and Bob initially share a random string of secret that is as long as the message. Therefore, the one-time-pad simply shifts the problem of secure communication to the problem of key distribution. This is the key distribution problem.

The one of possible solution to the key distribution problem is public key cryptography .Quantum mechanics can also provide a solution to the key distribution problem.

In quantum key distribution, an encryption key is generated randomly between Alice and Bob by using non orthogonal quantum states. In quantum mechanics there is a quantum no-cloning theorem, which states that it is fundamentally impossible for anyone including an eavesdropper to make an additional copy of an unknown quantum state. Therefore, any attempt by an eavesdropper to learn information about a key in a QKD process will lead to disturbance, which can be detected by Alice and Bob who can, for example, check the bit error rate of a random sample of the raw transmission data.

Quantum Cryptography:

Quantum cryptography solves the problem of key distribution by allowing the exchange of a cryptographic key between two remote parties with absolute security, guaranteed by the fundamental laws of physics. This key can then be used securely with conventional cryptographic algorithms. The more correct name for Quantum Cryptography is therefore Quantum key Distribution.

Let’ s now introduce the most known QKD protocol :

BB84 protocol:

BB84 is one of the most known protocols in QKD is the BB84 protocol introduced by Bennet and Brassard in 1984.

The procedure of the BB84 protocol is as follows :

Alice generates a random key , than she chooses randomly one of the two basis states (S-basis or H-basis ) , She encodes each bit in its quantum state according to the chosen basis . Bob randomly chooses the standard or the hadamard basis to measure each qubit in the chosen basis, Bob have 50% of chance to measure qubits in the correct basis. Bob tells Alice over the Classical Authenticated Channel (CAC) that he has received and measured all the qubits. Alice and Bob compare over the CAC their basis choices; they will discard all rounds in which they didn’t measure in the same basis. This step is commonly called basis reconciliation. Alice picks randomly 1/4 bits for testing, she tells Bob over the CAC the bits indices and their values. Both Alice and Bob will perform testing and compute the error rate in the sifted basis (where they measured correctly in the same basis). If the error rate exceed a given threshold, the protocol will be discarded, other-wise they will continue the next steps of the protocol. Alice and Bob perform information reconciliation through the CAC. they may use some error correction protocol like the cascade protocol. Finally, Alice and Bob perform privacy amplification to extract their private secret key.

What is privacy ampliﬁcation?

Privacy ampliﬁcation is a method for reducing Eve’s partial information about Alice and Bob’s key. This partial information that could have been gained by Eve on the quantum channel during the previous steps of the protocol. It uses Alice and Bob’s key to produce a new, shorter key, in such a way that Eve has only negligible information about the new key. This can be done using a universal hash function.

Authentication in the Classical Authenticated Channel:

Let us return to an important assumption that is always made when considering the BB’84 protocol: that the communication channel between Alice and Bob, that we’ve been calling the “CAC”, is what its name implies — an authenticated channel. This assumption is used to guarantee that, although the eavesdropper may intercept any communication, she cannot “impersonate” Alice or Bob by sending fake messages on the channel. You can easily imagine the catastrophic consequences that such an attack could have: for example Eve could lie to Bob about Alice’s choice of the test set T ,making it a bit bigger; then Bob would reveal his outcomes in the bigger set, and Eve could keep them as additional side information about Alice’s raw key. This assumption is usually considered “benign”, as indeed the access to an authenticated channel is a prerequisite for a large variety of cryptographic tasks. Thus in practice one imagines that any two parties Alice and Bob wanting to implement QKD have access to such a channel, that has been implemented by other means. It is still interesting to consider how reasonable the assumption is, and how an authentication channel can be constructed. There are two main methods to achieve authentication, and each has its drawbacks. The first is to use public-key cryptography, which requires computational assumptions on the power of the eavesdropper. The second is to use private-key cryptography, which requires Alice and Bob to share a secret key… precisely the task QKD is meant to solve in the first place!

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We can’t talk about quantum cryptography without mentioning the field of post-quantum cryptography: those are classical algorithms which are resilient to quantum attacks. NIST has initiated a process to solicit, evaluate, and standardize one or more quantum-resistant public-key cryptographic algorithms.

So, do we really need quantum cryptography to secure our data ?

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