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  • CG 8:34 am on August 20, 2010 Permalink | Reply
    Tags: , hardware architecture, , , optimal normal basis   

    Choosing n and m for composite field 

    Referring to “Efficient Normal Basis Multipliers in Composite Fields” – Sangho Oh, Chang Han Kim, Jongin Lim, and Dong Hyeon Cheon, there is classification of hardware-applicable composite fields:

    1. Type I composite field where a subfield GF(2^n) in ONB2 and an extension field GF(2^{nm}) in ONB1
    2. Type II composite field where a subfield GF(2^n) in ONB1 and an extension field GF(2^{nm}) in ONB2
    3. Type III composite field where a subfield GF(2^n) in ONB2 and an extension field GF(2^{nm}) in ONB2

    This is different with composite fields presented in “Efficient Methods for Composite Field Arithmetic” – E. Sava ̧s and C ̧. K. Koc, where the selection of n and m  does not put their normal basis types (ONB1 or ONB2) into consideration.

    Now the questions are:

    1. Would it be better if we choose n , m and nm in ONB1/ONB2?
    2. Which polynomial irreducible to be used? With degree = n , or degree = m or degree = nm ?

    [pounding headache, and without answering these questions i wouldnt be able to start the hw design.]

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  • CG 10:57 am on February 24, 2009 Permalink | Reply
    Tags: optimal normal basis   

    ONB Type II, why? 

    Why do we need to use two field elements from two different field to generate a Type II ONB?

    First pick an element \gamma of order 2m+1 in F_{2}^{2m} to find \beta in field F_{2}^{m} .

    Why?

     
    • Mike 5:46 pm on March 1, 2009 Permalink | Reply

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  • CG 4:27 pm on February 7, 2009 Permalink | Reply
    Tags: optimal normal basis   

    More answers for #2 

    Answering #2 from list of questions here:

    An ONB of Type I exists a given field GF(2^{m}) if:

    • m+1 is a prime
    • 2 is a primitive in GF(m+1)

    A Type II optimal normal basis exists in GF(2^{m}) if:

    • 2m+1 is prime
    • either 2 is a primitive in GF(2m+1) or 2m+1\equiv 3 \left (mod\; 4 \right ) and 2 generates the quadratic residues in GF(2m+1)

    Interesting notes:

    An ONB exists in GF(2^{m}) for 23% of all possible values of m

    said this paper. Hmmm, that’s something.

     
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