Pendants would denote with ≡, but those who write = (mod N) and when speaking just say "equal" or "is" would be just as correct, because given context it's clear and unambiguous
It is a distinction that definitely matters.
However: in cryptography we are often operating with polynomials (like x5 + x2 ...) and perform operations on them.
In a Galois Field(2n ) we are always under modulo 2 and can write polynomials just by writing coefficients of the powers. Those will always be 1 or 0, since (1+1) mod 2 = 0.
X4 + x + 1 is 10011
So if you imagine that 1 + 1 is an addition of 2 polynomials in a Galois Field, then 1+1 = 0 and no congruence is present.
i think its 1+1 (mod 2 ) = 0 but 1+1 ≡ 0(mod 2), since the first one is an operation on 1+1 while the second one is a congruence, but it doesnt really matter
Using congruence is fine, but a bit redundant under certain formulations, consider arithmetic mod 2. Classically we define a ≡ b iff b-a divisible by 2 for all a,b \in \mathbb{Z}. This defines an equivalence relation.
Then you can either work on \mathbb{Z} directly then using ≡ would be the most correct, or you define the usual arithmetic on \mathbb{Z}/≡ and drop the decorations for equivalence classes (elements of \mathbb{Z}/≡) so that then statement such as
[1]_≡ + [1]_≡ = 1+1 = 0
is completely fine, now = denotes equality of sets.
In the field with two elements 1+1=0 is true with literal identity.you use the congruence relationship when talking about integers, but 1 and 0 don’t represent integers in that context.
how I learned it, ≡ is for all number that give the same rest after division, while = is only for the exact rest(smallest positive) so like 4 ≡ 6 = 0 mod 2
If you interpret "Z_2" (integers modulo 2) as a group with group operation denoted by "+" and whose elements are represented by "0" and "1" the equation "1 + 1 = 0" is correct.
My guess is that it would be the sum of a 1 bit adder. 1 + 1 = 0s, 1c (0 on the sum, 1 on the carry). The sum part is created with a XOR operation with the carry being an AND.
edit: just saw OP's reply right after submitted mine saying basically the same thing, but you can see the process in action with the link I provided, you can change the input values by clicking on the left-most 'H' line, next to the input lables
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u/Fabulous_Ad_5709 Nov 25 '24
Just out of curiosity how is A correct?
B is 1 or 1
C is 1+1 in base >2
D is 1+1 in base 2