Division, scope finite fields polynomials in general vs. f.f. polynomials in ECC

Lilkp2

11/26/23, 5:42 PM

A cryptography course covered among others following questions:

arithmetic of polynomials over $GF(2^m)$ fields - polynomials division
elliptic curves over field $GF(2^m)$

In scope of former point students were told polynomials product needs subsequent division by irreducible polynomial in order to stay in given order of polynomial (this detail is not this question objective). Then proceeding had been shown how to conduct division of one polynomial - quite similar procedure as that one taught in secondary school math.

Subsequently students were taught how to do division for points on elliptic curve over finite field $GF(2^m)$. Students were told at this point to use XOR-operator. XOR-operation to be used when doing arithmetic on points on elliptic curve, curve kind as mentioned above. Divide product of two polynomials by irreducible polynomial (if multiplication delivered polynomial of higher order than operation input operands order) in order to go back to order $m$ of operands.

Where does that distinction come from: fashion learned with high school mathematics in one case but by XOR operator in another case?

0 + 6

elliptic-curves

finite-field

kodlu

11/26/23, 6:40 PM

The field with $2^m$ elements should be written as $GF(2^m)$ or $F_{2^m}.$ Please ask the question properly. The part about the EC operations "XOR but bitwise" is meaningless. how else can you do XOR? Also the operations that are used to implement ECC addition are over a traditional finite field, but normally a prime field $GF(p).$ The part about regular finite field operations is clearer, every finite field can be expressed as the residue classes of an irreducible polynomial over the ground field, e.g., $GF(2).$

Lilkp2

11/26/23, 8:13 PM

Thanks for comments - incorporated. GF(p) were taught too, prior to teaching $GF(2^m)$. However, I have no question regarding GF(p) this problem area. In case of GF(p) I learned to use modulo calculus to do reduction, not taught that case to use polynomial-form view. Hard to see how pointing out ECC addition details helps to get answer to question asked. Term "ground field" unclear too yet how last sentence helps to get answer. Me tried to put to question only those points which I mean be touching question directly, to not overload question text.

Lilkp2

11/26/23, 10:36 PM

Regarding wrong notation in Q first version, $\mathbb{F_{2^m}}$

kodlu

11/26/23, 10:51 PM

Sure, I won't quibble about \mathbb{F}, however your question is still NOT clear. What is the XOR operation that they were asked to use? Define explicitly.

poncho

11/27/23, 3:29 AM

For that matter, what is your question? Are you asking about the pedagogical approach that the material you're studying uses?

Lilkp2

11/29/23, 6:04 PM

Matter for which I ask question? I want simply understand statements I got from sources in my use yet presented in this question. I like to understand where discrepancy described here comes from, nothing more. I use it in my self-study. Myself placed all details to question I could only place. No clue which details else can be added while strictly related to question asked. Please explain which details can be added to question.

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