AES-BitDist.html

AES - Bit Distribution

ŠMike May, S.J., 2002, maymk@slu.edu

> restart:

Following the same pattern we used with Baby DES and DES, we would like to look at the distribution of zeroes and ones from a collection of words encrypted with AES. The digits should behave like independent random variables. First we need to load in the procedures created in AES-Encryption.mws.

> read `AES.m`:

We want a procedure for turning a 32 character hex string into a list of 128 binary digits.

>    hex1 := "66E94BD4EF8A2C3B884CFA59CA342B2E";
hexStringToBinList := hexString -> [seq(parse(substring(
    convert(convert(convert(hexString,decimal,hex)+2^128,binary),string),
    i)),i=2..129)]:
hexStringToBinList(hex1);

We return to our favorite example. For the key we will use the zero word. For our plaintexts, we will use words that are all zeroes except for a single bit that is turned on. In this situation it makes sense to expand the key only once.
The list bitChangeCounter gives us a place to record the number of times each bit is used.

> keyHex := intTo128Bits(0);
expandedKey := hexKeyExpander(hexkey):
bitChangeCounter := [seq(0,i=1..128)]:
testLine2 := intVal ->
encryptAESExpanded(intTo128Bits(2^(intVal-1)),expandedKey):

We are ready to count the number of times a bit is used in each of the first 30 test words.

>    for i from 1 to 30 do
  bitChangeCounter := zip(`+`,bitChangeCounter,
      hexStringToBinList(testLine2(i))):
end do:
bitChangeCounter;

> stats[transform,tally](bitChangeCounter);
evalf(stats[describe, mean](bitChangeCounter));
evalf(stats[describe, standarddeviation](bitChangeCounter));

The expected mean is 30*.5 = 15. The expected standard deviation is sqrt(30*.5*.5)=2.7386.

The behavior looks like what we expect from a random distribution of bits.

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