Crypto Flashcards
Define a finite field
A finite set and two operations (addition and multiplication), that satisfies:
- closed over those two operations
- identities and inverses exist for both operations
What does it mean (in finite fields) that the set is “closed” over its operators?
If a and b are in the set, then a + b and a * b are also in the set.
In a finite field, what are the additive and multiplicative identities?
0 and 1, eg:
a + 0 = a
and
a * 1 = a
In a finite field, what are the additive and multiplicative inverses?
-a is the additive inverse and a⁻¹ is the multiplicative inverse
The “size” of a set is better known as the …
“order” of the set
In finite fields, how is subtraction defined?
Using the additive inverse.
In finite fields, how is division defined?
Using the multiplicative inverse.
What set of integers is closed under normal multiplication?
{-1, 0, 1}
Give generalised form of a finite field of p
Fₚ = {0, 1, 2, … p-1}
It is 3 o’clock now… what hour was it 16 hours ago?
(3 - 16) % 12
It is 3 o’clock now, what hour will it be 47 hours from now?
(3 + 47) % 12
It is 12 minutes past the hour now… what minute will it be 834 minutes from now?
(12 + 834) % 60
We us _ arithmetic to define addition on a finite field set?
modulo
Write generalised form of the “closed over addition” rule for finite sets?
a ⊕ b ∈ Fₚ
we write ⊕ or +ƒ to avoid confusion with regular arithmetic
Define addition over Fₚ
a ⊕ b = (a + b) % p
where a, b ∈ Fₚ
(we write ⊕ or +ƒ to avoid confusion with regular arithmetic)
