Lattices (empty) → 0.0.1

raw patch · 10 files changed

+1006/−0 lines, 10 filesdep +HUnitdep +HaskellForMathsdep +arraysetup-changed

Dependencies added: HUnit, HaskellForMaths, array, base, test-framework, test-framework-hunit

Files

• LICENSE +674/−0
• Lattices.cabal +52/−0
• README +7/−0
• Setup.hs +5/−0
• TODO +5/−0
• src/Math/Lattices/LLL.hs +156/−0
• src/Math/LinearAlgebra/GramSchmidt.hs +32/−0
• tests/Math/Lattices/LLL/Tests.hs +40/−0
• tests/Math/LinearAlgebra/GramSchmidt/Tests.hs +22/−0
• tests/TestSuite.hs +13/−0

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@@ -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 <kde@bartcoppens.be>+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+

@@ -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+

@@ -0,0 +1,5 @@+#!/usr/bin/env runhaskell++import Distribution.Simple+main = defaultMain+

@@ -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! :-)

@@ -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

@@ -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)

@@ -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]+    ]+

@@ -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]+    ]+

@@ -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+    ]