## From one basis to another

Apparently converting from one basis to another like from polynomial to normal basis is not as easy as I thought, hmm… have spent days scribbling, thinking, frustated, madly curious, and end up browsing several papers about that and being succesfully diverted from the main target of doing paper on plaintext embedding, oh my!

Let me digest some more papers I have just downloaded 30secs ago, and will post something useful here as soon as possible π

• #### CG 2:38 pm on February 2, 2009 Permalink | Reply Tags: normal basis, polynomial basis ( 4 )

Again, from here:

1. How to convert from polynomial to normal bases?

GF element can be represented in Polynomial Basis (PB) or Normal Basis (NB).

For example we have polynomial $p(x)=x^{3}+x^{2}+1$

The PB representation in $GF\left (2^{3} \right )$ is

If $GF\left (p^{m} \right )$ be a field with $p^{m}$ elements and $\beta$ an element of it such that $m$ elements $\left \{\beta ,\beta ^{p}, ... , \beta ^{p^{m-1}} \right \}$ are linearly dependent. Then this set forms a normal basis for $GF\left (p^{m} \right )$

The NB representation of elements in $GF\left (2^{3} \right )$ will only use 3 elemens $\beta$, $\beta ^{2}$ dan $\beta ^{4}$.

• #### CG 1:58 pm on February 2, 2009 Permalink | Reply Tags: galois field ( 4 ), normal basis, polynomial basis ( 4 )

2. For what conditions ONB representation is available?

Answer: One condition is when the $p(x)$ generates GF elements which are linearly independent.

## Normal basis: squaring is fast

Yes, right. ONB (Optimal Normal Basis) is so appealing π

If we take some element $\beta$ in the field $F_{p^{m}}$, the polynomial representation is:
$\beta =a_{n}x^{n}+ ... + a_{1}x + a_{0}$
where $n < m$

A normal basis can be formed using the set:
$\left \{\beta ^{p^{m-1}},... \beta ^{p^{2}}, \beta ^{p}, \beta \right \}$

Any element in a field $F_{p^{m}}$ can be represented in a normal basis format. An element $e$, can be written in a normal basis as:
$e = e_{m-1}\beta ^{2^{m-1}} + ... + e_{2}\beta ^{2^{2}} + e_{1}\beta ^{2} + e_{0}\beta$

It seems complex but the implementation in the computer is as simple as using only AND, OR and ROTATE!

The most interesting thing is that this representation allow squaring with just a number amounts of rotation!

Here are some proves:

1. $\left (\beta ^{2^{i}} \right )^{2}=\beta ^{2^{i+1}}$
2. $\beta ^{2^{m}}=\beta$

• #### Budi Rahardjo 11:19 pm on January 29, 2009 Permalink | Reply

Choosing whether you want to use ONB (and other[s]) depends on many things; (1) the number of different operations that you want to do; (2) the cost of each operation. Squaring is fast, but what about addition and multiplication? Can they also be done easily (hardware wise)? And how easy can you transform from polynomial representation to ONB? …
sorry, lots of questions π

• #### CG 9:23 am on January 30, 2009 Permalink | Reply

botak deh! π

jawab pertanyaannya dicicil ya? good questions… and hard ones! π

• #### Minnalkodi 10:12 am on February 6, 2013 Permalink | Reply

How to convert binary values to normal basis plz anybody tell

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