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Elements of Quantum Computation and Quantum Communication by Anirban Pathak

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Chapter 8
Quantum cryptography
The word cryptography originates from two Greek words: kryptos (secret)
and graphein (writing). Thus cryptography is the art of communicating
a message in secret manner or in other words, it is the art of rendering
a message unintelligible to any unauthorized party. Its history is very
old and exciting. From the beginning of human civilization there was a
need to hide information from unauthorized people. This need gave birth
to cryptography. Initial cryptographic protocols were very simple. With
time they have become much more sophisticated and secure. However,
all the classical cryptographic protocols developed so far are secure under
some assumptions. In contrast, the protocols of quantum cryptography are
unconditionally secure. This particular aspect of quantum cryptography
makes it very interesting. This chapter aims to provide a flavor of the in-
trinsic beauty of various aspects of secure quantum communication. Before
we start a journey into the fascinating world of secure quantum communi-
cation, it would be apt to become familiar with the jargon used in this field
and also with some interesting protocols of classical cryptography. Let us
first do that.
8.1 Jargon related to cryptography
To develop an elementary perception about cryptography, we may consider
a practical situation in which the king of Atan wants to invade his neighbor-
ing state Acos. Now the king of Atan sends some information to his generals
present in the battlefield through an emissary. The king must not write his
instructions in normal language as he wants it to be understood by his gen-
erals only. So the king uses some kind of coding to hide his message. For ex-
ample, we may think of the following simple ideas: The king may shift each
letter of the English alphabet by 3 places, A → D, B → E and so on, or the
king may replace the letters randomly, say A → W, B → N, ···,Z→ Q.
265
266 Quantum cryptography
If the generals know the mapping (transformation) used by the king then
they can easily decode the message. This kind of encoding is known as
substitution cipher or Caesarian cipher as this technique was introduced
by Julius Caesar. Substitution cipher is one of the simplest techniques of
encoding. Today we use much more sophisticated techniques to encode our
message. In general, the art of encoding the message is known as cryptog-
raphy. If we assume that the emissary of Atan is caught and the king of
Acos has obtained the message, then the king of Acos has to try to under-
stand what is written in that secret message. To do so, he has to break
the code used by the king of Atan. This is important for the security of
his country. This practical need has developed the art of code breaking,
which is known as cryptanalysis. Every success in cryptanalysis (cryptog-
raphy) has motivated the cryptographers (crypto-analyzers) to improve the
encryption (code-breaking) techniques. Consequently, both cryptography
and cryptanalysis techniques have evolved with time and together they
form a subject called cryptology, which is the combination of cryptography
and cryptanalysis. We may say, cryptology=cryptography+cryptanalysis.
A cryptosystem or a cipher is an algorithm which combines the message
to be encrypted with some additional information known as the key and
produces a cryptogram. The most important thing in cryptology is the
key. If the key is secure then the cryptogram is secure. This fundamental
principle of cryptography is known as Kerckhoff’s principle. This principle
implies that the security of a cipher depends only on the secrecy of the
key and not on the secrecy of the protocol used for encryption. In other
words, a cryptosystem can be secure even if everything about the system,
except the key, is public knowledge. Now we may elaborate on the concept
of key through a very simple example. Consider that Alice has a message
string 1101001 and a key 0101101. To encrypt the message Alice does
bit-wise XOR operation between her key and the message and gets the
encrypted message as 1000100. To obtain the original message Bob uses
the same key and performs bit-wise XOR operation between his key and
the encrypted message. This decrypts the message and thus Bob obtains
the secret message encoded by Alice. The example is also shown in the box
below.
Alice’s message 1101001
Alice’s key 0101101
Encrypted message after bit-wise operation 1000100
Bob’s key 0101101
Decrypted message after bit-wise operation 1101001
In the above example, both Alice and Bob use the same key. If in a cryp-
tographic protocol the same key is used for both encryption and decryption
then such a protocol is called symmetric key cryptographic protocol. Sim-
ilarly, if in a cryptographic protocol different keys are used for encoding

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