加密密鑰等于脫密密鑰，或者由一個可以輕易的計算出另一個的密碼體制，稱為單密鑰密碼體制，亦或稱為對稱密碼體制或傳統密碼體制。其最具代表意義的當然屬于DES密碼體制了。

1、DES的設計背景

• 1973年5月 NBS(美國國家標準局)發布通告，征集一種加密算法
• 1974年8月 收到了IBM公司提交的算法
• 1976年11月 被推薦為聯邦標準
• 1977年1月 發布
• 服役了20年

2、DES加密算法

$$\begin{gathered} 迭代型分組算法 \\ 分組長度/密鑰長度：64bit \\ 有效密鑰長度：56bit(8bit奇偶校驗位) \\ 迭代圈數：16圈 \\ 圈密鑰長度：48bit(即16圈中每一圈所要用到的密鑰位數(bit)) \end{gathered}$$

形式化表達為：$DES(M)=I{P}^{?1}°T(16)°T(15)°...°T(1)°IP(m)（由后向前運算）$
IP是初始置換，IP^-1是初始逆置換，T表示16組迭代圈函數。

圖形化表達：

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.edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-aWkJIWXAooKcdspR .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-aWkJIWXAooKcdspR .marker{fill:#333}#mermaid-svg-aWkJIWXAooKcdspR .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-aWkJIWXAooKcdspR {color: rgba(0, 0, 0, 0.75);font: ;}64bit64bit64bit64bit明文m初始置換<IP>圈1圈2圈3....圈16初始逆置換密文c

$前15圈的算法結構相同，16圈中每一圈都有不同的密鑰值$

$$\begin{gathered} 由上圖可知，前15圈模2加（亦或）后，左右進行了互換，而在第16圈則沒有。 \\ 初始置換后，將64bit數據等分為2份32bit分別給予了L_0（Left）和R_0（Right） \end{gathered}$$

$2.1初始置換IP與逆初始置換I{P}^{?1}$

若某一位經過初始置換又經過了逆初始置換，則其位置并沒有發生改變。即，IP和IP^-1互逆

如IP轉換表中，第8位對應了數字2，即表示對應著第2位；
而在IP^-1轉換表中，第二位對應了數字8，即表示對應著第8位；
此時正好又將位置轉換了回去。$之前有提到過置換是一種古典密碼$ 置換密碼.
所以在設計置換表時可以參照此原則。

$2.2圈函數$

前15圈

右上角的$L_{i?1}$應該是$R_{i?1}$

• 本一圈的右部$R_{i?1}$ 直接作為下一圈的左部$L_i$ 進行后續的運算
• 本一圈的（左部$L_{i?1}$） 要與（經過 $f函數$ 的 $R_{i?1}$ 和本圈密鑰$k_i$ 所得到的結果，$f(R_{i?1},k_i)$）進行模二加（亦或）運算之后作為下一圈的右部$R_i$ 進行后續的運算

$$\begin{gathered} 表達式為：(左部，右部)=(L_i,R_i)=(R_i,L_{i?1}?f(R_{i?1},k_i)) \\ ?模2加：0+0=0；0+1=1+0=1；1+1=0 \\ f(R_{i?1},k_i):表示在密鑰k_i的影響下R_{i?1}的值。 \end{gathered}$$

第16圈

$(左部，右部)=(L_{16},R_{16})=(L_{15?1}?f(k_{16},R_{15?1}),R_{15})$
\;
同前15圈相比，其左右部，一定意義上講，沒有發生交換。

• $$\begin{gathered} 這里的k_i是由64bit密鑰產生的子密鑰，k_i是48bit。 \\ DES的核心在于f(R_{i?1},k_i)函數的功能。f函數是將32bit的輸入轉化為32bit的輸出。 \\ 其中涉及到了擴展和壓縮。擴展是為了能和k_i進行模2加，壓縮是為了最后要輸出32bit \end{gathered}$$

$2.3f函數的構造$

$$\begin{gathered} f函數的內部構造: \\ E盒擴展； \\ E盒擴展后的值與圈密鑰k進行模2加；（亦或運算） \\ S盒轉換；（分為8個部分） \\ P盒變換； \end{gathered}$$
$圖形化表示：$

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.edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-LEJCyLHFFCHvJps3 .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-LEJCyLHFFCHvJps3 .marker{fill:#333}#mermaid-svg-LEJCyLHFFCHvJps3 .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-LEJCyLHFFCHvJps3 {color: rgba(0, 0, 0, 0.75);font: ;}32bit48bit48bit6bit6bit6bit6bit6bit6bit6bit6bit4bit4bit4bit4bit4bit4bit4bit4bitR<32bit>E+圈密鑰kS1S2S3S4S5S6S7S8P最后得到的32bit數據

$E盒擴展：$

將32bit擴展到48bit，先將32bit的字符分為8組，每一組的前后，分別添加上一個字符和下一個字符：
$$將\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& 7& 8\\ 9& 10& 11& 12\\ 13& 14& 15& 16\\ & ......& & \\ 29& 30& 31& 32\end{array}?\begin{array}{cccccc}32& 1& 2& 3& 4& 5\\ 4& 5& 6& 7& 8& 9\\ 8& 9& 10& 11& 12& 13\\ 12& 13& 14& 15& 16& 17\\ ...& & ......& & & ...\\ 28& 29& 30& 31& 32& 1\end{array}$$
$$\begin{gathered} 如， \\ 在下標為1的字符前插入上一個字符。即，下標為32的字符； \\ 在下標為4的字符后插入下一個字符。即，下標為5的字符； \\ 在下標為32的字符后插入下一個字符。即，下標為1的字符； \\ 按照此規律擴展16bit。 \end{gathered}$$

$模2加$

就令經過E盒擴展后的48個2進制位逐一與密鑰的48bit進行亦或運算。

$S盒代替$

將48bit轉換為32bit

$S盒是唯一的非線性變換（非數學變換），其是DES中f函數的核心，S盒的好壞決定了該算法。$

$該S盒分為了8個子盒，對應著8張表(S_i)，每個表是4行16列，每個子盒的輸入為6bit，輸出為4bit$

先將輸入的48bit分為8組6bit的數據對應于8個子盒，
$$\begin{gathered} 假定輸入S_i盒的6個bit為b_1{b}_2{b}_3{b}_4{b}_5{b}_6（這里每一個b都是一個二進制位） \\ 另b_1{b}_6為十進制的n，令b_2{b}_3{b}_4{b}_5為十進制的m， \\ 在設計的表中找出對應的n行m列對應的數字，并轉換為2進制作為本s_i盒的輸出 \\ 下面有栗子： \end{gathered}$$

$$\begin{gathered} 栗子：如，向s_1盒中輸入的6bit數為：101101 \\ {b}_1{b}_6為11，即十進制的3 \\ {b}_2{b}_3{b}_4{b}_5為0110，即十進制的6 \\ 在s_1中，第3行6列對應著數字1 \\ 所以最后的輸出為0001 \\ 8組中每一個都會得到4bit的數據，最終會得到32bit的數據 \end{gathered}$$
$S盒作為DES的心臟，DES靠它實現非線性變換$

$P盒轉換:$

其本質是一種置換
\;
如原來的第1個bit被置換到第16個bit
該表格滿足分組擴散與混亂原則。

3、密鑰生成算法

加密的基本流程已經介紹完畢，那么如何將64bit的初始密鑰生成16組48bit的圈密鑰（子密鑰）供每一圈使用呢？

$圖形化表達為：$

$PC?1為，置換選擇算法1（由64位篩選出56位）；$
$C和D是兩個寄存器；$
$LS為移位運算（循環左移）$
$PC?2為，置換選擇算法2（由58位選出48位的圈密鑰）$

$置換選擇1（PC?1）$

拋去每個字節的最高位，即，第8，16，24，32，40，48，56，64位，只選擇剩余的56位，被拋去的字節本來就是奇偶校驗位。（用于驗證完整性）

將剩余的比特數，按照一定的規則，進行相應的置換。

置換后，分別放在兩個寄存器C和D中

置換規則：

如，將原來的第57bit換到第1位。置換后的前28bit放在C寄存器中，其余的放在D寄存器中。

$移位（LS）【兩個28位密鑰的寄存器循環左移】$

$$\begin{gathered} 規則： \\ 第1，2，9，16圈左移1位； \\ 其余都左移2位。 \end{gathered}$$

$L{S}_1、L{S}_2、L{S}_9、L{S}_{16}左移1位，其余L{S}_i左移2位$

$最后16圈左移結束后，移位數正好等于28$

$置換選擇（PC?2）【由56位得到48位】$

按照此規則進行置換，同時去除了8位

4、脫密算法

DES的解密過程和DES加密完全類似，只不過將16圈的子密鑰序列顛倒了一下，即，第一圈用$k_{16}$ 最后一圈用$k_1$。
$$\begin{gathered} 形式化表達為： \\ DE{S}^{?1}=I{P}^{?1}°T(1)°T(2)°...°T(16)°IP(c)（由后向前運算） \end{gathered}$$

所以脫密的算法就在于圈密鑰如何進行計算的

$圈函數的分解：$

一般地，圈函數可以表達為：$(左，右)=(x,y)=(y,x?f(y,k))$，第16圈顯然不能用這個表示。
可將其看作兩部分，

$(x,y)\to (x?f(y,k)，y)$，稱為對合交換1。（Fesihel函數）

$(x?f(y,k)，y)\to (y,x?f(y,k))$，稱為對合變換2。（左右塊交換）【顯然，第16圈并沒有進行這一步驟】

$步驟一樣，為了方便表示，我們將圈迭代看成3圈，最后一圈同第16圈一樣,僅有對合變換1$

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$可知，將C解密為M的過程為M加密為C的逆過程$

$將此3圈擴展為16圈即為DES的脫密過程$

總結

$最后附帶一張手繪總結圖$

總結

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