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2.5.1 Shannon information theory

One of the most important contributions of Claude Shannon to the area of information theory was contained in his paper “The Mathematical Theory of Communication” [3]

where he introduced the concept of entropy into the field of information theory. From the second law of thermodynamics entropy is a measure of the randomness in a system and its value always increases. When applied to information theory this meant that many sentences can be dramatically shortened without losing their meaning. He proved that in a noisy communication, a signal could always be sent without distortion. If the message is encoded in such a way that it is self-checking, signals can always be received with the same accuracy as if there was no interference on the communication channel. Shannon showed that if the entropy rate, the amount of information you wish to transmit, exceeds the channel capacity then there were unavoidable and un-correctable errors in the transmission. If the entropy rate is below the channel capacity then there is a way to encode the information so that it can be received without errors.

This is true even if the channel distorts the message during transmission. In Shannon’s paper he introduced two important theorems, the noisy coding theorem and the noiseless coding theorem. As a simple example of the noiseless theorem can be understood by considering the following message:

TXT MSSGS SHRTN NGLSH SNTNCS

It is still possible to correctly read the above message even though there are 11 vowels missing. The noiseless coding theorem tries to quantify how much redundancy can be introduced and still the message can be decoded without errors. In communications redundancy is used to combat errors. The noisy coding theorem quantifies how much redundancy is needed in a message in order to correct errors introduced by noise present in a communication channel [32].

In the area of cryptography Shannon showed that the one-time pad achieves perfect secrecy if Alice and Bob share a secret key that is as long as the message to be encrypted.

2.5.2 Error correction, reconciliation and privacy amplification

The basic quantum key distribution protocol is inadequate in practice because realistic detectors have noise which results in Alice’s and Bob’s data differing even without the presence of an eavesdropper. When the quantum transmission is complete Alice and Bob must exchange a public message which enables them to reconcile the differences between their data. One of the most common methods of error correction is the Cascade error correction code [33]. Cascade is easily implemented and has the added benefit of information leakage which is close to the theoretical limit shown in Figure 2.16. Other methods include a low density parity check (LDPC) and Winnow. Although Cascade requires a lot more two-way communications than Winnow, Cascade is more efficient at error rates up to 10% and is therefore more commonly used in QKD systems [15]. The first step of the Cascade error correction protocol is that Alice and Bob publically agree on a permutation of their bits. The shuffled strings are then portioned into blocks of size k in such a way that it is believed that it contains no more than one error. Next Alice and Bob publically compute and compare the parity bits of each block. If the block contains an error, a difference in parity is detected. With this method an odd number of errors are detected but even numbers remain undetected. If a block has an unequal public parity the block is searched using a bisective search to locate the error and corrected. In order to locate and correct blocks with an even number of errors in them Alice and Bob then repeat the randomising and partitioning step for several passes with increasing block sizes for each pass. To correct further errors Alice and Bob continue to compare the parities of randomly chosen subsets of bits. If a parity mismatch is detected a similar procedure described above is performed to locate and

correct the error. The last bit of each random subset is deleted to avoid information leakage [34].

Figure 2.16. The gain for privacy amplification against increasing error rate is shown for the Cascade error correction protocol and for the theoretical Shannon’s limit. The maximum error rate allowable for Cascade is about 10% while it is about 11% if the Shannon limit is obtained [34].

The efficiency of the error correction is given by the Shannon limit and it gives the minimum number of bits which must be revealed about the correct key to reconcile an error rate e [35]. The Shannon limit can be given in terms of the amount of Shannon information Is

( )

( )

( ) (

)

( )

Is e = +e e + -e -e Equation (2.12) The minimum number of bits needed to correct a key whose length is n is given by

( ( ) )

min 1 _s

n =n -I e Equation (2.13)

The last step in a quantum cryptography protocol is called privacy amplification which was first described by Bennett et al. in 1988 [36] and is an attempt to reduce Eve’s information about the key. For this step Alice randomly selects pairs of bits and computes their parity. Unlike in error correction she does not publically announce the parity value but only the bits which were used. Alice and Bob then replace the two bits

by their value of the parity. The key length is shortened but if Eve has only partial information on the two bits, her information on the value of the parity is even less [17].