I. Historical background

Shannon once proposed that on any channel, the requirements for trusted communication are as follows:

$R<C$

In the above formula, R represents the code rate, and C represents the channel capacity. The code rate calculation formula is as follows:

$R=\frac{K}{N}$

In the above formula, K represents the codeword length, and N represents the total length. Erdal Arikan was the first to discover what would make this possible: polar codes. Here is a typical picture of the polar code:

2. Channel transmission introduction

Using X to represent binary input (usually 0 or 1), Y to represent output, W(y|x) to represent transition probability, and W to represent discrete memoryless channels, the channel capacity can be calculated as follows:

$C=\underset{p(x)}{max}I(X,Y)=\underset{p(x)}{max}\sum_{x,y}p(x)W(y|x)log\frac{W(y|x)}{\sum_xp(x)W(y|x)}$

When transmitting information, there are usually two types. The transmission rules of the first binary symmetric channel are as follows:

From this, the transition probability matrix can be obtained as follows:

$\begin{bmatrix}1-\epsilon&\epsilon\\ \epsilon&1-\epsilon \end{}$

The second is the binary erasure channel, and the transfer diagram at this time is as follows:

The transition probability matrix obtained similarly is as follows:

$\begin{bmatrix}1-\epsilon&\epsilon&0\\0&\epsilon&1-\epsilon\end{}$

Specializing the channel as an output-output symmetric channel, the above channel capacity formula can be further simplified as follows:

$C\overset{\Delta}{=}\sum_{x,y}\frac{1}{2}W(y|x)log\frac{w(y|x)}{\frac{1}{2}w(y|0)+\frac{1}{2}w(y|1)}$

It is not difficult to conclude that the value range of the channel capacity is as follows:

$0\leq C(W)\leq 1$

Considering two extreme cases, the perfect channel is defined as follows:

$C(W)=1$

The unwanted channel is defined as follows:

$C(W)=0$

The purpose of the polar code is to polarize the ordinary channel W to these two extremes.

3. Analysis of polarization process

Multiplex one W channel into two, as follows:

$W_1$ The mutual information analysis of the channel is as follows :

Similarly, $W_2$ the mutual information analysis of the channel is as follows:

According to the chain rule of mutual information, as follows:

$I(U_1U_2;Y_1Y_2)=I(U_1;Y_1Y_2)+I(U_2;Y_1Y_2U_1)$

With the channel conservation law, as follows:

$C(W_1)+C(W_2)=2C(W)$

Finally, the result of the first polarization can be obtained as follows:

$C(W_1)\leq C(W)\leq C(W_2)$

Increase the two inputs to four inputs, as shown in the figure below:

After multiple polarizations, the result is as follows:

4. Example Analysis

Set the channel as a binary erasure channel with a probability of 0.5, that is, W=BEC(0.5), N=8, Rate=0.5, and analyze the output of the following $U_1\backsim U_8$ examples with the values you can set yourself.

5. Cryptographic system based on polar code

Here we will take McEliece as an example for analysis.

5.1 Key generation algorithm

The error correction capacity is set to be related to t, G represents the transition matrix, and P represents the correlation probability. From this, the private key can be obtained as follows:

$Private\ key:G,S,P$

Multiply the above three elements to get the public key, as follows:

$Public\ key: G'=SGP$

5.2 Encryption algorithm

The error information in the channel satisfies the following:

$W_t(e)\leq t$

With the help of the public key, the resulting ciphertext is as follows:

$c=mG'+e$

5.3 Decryption algorithm

The first step of decryption, using the ciphertext and probability P, is as follows:

$cP^{-1}=mSG+eP^{-1}$

The second step uses the fast decoding algorithm to calculate mS. The third step is to recover the plaintext m as follows:

$m=(mS)S^{-1}$

6. Security communication analysis

With the above polarization process, the channels of the legal receiver Bob and the eavesdropper Eve will appear as follows:

To achieve secure communication, it is necessary to use the part that is noiseless for Bob and full noise for Eve. Analysis in the form of blocks can be obtained as follows:

Polar codes can mainly realize secure communication at the physical layer, including weak security and strong security. Here are some important references:

The message is first coded by the source to form x, then c is formed by means of polar code, and then transmitted to the receiving end through the channel to form y. The flow chart of secure communication formation is as follows:

Encryption based on Polar Code