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If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/Lattices.cabal b/Lattices.cabal new file mode 100644 --- /dev/null +++ b/Lattices.cabal @@ -0,0 +1,52 @@ +Name: Lattices +Version: 0.0.1 +Category: Math +Synopsis: A library for lattices +Description: A library for lattices, in particular for computing an LLL reduced basis for a lattice and finding a close lattice vector +Author: Bart Coppens +Maintainer: Bart Coppens +Build-Type: Simple +Cabal-Version: >=1.8 + +License: GPL +License-file: LICENSE + +Extra-Source-Files: + README + TODO + tests/TestSuite.hs + tests/Math/LinearAlgebra/GramSchmidt/Tests.hs + tests/Math/Lattices/LLL/Tests.hs + + +Source-Repository head + type: git + location: git://github.com/bcoppens/Lattices.git + +Library + Build-Depends: + base >= 4 && < 5, + HaskellForMaths >= 0.4.0, + array >= 0.3 + + Exposed-modules: + Math.LinearAlgebra.GramSchmidt + Math.Lattices.LLL + + Hs-Source-Dirs: src + + ghc-options: -w + +Test-Suite test-all + Hs-Source-Dirs: src, tests + Main-Is: TestSuite.hs + Type: exitcode-stdio-1.0 + + Build-Depends: + base >= 4 && < 5, + HaskellForMaths >= 0.4.0, + array >= 0.3, + test-framework >= 0.4, + test-framework-hunit >= 0.2, + HUnit >= 1.2 + diff --git a/README b/README new file mode 100644 --- /dev/null +++ b/README @@ -0,0 +1,7 @@ +A very basic LLL (Lenstra-Lenstra-Lovász) lattice reduction algorithm in Haskell. + +* As of now, it seems to work on 2 small test cases and on some larger ones. I need to generate tests for it, for example comparing it to NTL's LLL output +* There are *no* optimizations implemented yet +* The interface is currently not well-designed, it should of course take at least Integer instead of Rational +* etc + diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,5 @@ +#!/usr/bin/env runhaskell + +import Distribution.Simple +main = defaultMain + diff --git a/TODO b/TODO new file mode 100644 --- /dev/null +++ b/TODO @@ -0,0 +1,5 @@ +* in lllDelta, reuse the GSO for filling in mu_arr +* see if there are even more efficient arrays I should use +* maybe let the GSO / LLL algorithms return a Data.Array instead of converting back/from lists? +* use Array in GSO's gs function +* implement (algorithmic/implementational) speedups! :-) diff --git a/src/Math/Lattices/LLL.hs b/src/Math/Lattices/LLL.hs new file mode 100644 --- /dev/null +++ b/src/Math/Lattices/LLL.hs @@ -0,0 +1,156 @@ +-- | Implements a *very* basic LLL (Lenstra-Lenstra-Lovász) lattice reduction algorithm. This version uses exact arithmetic over the rationals. +-- References for the LLL algorithm: +-- +-- * Factoring Polynomials with Rational Coefficients, Arjen K Lenstra, Hendrik W Lenstra Jr, and László Lovász. Mathematische Annalen 261, 515-534 (1982) +-- +-- * Mathematics of Public Key Cryptography, Steven Galbraith. Chapter 17 of draft 1.0 +-- +-- * Modern Computer Algebra, second edition, Joachim von zur Gathen and Jürgen Gerhard. Chapter 16. +-- +-- References for Babai's Nearest Plane Method for the Closest Vector Problem: +-- +-- * On Lovász' Lattice Reduction And The Nearest Lattice Point Problem, László Babai. Combinatorica 6 (1), 1-13 (1986). +-- +-- * Mathematics of Public Key Cryptography, Steven Galbraith. Chapter 18 of draft 1.0 +-- +module Math.Lattices.LLL ( + lll, + lllDelta, + closeVector, + Basis(..) +) where + +import Data.Array +import Data.Ratio +import Math.Algebra.LinearAlgebra hiding ((!)) +import Math.LinearAlgebra.GramSchmidt + +-- | A matrix representing a basis +type Basis = Array Int [Rational] +type GSO = Array (Int, Int) Rational + +-- The $B_i$ set is called 'bb in this file, because of course we cannot call it 'B in Haskell. + +-- | Just an easy way to write $||v||^2$ +norm2 v = v <.> v + +-- | Closest 'Integral to the given n, rounding up. $\lfloor n\rceil$ +rnd x = floor $ x + 1%2 + +-- |Return an LLL reduced basis. This calls 'lllDelta with a default parameter $\delta = 3/4$ +lll :: [[Rational]] -> Basis +lll basis = lllDelta basis $ 3%4 + +-- | Return an LLL reduced basis, with reduction parameter $\delta$. This is the conventional flavor of the algorithm using Gram-Schmidt, no fancy speedups yet +lllDelta :: [[Rational]] -> Rational -> Basis +lllDelta basis delta = lllLoop b' delta bb' mu_arr 1 n + where + n = length basis - 1 + (b, mu) = gramSchmidtOrthogonalization basis + bb = map norm2 b + + b' = listArray (0, n) basis + bb' = listArray (0, n) bb + + -- TODO: reuse mu from GSO!!! + mu_arr = array ( ((0, 0), (n, n) ) ) [ ( (i,j), m ) | i <- [0..n], + j <- [0..n], + let m = (basis !! i <.> (b !! j)) / (norm2 $ b !! j) ] + +-- | Perform a size reduction. Returns the new $b_k$, the new $\mu_k$. +sizeReduction :: Int -> Basis -> GSO -> (Basis, GSO) +sizeReduction k b mu = sizeReduction' indices k b mu + where + indices = reverse $ [0..k-1] + +sizeReduction' (l:ls) k b mu = sizeReduction' ls k b' mu' + where + (b', mu') = sizeReduction'' k l b mu +sizeReduction' [] _ b mu = (b, mu) + +-- | Size reduction of the basis for a single index (k, l). Returns a tuple of the new $b_k$ and the new $\mu_k$. +sizeReduction'' k l b mu = (b', mu'') + where + r = toRational $ round $ mu ! (k, l) + + b_k = b ! k + b_l = b ! l + b_k' = b_k <-> (r *> b_l) + b' = b // [ (k, b_k') ] + + mu' = mu // [ update | j <- [0..l-1], + update <- [ ( (k, j), mu ! (k,j) - (r * mu ! (l,j)) ) ] ] + mu'' = mu' // [ ( (k, l), mu' ! (k, l) - r) ] + +-- | Returns whether the Lovász Condition holds: $B_k \geq \delta - \mu^2_{k,k-1}B_{k-1}$ +lovaszCondition :: Array Int Rational -> Int -> Rational -> GSO -> Bool +lovaszCondition bb k delta mu = (bb ! k) >= (delta - m^2)*(bb ! (k-1)) + where + m = mu ! (k, k-1) + +-- | Swaps $b_k$ and $b_{k-1}$, returns a triple: (new $b$, new $B$, new $\mu$) +swapBaseVectors b bb mu_ k n = (b', bb', mu'') + where + b' = b // [ (k - 1, b ! k), (k, b ! (k-1)) ] + + m = mu_ ! (k, k-1) + bb_k1 = bb ! (k-1) + bb_k = bb ! k + + btmp = bb_k + m^2*bb_k1 + + bb' = bb // [ (k, bb_k1*bb_k/btmp), (k-1, btmp) ] + + mu = mu_ // [ ( (k, k-1), m*bb_k1/btmp ) ] + + mu' = mu // [ update | j <- [0..k-2], + update <- [ ( (k-1, j), mu!(k,j) ), ( (k, j), mu!(k-1, j)) ] ] + mu'' = mu' // [ update | i <- [k+1..n], + update <- [ ( (i, k-1), update_i_k1 i), ( (i, k), update_i_k i) ] ] + where + update_i_k1 i = (mu' ! (k, k-1)) * (mu' ! (i, k-1)) + (mu' ! (i, k)) - m*(mu' ! (i, k)) * (mu' ! (k, k-1)) + update_i_k i = (mu' ! (i, k-1)) - m * (mu' ! (i, k)) + +-- | The main loop of the LLL algorithm. We reduce basis 'b with $\delta$ 'delta, with a Gram-Schmidt basis $b^*$ as 'b and the $\mu_{i,j}$ coefficients in 'mu. +-- The current iteration of the loop is 'k out of a maximum of 'n +lllLoop :: Basis -> Rational -> Array Int Rational -> GSO -> Int -> Int -> Basis +lllLoop b delta bb mu k n | k > n = b + | isLovasz = lllLoop b' delta bb mu' (k+1) n + | otherwise = lllLoop b'' delta bb' mu'' nextk n + where + (b', mu') = sizeReduction k b mu + isLovasz = lovaszCondition bb k delta mu' + + (b'', bb', mu'') = swapBaseVectors b' bb mu' k n + nextk = max 1 $ k - 1 + + +-- Two small test cases (will put into unit tests): +-- lll $ [ [12, 2], [13, 4] ] +-- lll $ [ [1, 0, 0], [4, 2, 15], [0, 0, 3] ] + +-- Babai's Algorithm for CVP + +-- | Find a lattice vector in 'basis close to 'x'. 'basis' is assumed to be LLL-reduced +closeVector :: [[Rational]] -> [Ratio Integer] -> [Rational] +closeVector basis x = foldl1 (<+>) $ babaiNP (reverse $ [0..d]) basis' b' x + where + d = length basis - 1 + b' = listArray (0, d) $ gramSchmidtBasis basis + basis' = listArray (0, d) basis + +projectTo v b = (v <.> b) / (norm2 b) + +vsum zero = foldl (<+>) zero + +-- | Find a close vector to 'x using Babai's Nearest Plane Method. 'b is an LLL-reduced basis, 'b'' is its Gram-Schmidt basis d is the size of the (sub)space. +babaiNP [] _ _ _ = [] +babaiNP (i:is) b b' w = y_i : recurse + where + l_i = projectTo w $ b' ! i + delta = toRational $ rnd $ l_i + y_i = delta *> b ! i + + w_i1 = w <-> (l_i - delta) *> (b' ! i) <-> y_i + + recurse = babaiNP is b b' w_i1 diff --git a/src/Math/LinearAlgebra/GramSchmidt.hs b/src/Math/LinearAlgebra/GramSchmidt.hs new file mode 100644 --- /dev/null +++ b/src/Math/LinearAlgebra/GramSchmidt.hs @@ -0,0 +1,32 @@ +-- | Compute a Gram-Schmidt orthogonal basis +module Math.LinearAlgebra.GramSchmidt ( + gramSchmidtBasis, + gramSchmidtOrthogonalization +) where + +import Math.Algebra.LinearAlgebra + +-- | Given a basis, return the Gram-Schmidt orhthogonal basis +gramSchmidtBasis :: Fractional a => [[a]] -> [[a]] +gramSchmidtBasis a = fst $ gramSchmidtOrthogonalization a + +-- | Given a basis, return the Gram-Schmidt orthogonalization, which is a tuple with the Gram-Schmidt orthogonal basis first, and the +-- $\mu_{i,j} = \langle b_i, b^*_j \rangle / \langle b^*_j, b^*_j \rangle$ triangular matrix second, for $1 \leq j < i < n$. +gramSchmidtOrthogonalization :: Fractional a => [[a]] -> ([[a]], [[a]]) +gramSchmidtOrthogonalization (b0:bs) = gs bs [b0] [] + +-- TODO get rid of the (++) used like this, to make it faster +-- | Perform actual Gram-Schmidt reduction +gs [] b' mu = (b', mu) +gs (b_i:bs) b' mu = gs bs b'' mu' + where + mu_i = mu_row b' b_i + mu' = mu ++ [mu_i] + tosum = zipWith (*>) mu_i b' + offset = foldl1 (<+>) tosum + b'_i = b_i <-> offset + b'' = b' ++ [b'_i] + +-- | Compute a (partial) row of the $\mu_{i,j}$ matrix. This is based on the previously orthogonalized vectors $b^*_j$, and the current vector $b_i$. +-- This assumes that 'b_i is of length 'i +mu_row b' b_i = flip map b' $ \b'_j -> (b_i <.> b'_j) / (b'_j <.> b'_j) diff --git a/tests/Math/Lattices/LLL/Tests.hs b/tests/Math/Lattices/LLL/Tests.hs new file mode 100644 --- /dev/null +++ b/tests/Math/Lattices/LLL/Tests.hs @@ -0,0 +1,40 @@ +module Math.Lattices.LLL.Tests ( + tests +) where + +import Test.Framework +import qualified Test.HUnit as H +import Test.Framework.Providers.HUnit + +import Data.Ratio +import Data.Array +import Math.Lattices.LLL + +equalsArray computed ok = H.assert $ elems computed == ok + +simpleLLLTest1 = equalsArray computed ok + where + ok = [ [1 % 1,2 % 1], [9 % 1,(-4) % 1] ] + computed = lll $ [ [12, 2], [13, 4] ] + +simpleLLLTest2 = equalsArray computed ok + where + ok = [ [1 % 1,0 % 1,0 % 1], [0 % 1,2 % 1,0 % 1], [0 % 1,0 % 1,3 % 1] ] + computed = lll $ [ [1, 0, 0], [4, 2, 15], [0, 0, 3] ] + + +closeVectorTest = H.assert $ ok == closeVector basis x + where + basis = map (map toRational) [ [1, 2, 3], [3, 0, -3], [3, -7, 3] ] + x = [10 % 1, 6 % 1, 5 % 1] + ok = [10 % 1,8 % 1,6 % 1] + + +tests :: [Test] +tests = concat + [ + [testCase "Simple LLL test 1" simpleLLLTest1], + [testCase "Simple LLL test 2" simpleLLLTest2], + [testCase "Simple CVP test" closeVectorTest] + ] + diff --git a/tests/Math/LinearAlgebra/GramSchmidt/Tests.hs b/tests/Math/LinearAlgebra/GramSchmidt/Tests.hs new file mode 100644 --- /dev/null +++ b/tests/Math/LinearAlgebra/GramSchmidt/Tests.hs @@ -0,0 +1,22 @@ +module Math.LinearAlgebra.GramSchmidt.Tests ( + tests +) where + +import Test.Framework +import qualified Test.HUnit as H +import Test.Framework.Providers.HUnit + +import Data.Ratio +import Math.LinearAlgebra.GramSchmidt + +simpleTest = H.assert $ computed == correct + where + computed = gramSchmidtOrthogonalization $ map (map toRational) [ [1, 1, 0], [1, 0, 1], [0, 1, 1] ] + correct = ([[1 % 1,1 % 1,0 % 1],[1 % 2,(-1) % 2,1 % 1],[(-2) % 3,2 % 3,2 % 3]],[[1 % 2],[1 % 2,1 % 3]]) + +tests :: [Test] +tests = concat + [ + [testCase "Simple G-S test on Rationals" simpleTest] + ] + diff --git a/tests/TestSuite.hs b/tests/TestSuite.hs new file mode 100644 --- /dev/null +++ b/tests/TestSuite.hs @@ -0,0 +1,13 @@ +module Main where + +import Test.Framework (defaultMain, testGroup) + +import qualified Math.LinearAlgebra.GramSchmidt.Tests +import qualified Math.Lattices.LLL.Tests + +main :: IO () +main = defaultMain + [ + testGroup "Math.LinearAlgebra.GramSchmidt.Tests" Math.LinearAlgebra.GramSchmidt.Tests.tests, + testGroup "Math.Lattices.LLL.Tests" Math.Lattices.LLL.Tests.tests + ]