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