From db6cffe92c6d3eacc8ead34b6e015748850f536f Mon Sep 17 00:00:00 2001 From: Peter Harpending Date: Sat, 1 Apr 2023 23:37:59 -0600 Subject: [PATCH] wrote up rho step --- docs/kek/README.md | 159 +++++++++++++++++++++++++++++++++++ docs/kek/keccak-f.png | Bin 0 -> 111357 bytes docs/kek/rho-table-wojak.png | Bin 0 -> 38979 bytes docs/kek/rho-table.png | Bin 0 -> 28806 bytes docs/kek/rho.png | Bin 0 -> 77691 bytes 5 files changed, 159 insertions(+) create mode 100644 docs/kek/keccak-f.png create mode 100644 docs/kek/rho-table-wojak.png create mode 100644 docs/kek/rho-table.png create mode 100644 docs/kek/rho.png diff --git a/docs/kek/README.md b/docs/kek/README.md index 702aab3..5ece875 100644 --- a/docs/kek/README.md +++ b/docs/kek/README.md @@ -34,6 +34,17 @@ special cases of Keccak. ### Pitfalls +#### Pitfall: This is only keccak-f-1600 + +Keccak has an "outer keccak" and an "inner keccak". The standard specifies +this free parameter called `b`, which is the bit size of the algorithm's state. +For what I needed, this can be assumed to be `1600`. You may need to +generalize this code. Hopefully this explainer is clear enough that you can do +that. + +![NIST standard, pp. 17](./keccak-f.png) + + #### Pitfall: "fast keccak" versus "clear keccak" The main reference is the "clear" Erlang code. The "fast" version is the @@ -722,11 +733,159 @@ parity(<<>> , NOnes) -> NOnes rem 2. ``` ### Inner Keccak: rho stage + +This stage iterates over the `{X, Y}`-coordinate pairs, and applies an affine +shift to the remaining `Z`-coordinate. + +![NIST standard, pp. 13](./rho.png) + +The shifts are given by a table which is in a janky order because look how +smart we don't you get it it's modular arithmetic it means 4 and 3 are like +negative numbers I am so smart please tell me how smart I am + +![NIST standard, pp. 13](./rho-table-wojak.png) + +The outer part of this code `rho/1` and `rho/2` just contains the loopy part. +The actual transformation is in `rhoxy/2`. Erlang doesn't have loops. Instead +we pick a starting point... + +```erlang +%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L459-L467 + +-spec rho(Array) -> NewArray + when Array :: <<_:1600>>, + NewArray :: <<_:1600>>. +%% @private +%% do the rho step +%% @end + +rho(Array) -> + rho(Array, {xy, 0, 0}). +``` + +Then on each step, update the `{X, Y}` value, with our branching logic to test +if we're done... + +```erlang +%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L471-L492 + +-spec rho(Array, LaneXY) -> NewArray + when Array :: <<_:1600>>, + LaneXY :: {xy, 0..4, 0..4}, + NewArray :: <<_:1600>>. +%% @private +%% do the rho step to each lane +%% @end + +% terminal case +rho(Array, XY = {xy, 4, 4}) -> + Result = rhoxy(Array, XY), + Result; +% need to reset Y and increment X +rho(Array, XY = {xy, X, 4}) -> + NewArray = rhoxy(Array, XY), + NewXY = {xy, X + 1, 0}, + rho(NewArray, NewXY); +% need to increment Y and leave X +rho(Array, XY = {xy, X, Y}) -> + NewArray = rhoxy(Array, XY), + NewXY = {xy, X, Y + 1}, + rho(NewArray, NewXY). +``` + +The actual transformation looks up the offset for the `{X, Y}` coordinate in a +table and applies it. `xyset/3` is a function that replaces a lane. This is +explained in the [coordinate system section.](#inner-keccak-coordinate-system). + +```erlang +%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L496-L517 + +-spec rhoxy(Array1600, LaneXY) -> NewArray1600 + when Array1600 :: <<_:1600>>, + LaneXY :: {xy, 0..4, 0..4}, + NewArray1600 :: <<_:1600>>. +%% @private +%% do the rho step to a given lane +%% @end + +rhoxy(Array, ThisXY = {xy, ThisX, ThisY}) -> + ThisOffset = offset(ThisX, ThisY), + ThisLane = xyth(ThisXY, Array), + % we increase the z coordinate by the offset + % Suppose the offset is 2 + % bits = A B C D E + % z = 0 1 2 3 4 + % newbits = D E A B C + % in other words, we take Offset number of bits off the tail of the lane + % put them on the front + <> = ThisLane, + NewLane = <>, + NewArray = xyset(ThisXY, Array, NewLane), + NewArray. +``` + +The `offset/2` table is copied from the table on pp. 13 of the NIST +specification. The reason for the `rem 64` is that our lane depth is `64` +because [we're not implementing full generalized +keccak](#pitfall-this-is-only-keccak-f-1600), we're only implementing the +keccak that is actually used in SHA-3 and SHAKE. + +```erlang +%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L521-L558 + +-spec offset(X, Y) -> Offset + when X :: 0..4, + Y :: 0..4, + Offset :: 0..63. +%% @private +%% See NIST specification, pg. 21 +%% @end + +offset(3, 2) -> 153 rem 64; +offset(3, 1) -> 55 rem 64; +offset(3, 0) -> 28 rem 64; +offset(3, 4) -> 120 rem 64; +offset(3, 3) -> 21 rem 64; + +offset(4, 2) -> 231 rem 64; +offset(4, 1) -> 276 rem 64; +offset(4, 0) -> 91 rem 64; +offset(4, 4) -> 78 rem 64; +offset(4, 3) -> 136 rem 64; + + +offset(0, 2) -> 3 rem 64; +offset(0, 1) -> 36 rem 64; +offset(0, 0) -> 0 rem 64; +offset(0, 4) -> 210 rem 64; +offset(0, 3) -> 105 rem 64; + +offset(1, 2) -> 10 rem 64; +offset(1, 1) -> 300 rem 64; +offset(1, 0) -> 1 rem 64; +offset(1, 4) -> 66 rem 64; +offset(1, 3) -> 45 rem 64; + +offset(2, 2) -> 171 rem 64; +offset(2, 1) -> 6 rem 64; +offset(2, 0) -> 190 rem 64; +offset(2, 4) -> 253 rem 64; +offset(2, 3) -> 15 rem 64. +``` + ### Inner Keccak: pi stage + + ### Inner Keccak: chi stage + + ### Inner Keccak: iota stage + + ### Inner Keccak: coordinate system + + [coord-system]: #inner-keccak-coordinate-system [greek-letter-pitfall]: #pitfall-greek-letter-steps-require-two-copies-of-the-sponge-to-compute [german-lecture]: https://www.youtube.com/watch?v=JWskjzgiIa4 diff --git a/docs/kek/keccak-f.png b/docs/kek/keccak-f.png new file mode 100644 index 0000000000000000000000000000000000000000..a682e297aa07f24f25524ac14522e6fe9ecb3d16 GIT binary patch literal 111357 zcmeFZWmH?y*Eg6_pg?hGaf-W@;!e@v4#C~s309zm0>w*!V8tDZ7cZq032wnD?iySt z{Xft1&U~GB&8+z{cdaBhxi|aVyM3SEz9(8kO#ugs918#d-~bh6wEzIr0ssI-6$1tN zizPa3Jn{q6T@mC30K6jn_j-o-Nb(8*paB47rFHzX4wg9mvu5vv?gMl8{Y+$ZOR;qG 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