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I've some difficulties about sums in the field $\mathbb{F}_{32}$. In particular I'm studying an example of a cryptographic attack, where there are a lot of sums in this field, which I don't understand. One of these is:

$1+7=6$ in $\mathbb{F}_{32}$.

Anyone coud clarify me the reason of this result?

Clà
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1 Answers1

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You are actually adding polynomials of degree less than $5$ with coefficients in $\mathbb F_2$. Thus $$1+7 = 1 + (1+x+x^2) = (1+1) + x + x^2 = x+x^2 = 6$$ Note that this operation is equivalent to binary XOR of the numbers. $$1 \oplus 7 = 00001_2 \oplus 00111_2 = 00110_2 = 6$$

Dilip Sarwate
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AlexR
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  • Thank you, your answer is very detailed, but why $7=1+x+x^2$ and $6=x+x^2$? I know the construction of $mathbb{F}_{32}$, but I'm not understanding the relation with this. Sorry if my questions are too stupid :-S – Clà Sep 08 '14 at 11:00
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    @Clà See $7$ in its base-$2$ representation: $7 = 111_2 = 100_2 + 10_2 + 1_2$. Now canonically $$1_2 = 1, 10_2 = x, 100_2 = x^2, \ldots, 1\underbrace{0\ldots0}{n \text{ times}}{}_2 = x^n$$ This comes in particularly handy for the definition of multiplication in $\mathbb F{p^n}$ – AlexR Sep 08 '14 at 11:03
  • Oh great! I've never seen such a construction! Thank you very much! Can I ask you something else? Have you some books to suggest me to learn more about these fields and operations in a cryptographic context? – Clà Sep 08 '14 at 11:09
  • Any introductory book on galois theory should do. Unfortunately I don't know anymore wich one I read. You should ask a [tag:reference-request]-Question on Galois Theory in conjunction with cryptography, I'm sure someone around here has some decent references for you :) – AlexR Sep 08 '14 at 13:31
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    @Clà See this answer for a detailed exposition. – Dilip Sarwate Sep 10 '14 at 14:02