## Polynomial Basis Squaring

Finally have successfully found some spare time to do coding to solve this polynomial squaring:

And this is the result, x^5 + x + 1 :

• #### rudi 9:37 pm on May 30, 2013 Permalink | Reply

bu, akan lebih cantik kalo nulis polinomnya pake latex,
$x^5 + x+1$

• #### CG 11:00 am on May 31, 2013 Permalink | Reply

iya belum sempet dirapihin 😀 biasanya saya pake latex for wordpress

## 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 😉

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

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 ( 4 ), polynomial basis

2. For what conditions ONB representation is available?

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

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