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Galois Finite Fields and the Advanced Encryption Standard (AES)

72 points - last Saturday at 8:34 AM

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  • benlivengood

    today at 9:39 PM

    Hey, one of my university professors! GF2^8 is also great for RAID6+ and Reed Solomon codes.

    Back in the day it was tractable to build AES lookup tables for computing MixColumns and SubBytes similar to the site's multiplication table. Then djb found a remote timing attack, so processors started implementing hardware AES instructions.

      • _peeley

        today at 10:01 PM

        I was also pleasantly surprised to see a `uaf.edu` link on the front page of HN, and even more so when I saw it was written by Dr. Lawlor. His course on x86 assembly is easily one of the best classes I took in college.

    • mmozeiko

      today at 9:21 PM

      > Inverses exist for each operation, which makes decryption possible.

      It is worth mentioning that in modern AES usage this feature does not really matter, because block cipher is used in CTR mode. In CTR block cipher is used to create long keystream bytes (pretty much one-time-pad key). This keystream gets xor'ed with plaintext to get ciphertext. Then getting from ciphertext back to plaintext is exactly the same xor operation with same keystream, no inverse AES operations needed.

        • orlp

          today at 9:27 PM

          AES is used for a whole lot more than just encryption in CTR mode.

      • ceroxylon

        today at 9:56 PM

        >Weirdly, members of a field of size pn are equivalent to polynomials of degree up to n

        Wouldn't it be equivalent to polynomials of degree less than n (up to nāˆ’1)?

        • prideout

          today at 10:01 PM

          Is this post popular because of a recent mathematical discovery in this problem space?

          • dboreham

            today at 6:45 PM

            I don't know that "Weirdly" is right for the polynomial equivalence -- isn't that what Galois was trying to do in the first place (prove things about polynomials such as the impossibility of analytic solutions to higher order polynomials, and the impossibility of constructing a heptagon)?

              • tomsmeding

                today at 8:51 PM

                I read the "weirdly" as "unexpectedly", i.e. "you might not see this coming, but it's still true (and there's deep theory behind it)".

                • bananaflag

                  today at 7:26 PM

                  A bit, yeah. I deleted a comment where I said it's not quite the same but it kinda is, and figuring it out is very rewarding.

                  • today at 7:24 PM