cleanup RSA

Cryptography/RSA.md

@@ -2,43 +2,24 @@
 
 What is interesting about an RSA key:
 
-`e` is a constant, often it is 65537
+The modulus is `N` and it is `p * q = N` through factorization. `p` and `q` are primes.
 
-`n` is the modulus, `p * q = n` through factorization
+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).
-
-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)
 
+There is:
 $$
-\phi(n) = (p-1)(q-1)
+\phi(N) = (p-1)(q-1)
 $$
 
 and further
 
 $$
-1 < \phi < n
+1\ <\ \phi < N
 $$
 
+The public key is `(N, e)`. If you create a real key e.g. through OpenSSH, the default for `e` (encryption) is `65537` or `0x10001` in hex.
 
-Encryption, public key `e` is a prime between 2 and phi  
+The private key is `(N, d)` and `d` (decryption) is the modular multiplicative inverse of `e` and $\phi$.
-$$
-2 < e < \phi
-$$
-
-
-Decryption, private key `d`
-$$
-d\ e\ mod\ \phi(n) \equiv 1
-$$
-
-$$
-d\ e \equiv 1\ (mod\ \phi(n))
-$$
-
-`d` is the modular inverse of e and phi and makes the private key.
 
 $$
 Cipher = msg^{d}\ mod\ \phi
@@ -48,6 +29,31 @@ $$
 Cleartext = cipher^{e}\ mod\ \phi
 $$
 
+
+Further properties:
+
+public key `e` is a prime between 2 and phi
+
+$$
+2 < e < \phi
+$$
+
+Private key `d` is the multiplicative inverse modulo $\phi(N)$
+
+$$
+1\ \equiv de\ mod\ \phi(N)
+$$
+
+$$
+de\ \equiv 1\ mod\ \phi(N)
+$$
+
+This means `d` can be calculated by
+
+$$
+d \equiv\ e^{-1}\ mod\ \phi(N)
+$$
+
 ---
 
 `e` and `d` may be found through the following Python snippets
@@ -68,7 +74,7 @@ for i in range (phi + 1, phi + foo):
 
 ## Euklid
 
-Just a short excourse:  
+Just a short excourse:
 A greatest common divisior out of an example a = 32 and b = 14 would be the
 groups of the following divisors
 
@@ -120,10 +126,10 @@ $$
 ### Modular Inverse
 
 Coming back to the modular inverse $n$, it can be found in the following way
-$n^{p-1} \equiv 1\ mod\ p$  
+$n^{p-1} \equiv 1\ mod\ p$
-$n^{p-1} * n^{-1} \equiv n^{-1}\ mod\ p$  
+$n^{p-1} * n^{-1} \equiv n^{-1}\ mod\ p$
-$n^{p-2} * n * n^-1 \equiv n^{-1}\ mod\ p$  
+$n^{p-2} * n * n^-1 \equiv n^{-1}\ mod\ p$
-$n^{p-2} * 1 \equiv n^{-1}\ mod\ p$  
+$n^{p-2} * 1 \equiv n^{-1}\ mod\ p$
 $n^{p-2} \equiv n^{-1}\ mod\ p$
 
 ## Quadratic Residue