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docs/lectures/cryptography/09_public_key_maths.md

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+# Public Key Mathematics
+
+- Recap: in rings and fields, multiplicative inverse might exist such that
+
+$$
+a\cdot a^{-1} \equiv 1 \space (mod \space p)
+$$
+
+- Modular inverse exists when $gcd(a,p)=1$
+- For prime fields, $gcd(a,p)=1, \forall a \neq 0 \in GF(p)$
+
+#### Euclidean Algorithm
+
+- The euclidean algorithm calculates the greatest common divisor of two numbers $gcd(r_0, r_1)$
+  - This is the largest number that divides both $r_0$ and $r_1$
+- If $gcd(x,y)=1$ then $x$ and $y$ are **coprime** (sometimes called relatively prime)
+- The Euclidean algorithm is based around the fact:
+  - $gcd(r_0, r_1) = gcd(r_1, r_0 - r_1)$
+
+![1646755531.png](img/1646755531.png)
+
+- Computing $(x-y)\cdot gcd(r_0, r_1)$ is easier as its a smaller number
+- Doing this repeatedly is slow, we can use $gcd(r_0,r_1) = gcd(r_1, r_0\space mod \space r_1)$
+
+![1646755650.png](img/1646755650.png)
+
+##### Example
+
+$r_0 = 57 \\
+r_1 = 12$
+
+- At each step we convert $r_0$ and $r_1$ into the form $r_0=q\cdot r_1 + r_2$
+
+$r_0=q\cdot r_1 + r_2 \\57=4\cdot 12 + 9\\ r_1=q\cdot r_2 + r_3 \\ 12=1\cdot 9 + 3 \\ 9 = 3\cdot 3 + 0$
+
+- When the algorithm gets to 0, it is finished, therefore $gcd(57,12)=3$
+
+![1646756075.png](img/1646756075.png)
+
+#### Bezout’s Identity
+
+- Bezout’s identity tells us that the greatest common divisor of two numbers can be expressed as the sum of multiples of these numbers
+- $gcd(r_0,r_1) = s\cdot r_0 + t\cdot r_1$
+  - e.g. $gcd(99,20)=-1\cdot 99+5\cdot 20=1$
+  - $gcd(141,50)=11\cdot 141+-31\cdot 50=1$
+
+##### Extended Euclidean Algorithm
+
+- The extended euclidean algorithm calculates the $gcd(r_0,r_1)$ as normal, and in addition calculates $s$ and $t$.
+
+| Euclidean Algorithm                | Extended Euclidean Algorithm                                 |
+| ---------------------------------- | ------------------------------------------------------------ |
+| $r_0=q_1\cdot r_1+r_2$             | $r_2=r_0-q_1\cdot r_1 \quad \rightarrow \quad r_2=s_2\cdot r_0-t_2\cdot r_1$ |
+| $r_1=q_2\cdot r_2+r_3$             | $r_3=r_1-q_2\cdot r_2  \quad \rightarrow \quad r_3=s_3\cdot r_0-t_3\cdot r_1$ |
+| $r_2=q_3\cdot r_3+r_4$             | $r_4=r_2-q_3\cdot r_3   \quad \rightarrow \quad r_4=s_4\cdot r_0-t_4\cdot r_1$ |
+| …                                  | …                                                            |
+| $r_{l-2}=q_{l-1}\cdot r_{l-1}+r_l$ | $r_l=r_{l-2}-q_{l-1}\cdot r_{l-1}   \quad \rightarrow \quad r_l=s_l\cdot r_0-t_l\cdot r_1$ |
+| $r_{l-1}=q_{l}\cdot r_{l}+0$       |                                                              |
+
+###### Example
+
+![1646757043.png](img/1646757043.png)
+
+###### Formula
+
+![1646757908.png](img/1646757908.png)
+
+![1646758001.png](img/1646758001.png)
+
+#### Modular Inverses
+
+$$
+\begin{split}
+& a\cdot a^{-1} \equiv 1 \mod n \\
+& gcd(n,a) = s\cdot n + t\cdot a = 1 \\
+& s\cdot n + t\cdot a = 1 \\
+& s\cdot 0 + t\cdot a \equiv 1 \space mod \space n \\
+& t\cdot a \equiv 1 \space mod \space n \\
+& t \equiv a^{-1} \space mod \space n
+\end{split}
+$$
+
+Where $t$ is our multiplicative inverse
+