README.md

Binary Fields

Binary fields denoted as $GF(2)=Z/2Z$, i.e. quotient ring of integers modulo ring of 2 integers $\{0,1\}$, are a special class of Finite Fields, with modulus = $2$. Main properties exhibited by Binary fields are:

• Addition corresponds to bitwise XOR
• Multiplication corresponds to bitwise AND
• since, $x+x=0\implies x=-x$, i.e. negation of a number is itself

This allows for extremely efficient arithmetic that is much more hardware friendly than fields based on other primes.

Binary Extension fields

Finite field with $2^{k}$elements represented as$GF(2^k)=F(2)[X]/f(X)$, where$f(X)$is an irreducible of degree$k$. Used extensively in cryptography like AES block cipher and error-correcting codes.

Two ways of representing $GF(2^{k})$:

• univariate basis - two ways of representing in univariate basis as well, namely:
  • polynomial basis: elements are represented as degree k-1 polynomial $A_{k-1}\alpha^{k-1}+\dots+A_1\alpha+A_0$ by equivalence class $GF(2)/f(x)$, where f(x) is any irreducible in the kth power.
  • normal basis: elements are represented as taking powers of an element from the field
• multilinear basis: there’s one other way of representing elements, i.e. Multilinear basis, where elements are represented by monomials: $1,X_0,X_1\cdot X_0,X_2\cdot X_1\cdot X_0,\dots,X_0\dots X_{l-1}$, with each coefficient in$GF(2)$.

Extension field using towers

Binius realises binary extension field using towers formalised in Weidemann et al..

Basic idea is to derive sequence of polynomial rings inductively

• start with $\tau_{0}=\mathbb{F}_{2}=\{ 0,1 \}$
• set $\tau_{1}=\tau_{0}[X_{0}]/(X_{0}^{2}+X_{0}+1)=\mathbb{F}_{2^{2}}$, namely $\{ 0,1,X_{0},1+X_{0} \}$.
• set $\tau_{2}=\tau_{1}[X_{1}]/(X_{1}^{2}+X_{0}X_{1}+1)=\mathbb{F}_{2^{2^{2}}}$
• continue this further with $\tau_{0}\subset \tau_{1}\subset\dots \subset \tau_{i-1}\subset \tau_{i}=\tau_{i-1}[X_{i-1}]/f_{i-1}(X_{i-1})$, where $f_{i-1}(X_{i-1})=X_{i-1}^{2}+X_{i-1}X_{i-2}+1$ is an irreducible in $\tau_{i}$

In practice, Extension Field elements are represented in vector of binary field $GF_2$ components of length $2^K$. Forms a multilinear basis of the form$\beta_v=\prod_{i=0}^{K-1}(v_iX_i+(1-v_i))$, where$v_i={0,1}$.

Let's take an example of K=2, this forms a field extension of $GF(2^{2^2})=GF($2^{4}$)$. Let's form our basis vector, with $v=\{0,1,2,3\}$:

• $v=0$: representing in binary form, $v=\{0,0\}$, $\beta_v=\prod_{i=0}^{k-1}(0\cdot X_i+(1-0))=1$
• $v=1=\{0,1\}\implies\beta_v=X_0$
• $v=\{1,0\}\implies\beta_v=X_1$
• $v=\{1,1\}\implies\beta_v=X_0X_1$

Now, we have our basis to represents numbers in $GF($2^{4}$)$, taking few examples:

• $10={1,0,1,0}\implies X_0+X_0X_1$
• $7={0,1,1,1}\implies 1+X_0+X_1$

A very nice property of binary fields is defining an element using it's subfield, using it's first and second halves in the subfield: $a=a_0+a_1 X_{i-1}$

Arithmetic in Binary Extensions

• Addition, Subtraction is just bitwise XOR
• Negation is the element itself
• Multiplication is done using a hybrid of Karatsuba multiplication
• Inversion is $x^{(p-2)}$, using Fermat's little theorem