Query about arithmetic finite fields

A field $(\mathbb F,+,*)$ is a set closed under two operations $+$ and $*$, with $(\mathbb F,+)$ a commutative group having identity element noted $0$, $(\mathbb F\setminus\{0\},*)$ a commutative group with identity noted $1$, and $*$ distributive w.r.t. $+$. Equivalently, it's a commutative ring $(\mathbb F,+,*)$ in which every element except $0$ has a multiplicative inverse.

A Galois field is a field with a finite number of elements $q$. It can be shown that:

• It must be that $q=p^k$ with $p$ prime and $k$ a strictly positive integer.
• Any two Galois fields with the same number $q$ of elements are isomorphic. The set of such Galois fields (or for short, the Galois field with $q$ elements) is noted $\operatorname{GF}(q)$ or $\mathbb F_q$.
• When $k=1$, the isomorphism between two fields of $\operatorname{GF}(p)$ with $p$ prime is uniquely defined by the mere correspondence of each field's identity elements $0$ and $1$. The fields then are isomorphic to the ring of integers modulo the prime $p$, noted $\mathbb Z/p\mathbb Z$ or (somewhat less properly) $\mathbb Z_p$. That field is assimilated to $\mathbb F_p$.
• More generally, any field of (or the field) $\operatorname{GF}(p^k)$ is isomorphic to the set of polynomials of degree less than $k$ with coefficients in $\mathbb F_p$ (equivalently, the set of tuples with $k$ elements each an integer modulo $p$), with addition $+$ defined by polynomial addition (equivalently, addition modulo $p$ of tuple elements of the same indexes), and multiplication $*$ defined by polynomial multiplication followed by taking the remainder by Euclidean polynomial division of that product by some irreducible polynomial $P$ of degree exactly $k$.

When $p=2$ as is the case in $\operatorname{GF}(2^{256})$ of the question or $\operatorname{GF}(2^{128})$ of the code linked in the question (with $k=256$ or $k=128$ respectively), it's convenient to assimilate an element of the field to an integer in $[0,2^k)$. The most natural mapping, used by the linked code, is to assimilate a binary polynomial to the integer obtained by evaluating the polynomial in $\mathbb Z$ at $x=p=2$. That is, the polynomial $\sum x^j$ for $j$ in some finite subset of $\mathbb N$ is represented by the integer $\sum2^j$ with $j$ in that same subset. Addition is then bitwise eXclusive OR, implemented by the Python operator ^. Much like this is is addition without carry, multiplication is carry-less product, and modular reduction also is bitwise. These can be implemented as two separate operations like _mult_gf2 and _div_gf2 in the linked code, or they can be interleaved, as in __mul__.

when I compare my multiplication results with this it doesn't matches, what could be the reason?

• Confusing * or % as implemented by Python for int operands with polynomial multiplication or polynomial remainder as applicable to operands that are representation of polynomials as int.
• Not using the same $k$: the question is about $\operatorname{GF}(2^{256})$, the linked code is about $\operatorname{GF}(2^{128})$ and has $128$ hardcoded in multiple places.
• Not using the same irreducible polynomial of degree $k$ as reduction polynomial. The linked code uses $1+x+x^2+x^7+x^{128}$.
• Not using the same mapping of polynomial to integer, for the field elements or the reduction polynomial. An alternate representation for polynomial $\sum x^j$ of degree less than $k$ could use $\sum2^{k-j-1}$, but that's highly unusual, and that convention can't apply for the reduction polynomial itself.