Channel capacity $C = C(A,B) = max I(A,B)$

Max - when the differential is 0

Channel capacity of a binary symmetric channel, crossover probability $p = 1 - H(p)$

Shannon’s Noisy Channel Coding Theorem

Ch 5

– Euclidean Algorithm - find the gcd

Bezout’s Identity

for x,y in Z, $d = gcd(a,b) = ax + by$

Chinese remainder theorem

For any co-prime integers $p$ and $q$ and any $m_1$, $m_2$ in $Z$, the congruences $x === m_1 mod p$ and $x === m_2 mod q$ have a unique common solution modulo $pq$

The element $a$ in $Z_M$ is invertible (is a unit) if it has an inverse $a^-1$ in $Z_m$

$a$ is invertible only if $gcd(a,m) = 1$

$U_m$ is the set of units of $Z_m$
aka, the set where $gcd == 1$ aka $U_m = {a in Zm : gcd(a,m) = 1}$ if $(m = p) is prime then Um = {1,2,…,p-1}$

Euler’s phi-function: $\phi(m) = |U_m|$
if $p$ is prime, then $\phi(p) = p - 1$

if $gcd(m,n) = 1$ (aka if two numbers are co-prime) then $phi(mn) = phi(m) * phi(n)$ $phi(p^\alpha) = p^\alpha - p^{\alpha-1}$

To find for any, factorise into their prime factors, then we can find the product (as $phi(mn) = phi(m)phi(n))$
$\phi(m) = p_1^{\alpha_1}(1-1/p_1)$

Fermat’s Little Theorem

Let $p$ be a prime number and $a$ be an integer

$a^p === a mod p$
$a^{p-1} === 1 mod p whenever p !/ a$

–

Calculate $3^125 (mod 7)$
… = 5 mod 7

Euler's theorem

If $a$, $n$ are positive integers with $gcd(a,n) = 1$ then $a^{\phi(n)} === 1 mod n$

Calculate $7^131 mod 12$

$phi(12) = 4$, so $7^4 mod 12 = 1$