Finite field simply refer to a set of numbers which have a beginning and end, that is they are finite which basically means that the numbers in the set are countable.
For a set to be a finite field it
1. Has to be closed under two binary operations; Addition and Multiplication.
This means that if you add two numbers in the set, the result must be in the set also.
Also if you multiply two numbers in the set the result has to be in the set.
Consider a set {0,1,2} to check whether this set is closed under addition and multiplication you perform the binary operation on all possible combinations.
if the results are elements of the set, it's closed else it's open and not a finite field.
The result of multiplication in third row; 3 and 4 are not elements of the set hence the set is not closed under multiplication.
Also the result of 2 + 2 results 4, in the third row above. which is not element of the set.
This set is not closed under addition, and multiplication binary operations.
Note : we can define addition a little differently
to make this work, but using “normal” addition, this set is not closed.
In finite field usually the addition and multiplication is not the same with normal integer addition or multiplication.
2 There is 0 in the set, such that if you add it to any element in the set, the result is the element.
e.g 0 + a = a.
3 There is 1 in the set such 1.a = a.
4 For each element in the set there must be it inverse.
e.g a . a-1 = a;
5 For Each element in the set there must be equivalent negative of it.
e.g -a + a = 0
If a set of element satisfy all the above conditions then its a finite field.
Order of a finite field
Order of a finite field is the number of elements in the set.
e.g FieldA = {a,b,c,d} has 4 element therefore its order is 4.
e.g FieldB = {0,1,2,3,4...,n} has n-1 element therefore its order is n-1.
finite field over a prime order
This means that the number of elements in the set is a prime number.
e.g FieldC = {0,1,2,3,4,5,6} the order is 7, which is a prime number and hence the Finite field is over a prime order.
Thank you for reading, I hope you understand what finite fields are, its required to understand some cryptographic topics like Eliptic curve digital signature.
Drop any comments or corrections.
references
- Programming Bitcoin from scratch by jimmy song
- Finite fields Wikipedia
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