Rings

The Integer Ring $Z_m$ consists of -

• The set of {0,1,2,3… m-1}
• Two operators - plus and multiply, such that
  • for all a, b,c in $Z_m$
    • a+b === c mod m
    • a.b === d mod m

• Closure of the ring - Multiply the ring, the result is in the ring
• Addition and multiplication are associative
• Additive inverse exits a+ (-a) = 0, exists
• multiplicative inverse exists, a. a^-1 === 1 mod m
• If gcd(x,y) == 1, then the inverse exists
• if gcd(x,y) != 1, then the inverse does not exists

Affine cipher

key - (a,b)

Encryption

$y = ax+b (mod 26)$