# ElGamal 公開鍵暗号システム

ガロア体配列の関数 `gf` を使用して、ElGamal 公開鍵暗号システムを実装します。

### キーの生成

```m = 15; q = 2^m;```

```poly = primpoly(m,'nodisplay'); primeFactors = unique(factor(2^m-1)); rng(123456); while 1 g = gf(randi([1,q-1]),m,poly); isPrimitive = true; for i = 1:length(primeFactors) if g^((q-1)/primeFactors(i)) == gf(1,m,poly) isPrimitive = false; break; end end if isPrimitive break; end end```

```privateKey = 12; publicKey = {g,g^privateKey,poly};```

### 暗号化

```text = ['The Fourier transform decomposes a function of time (a signal)' newline ... 'into the frequencies that make it up, in a way similar to how a' newline ... 'musical chord can be expressed as the amplitude (or loudness) of' newline ... 'its constituent notes.']; disp(text);```
```The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. ```

メッセージをバイナリに変換して、m ビットごとにグループ化します。メッセージは ASCII 文字を使用します。ASCII テーブルは 128 文字であるため、1 文字あたり 7 ビットで十分です。

```bitsPerChar = 7; binMsg = int2bit(int8(text'),bitsPerChar); numPaddedBits = m - mod(numel(binMsg),m); if numPaddedBits == m numPaddedBits = 0; end binMsg = [binMsg; zeros(numPaddedBits,1)]; textToEncrypt = bit2int(binMsg,m);```

```cipherText = gf(zeros(length(textToEncrypt),2),m,poly); for i = 1:length(textToEncrypt) k = randi([1 2^m-2]); cipherText(i,:) = [publicKey{1}^k, ... gf(textToEncrypt(i),m,poly)*publicKey{2}^k]; end```

```tmp = cipherText.x; %disp(de2char(tmp(:,2),bitsPerChar,m));```

### 復号化

```decipherText = gf(zeros(size(cipherText,1),1),m,poly); for i = 1:size(cipherText,1) decipherText(i) = cipherText(i,2) * cipherText(i,1)^(-privateKey); end```

`disp(de2char(decipherText.x,bitsPerChar,m));`
```The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. ```

### サポート対象の関数

`de2char` はビットを char メッセージに変換します。

```function text = de2char(msg,bitsPerChar,m) binDecipherText = int2bit(msg,m); text = char(bit2int(binDecipherText(1:end-mod(numel(binDecipherText), ... bitsPerChar)),bitsPerChar))'; end```