145 lines
3.3 KiB
Markdown
145 lines
3.3 KiB
Markdown
# RSA
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* `p * q = n`
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* Coprime Phi is calculated either by [Euler Totient](https://en.wikipedia.org/wiki/Euler's_totient_function) or [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) via [euclidean algorithm](https://crypto.stanford.edu/pbc/notes/numbertheory/euclid.html)
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$$
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1 < \phi < n
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$$
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* There is also
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$$
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\phi = (p-1) * (q-1)
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$$$
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* Encryption, public key `e` is a prime between 2 and phi
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$$
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2 < e < \phi
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$$
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```python
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possible_e = []
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for i in range (2, phi):
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if gcd(n, i) == 1 and gcd(phi, i) == 1:
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possible_e.append()
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```
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* Decryption, private key `d`
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$$
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d * e mod \phi = 1
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$$
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```python
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possible_d = []
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for i in range (phi + 1, phi + foo):
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if i * e mod phi == 1 :
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possible_d.append()
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```
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* \\( Cipher = msg ** d mod $\phi$ \\)
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* \\( Cleartext = cipher ** e mod $\phi$ )
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## Euklid
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Just a short excourse:
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A greatest common divisior out of an example a = 32 and b = 14 would be the groups of the following divisors
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```sh
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a = 32, b = 24
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a = {1, 2, 4, 8, 16}
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b = {1, 2, 3, 8, 12}
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gcd(a,b) = 8
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```
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### Greatest Common Divisor (GCD)
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Two values are prime and have themselves and only `1` as a divisor are called coprime.
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To check if a and b have a greatest common divisor do the euclidean algorithm.
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```python
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def gcd(a, b):
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if b == 0:
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return a
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return gcd(b, a % b)
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```
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### Extended GCD
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#TODO
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## Fermat's Little Theorem
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If modulus $p$ is a prime and and modulus $n$ is not a prime, p defines a finite field (ring).
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$$
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n \in F_{p} \{0,1,...,p-1\}
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$$
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The field consists of elements $n$ which have an inverse $m$ resulting in $n + m = 0$ and $n * m = 1$.
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So , $n^p - n$ is a multiple of p then $n^p \equiv n\ mod\ p$ and therefore $ n = n^p\ mod\ p$. An example
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$$
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4 = 4^{31}\ mod\ 31
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$$
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Further, $p$ while still a prime results in $1 = n^{p-1} mod\ p$. An example
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$$
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1 = 5^{11-1}\ mod\ 11
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$$
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### Modular Inverse
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Coming back to the modular inverse $n$, it can be found in the following way
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$n^{p-1} \equiv 1\ mod\ p$
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$n^{p-1} * n^{-1} \equiv n^{-1}\ mod\ p$
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$n^{p-2} * n * n^-1 \equiv n^{-1}\ mod\ p$
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$n^{p-2} * 1 \equiv n^{-1}\ mod\ p$
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$n^{p-2} \equiv n^{-1}\ mod\ p$
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## Quadratic Residue
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$m$ is a quadratic residue when $\pm n^2 = m\ mod\ p$ with two solutions.
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Otherwise it is a quadratic non residue.
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So a porperty of quad res are, if Quadratic Residue $QR = 1$ and Quadratic NonResidue $QN = -1$
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$$
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QR * QR = QR\\
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QR * QN = QN\\
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QN * QN = QR\\
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$$
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## Legendre
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$$
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\frac{a}{p} =
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\begin{cases}
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1, & if\ a\ quadratic\ residue\ mod\ p\ and\ not\ a\ \equiv\ 0\ (mod\ p),\\
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-1, & if\ a\ is\ a\ non\ residue\ mod\ p,\\
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0, & if\ a\ \equiv 0\ (mod\ p)\\
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\end{cases}
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$$
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$$
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\frac{a}{p} \equiv a^{p-1/2}\ (mod\ p)\ and\ \frac{a}{p} \in \{-1,0,1\}
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$$
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* Legendre Symbol test via Python with
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```python
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pow(a,(p-1)/2,p)
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```
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[Finding the square root of integer a which is quadratic residue](http://mathcenter.oxford.emory.edu/site/math125/findingSquareRoots/)
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* Given $p \equiv 3\ mod\ 4$ the square root is calculated through
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```python
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pow(a,((p+1)//4),p)
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```
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## Tonelli-Shanks - Modular Square Root
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* Find elliptic curve co-ordinates
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* Precondition: modulus is not a prime
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* TBD
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## Links
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* [Encryption+Decryption](https://www.cs.drexel.edu/~jpopyack/Courses/CSP/Fa17/notes/10.1_Cryptography/RSA_Express_EncryptDecrypt_v2.html)
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* [Extended GCD](http://www-math.ucdenver.edu/~wcherowi/courses/m5410/exeucalg.html)
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