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Here is a sample; alter the names: + + Yoyodyne, Inc., hereby disclaims all copyright interest in the + library `Frob' (a library for tweaking knobs) written by James Random Hacker. + + , 1 April 1990 + Ty Coon, President of Vice + +That's all there is to it! + + + diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/data-dispersal.cabal b/data-dispersal.cabal new file mode 100644 --- /dev/null +++ b/data-dispersal.cabal @@ -0,0 +1,132 @@ +name: data-dispersal + +-- The package version. See the Haskell package versioning policy (PVP) +-- for standards guiding when and how versions should be incremented. +-- http://www.haskell.org/haskellwiki/Package_versioning_policy +-- PVP summary: +-+------- breaking API changes +-- | | +----- non-breaking API additions +-- | | | +--- code changes with no API change +version: 1.0.0.0 + +synopsis: Space-efficient and privacy-preserving data dispersal algorithms. + +description: + This library provides space-efficient (m,n)-information dispersal algorithms (IDAs). + . + Given a ByteString @bstr@ of length @D@, we encode @bstr@ as a list @fs@ of @n@ + 'Fragment's, each containing a ByteString + of length @O(D/m)@. Then, each fragment in @fs@ could be stored on a separate + machine for fault-tolerance. + Even if up to @n-m@ of these machines crash, we can still reconstruct the original + ByteString out of the remaining m fragments. + The total space required for the n fragments is @O((n/m)*D)@. + Note that @m@ and @n@ are roughly in the same order, so the actual storage overhead + for getting good fault-tolerance increases only by a constant factor. + . + The module @Data.IDA@ contains the basic information dispersal algorithm. The module + @Crypto.IDA@ augments the dispersal scheme by combining it with secret sharing, i.e., + the knowledge of up to @m-1@ fragments does not leak any information about + the original data. See "Crypto.IDA" for details. + . + /GHCi Example:/ + . + > > :m + Data.IDA + > > let msg = Data.ByteString.Char8.pack "my really important data" + > > let fragments = encode 5 15 msg + > -- Now we could distributed the fragments on different sites to add some + > -- fault-tolerance. + > > let frags' = drop 5 $ take 10 fragments -- let's pretend that 10 machines crashed + > > decode frags' + > "my really important data" + . + /Fault-Tolerance:/ + . + Suppose that we have @N@ machines and encode our data as @2log(N)@ fragments + with reconstruction threshold m = @log(N)@. + Let's assume that we store each fragment on a separate machine and each + machine fails (independently) with probability at most 0.5. + . + * What is the probability of our data being safe? + @Pr[ at most n-m machines crash ] >= 1-0.5^(log(N)) = 1-N^(-1).@ + . + * What is the overhead in terms of space that we pay for this level of fault-tolerance? + We have n fragments, each of size D\/m, so the total space is @n * D\/ m = + 2D.@ + In other words, we can guarantee that the data survives with high probability + by increasing the required space by a constant factor. + . + This library is based on the following works: + . + * \"Efficient Dispersal of + Information for Security, Load Balancing, and Fault Tolerance\", by Michael O. + Rabin, JACM 1989. + . + * \"How to share a secret.\" by Adi Shamir. + In Communications of the ACM 22 (11): 612–613, 1979. + . + * \"Secret Sharing Made Short\" Hugo Krawczyk. + CRYPTO 1993: 136-146 + + +license: LGPL-2.1 + +license-file: LICENSE + +author: Peter Robinson + +maintainer: peter.robinson@monoid.at + +copyright: Peter Robinson 2014 + +category: Data, Cryptography + +build-type: Simple + +cabal-version: >=1.8 + +homepage: http://monoid.at/code + + +library + hs-source-dirs: src + + exposed-modules: Data.IDA + Data.IDA.Internal + Data.IDA.FiniteField + Crypto.IDA + + build-depends: base ==4.6.* + ,array >= 0.4.0.1 + ,vector >= 0.10.11.0 + ,binary >= 0.7.2.1 + ,bytestring >= 0.10.0.2 + ,syb >= 0.4.0 + ,binary >= 0.5.1.1 + ,finite-field >= 0.8.0 + ,matrix >= 0.3.4.0 + ,AES >= 0.2.9 + ,entropy >= 0.3.2 + ,secret-sharing >= 1.0.0.0 + + ghc-options: -Wall + +test-suite Main + type: exitcode-stdio-1.0 + + x-uses-tf: true + + build-depends: base >= 4 && < 5 + ,QuickCheck >= 2.4 + ,test-framework >= 0.4.1 + ,test-framework-quickcheck2 + ,array >= 0.4.0.1 + ,vector >= 0.10.11.0 + ,spool >= 0.1 + ,binary >= 0.7.2.1 + ,bytestring >= 0.10.0.2 + ,syb >= 0.4.0 + + hs-source-dirs: src, tests + + main-is: Tests.hs + diff --git a/src/Crypto/IDA.hs b/src/Crypto/IDA.hs new file mode 100644 --- /dev/null +++ b/src/Crypto/IDA.hs @@ -0,0 +1,125 @@ +----------------------------------------------------------------------------- +-- | +-- Module : Crypto.IDA +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : experimental +-- Portability : portable +-- +-- This module provides an (m,n)-information dispersal scheme that provides +-- data redundancy while preserving secrecy. +-- In other words, this module combines the best of 2 worlds: secret sharing +-- algorithms with low-overhead information dispersal. +-- +-- Function 'encode' splits a given bytestring into @n@ fragments with the +-- following properties: +-- +-- 1. Any @m@ of the @n@ fragments are sufficient for reconstructing the original +-- bytestring via 'decode', and +-- 2. the knowledge of up to @m-1@ fragments does /not/ reveal any information +-- about the original bytestring. +-- +-- +-- In more detail, suppose that we have some bytestring @b@ that we want to +-- (securely) disperse and parameter @m@, @n@. +-- Running 'encode' @m n b@ does the following: +-- +-- * Generate a randomly chosen key of 32 bytes, called @key@. +-- * Encrypt the bytestring @b@ using @key@ via AES. +-- * Generate @n@ shares using the perfect secret sharing algorithm implemented +-- in module "Crypto.SecretSharing"; see package secret-sharing +-- * Generate @n@ fragments of the encrypted data using the information +-- dispersal algorithm in "Data.IDA". +-- * Finally, we pair up these shares and fragments as +-- a list of 'EncryptedFragment's. +-- +-- The size of each encrypted fragment is @O(|b|\/m + |key|)@. +-- For sufficiently large bytestrings, the @O(|b|\/m)@ factor dominates and thus +-- the scheme is space-optimal. +-- +-- The secret sharing algorithm guarantess that the knowledge of up to @m-1@ of +-- the fragments does not leak any information about the encryption key (and +-- hence the encrypted data). +-- +----------------------------------------------------------------------------- +{-# LANGUAGE DeriveDataTypeable, ScopedTypeVariables, DeriveGeneric #-} +module Crypto.IDA( EncryptedFragment(fragmentId,keyShare,aesIV,fragment) + , encode + , decode + ) +where +import Data.IDA.Internal( Fragment(..) ) +import qualified Data.IDA.Internal as IDA + +import Crypto.SecretSharing( Share ) +import qualified Crypto.SecretSharing as PSS + +import Data.ByteString.Lazy( ByteString ) +import qualified Data.ByteString.Lazy as BL +import qualified Data.ByteString as B +import System.Entropy( getEntropy ) +import Codec.Crypto.AES +import Control.Exception +import Data.Typeable +import Data.Binary( Binary ) +import GHC.Generics + +data EncryptedFragment = EncryptedFragment + { fragmentId :: Int -- ^ the id of the encrypted fragment, ranging from 1 to n. + , keyShare :: Share -- ^ the list of (bytewise) shares of the AES key + , aesIV :: B.ByteString -- ^ the initialization vector of the AES encryption + , fragment :: Fragment -- ^ the encrypted fragment of the original data + } + deriving(Typeable,Show,Eq,Generic) + +instance Binary EncryptedFragment + + +aesKeyLength, aesIVLength :: Int +aesKeyLength = 32 +aesIVLength = 16 + +-- | Space efficient and secrecy-preserving (m,n)-information dispersal: +-- Generates @n@ fragments out +-- of a given bytestring @b@. Each fragment has size @length b \/ m + O(1)@. +-- At least m fragments are required for reconstruction. +-- Preserves secrecy: Assuming that these fragments are distributed +-- among different sites, the knowledge of less than m +-- fragments provides /no/ information about the original data whatsoever. +encode :: Int -- ^ m: number of fragments required for reconstruction + -> Int -- ^ n: total number of fragments (@n ≥ m@) + -> Maybe ByteString -- ^ the initialization vector for the AES encryption + -> ByteString -- ^ the information that we want to disperse + -> IO [EncryptedFragment] -- ^ a list of n encrypted fragments. +encode m numFragments mIV msg = do + key <- getEntropy aesKeyLength + iv <- maybe (getEntropy aesIVLength) (return . BL.toStrict) mIV + keyShareList <- PSS.encode m numFragments (BL.fromStrict key) + let headers = zip keyShareList (replicate numFragments $ BL.fromStrict iv) + let fs = IDA.encode m numFragments $ BL.toStrict $ crypt CTR key iv Encrypt msg + return [ EncryptedFragment i ks (BL.toStrict iv) f + | (i,(ks,iv),f) <- zip3 [1..] headers fs + ] + +-- | Reconstruct the original data from (at least) @m@ fragments. +-- Throws an 'AssertionFailed' exception if an insufficient number fragments are +-- given or if a decoding error occurs. +decode :: [EncryptedFragment] + -> ByteString +decode [] = BL.pack [] +decode pss@(p:_) + | length pss < IDA.reconstructionThreshold (fragment p) = throw $ AssertionFailed + "decode: not enough fragments for reconstruction." + | otherwise = + let m = IDA.reconstructionThreshold $ fragment p in + let efs = take m pss in + let iv = aesIV p in + let fs = map fragment efs in + let ss = map keyShare efs in + let emsg = IDA.decode fs in + let key = PSS.decode ss in + crypt CTR (BL.toStrict key) iv Decrypt $ BL.fromStrict emsg + + diff --git a/src/Data/IDA.hs b/src/Data/IDA.hs new file mode 100644 --- /dev/null +++ b/src/Data/IDA.hs @@ -0,0 +1,31 @@ +----------------------------------------------------------------------------- +-- | +-- Module : Data.IDA +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : experimental +-- Portability : portable +-- +-- This module provides an implementation of a space efficient (m,n)-threshold information +-- dispersal algorithm (IDA) as described in +-- "Efficient Dispersal of Information for Security, Load Balancing, and Fault +-- Tolerance", by Michael O. Rabin, JACM 1989. +-- +-- Given a ByteString bstr of length D +-- >>> f = encode m n bstr +-- encodes bstr as a list of n 'Fragment's, each containing a ByteString +-- of length approximately @D/m@. For reconstructing the original ByteString out +-- of a sublist fssub of fs, +-- consisting of at least m fragments, use +-- >>> decode fssub +-- +----------------------------------------------------------------------------- +module Data.IDA(encode,decode,Fragment(fragmentId)) +where +import Data.IDA.Internal + + + + diff --git a/src/Data/IDA/FiniteField.hs b/src/Data/IDA/FiniteField.hs new file mode 100644 --- /dev/null +++ b/src/Data/IDA/FiniteField.hs @@ -0,0 +1,129 @@ +{-# LANGUAGE DeriveDataTypeable, DeriveGeneric, GeneralizedNewtypeDeriving, TemplateHaskell, Haskell2010, TypeFamilies, FlexibleContexts, Trustworthy, FlexibleInstances, TypeSynonymInstances, OverlappingInstances, UndecidableInstances #-} +----------------------------------------------------------------------------- +-- | +-- Module : Data.IDA.FiniteField +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : experimental +-- Portability : portable +-- +-- Computations in a finite prime field +-- +----------------------------------------------------------------------------- + +module Data.IDA.FiniteField +where + +import Control.Exception +import Data.Typeable +import GHC.Generics +import qualified Data.FiniteField.PrimeField as PF +import Data.FiniteField.Base +import Data.IDA.Prime + +import qualified Data.Vector as V +import Data.Vector(Vector) +import Data.Matrix +import Data.Monoid +import Data.Binary + + + +-- | Our finite prime field. All computations are performed in this field. +newtype FField = FField { number :: $(PF.primeField $ fromIntegral prime) } + deriving(Show,Read,Ord,Eq,Num,Fractional,Generic,Typeable,FiniteField) + +instance Monoid FField where + mempty = 0 + mappend = (+) + + +instance Enum FField where + toEnum = FField . fromIntegral + fromEnum = fromEnum . PF.toInteger . number + +instance Binary FField where + get = do + n <- get :: Get Integer + return $ FField { number = fromInteger n } + put f = put (PF.toInteger $ number f) + + +-- | A matrix over the finite field. +type FMatrix = Matrix FField + + +dotProduct :: Num a => Vector a -> Vector a -> a +dotProduct v1 v2 = V.sum $ V.zipWith (*) v1 v2 + + +-- | Solves a linear equality system @A x = b@ given by a lower triangular matrix via +-- forward substitution. +forwardSub :: Fractional a => Matrix a -> Vector a -> Vector a +forwardSub lower bV = + forwardSub' lower bV (V.empty) + where + forwardSub' lower bV xV + | nrows lower == 0 = xV + | otherwise = + let curRow = getRow 1 lower + offset = V.length xV + lm = getRow 1 lower V.! offset + curB = V.head bV + negSum = curRow `dotProduct` xV + curX = (curB - negSum) / lm + in + forwardSub' (submatrix 2 (nrows lower) 1 (ncols lower) lower) + (V.tail bV) + (V.snoc xV curX) + + +-- | Solves a linear equality system @A x = b@ given by an upper triangular matrix via +-- backward substitution. +backwardSub :: Fractional a => Matrix a -> Vector a -> Vector a +backwardSub upper bV = + backwardSub' upper bV V.empty (nrows upper) + where + backwardSub' upper bV xV i + | nrows upper == 0 = xV + | i<=0 = xV + | otherwise = + let curRow = snd $ V.splitAt i $ getRow i upper + lm = (getRow i upper) V.! (i-1) + curB = bV V.! (i-1) + negSum = xV `dotProduct` curRow + curX = (curB - negSum) / lm + in + backwardSub' upper bV (curX `V.cons` xV) (i-1) + +-- | Compute the inverse of matrix. Throws 'AssertionFailed' if the matrix is +-- not invertible. +inverse :: (Ord a,Fractional a) => Matrix a -> Matrix a +inverse mat = + let mInv = luDecomp mat + in + case mInv of + Nothing -> throw $ AssertionFailed "inverse: matrix is not invertible!" + Just (upper,lower,pmatrix,_) -> + let m = nrows mat + bVecs = [ getCol i (identity m) | i <- [1..m] ] + columnsOfInverse = flip map bVecs $ \bV -> + let yV = forwardSub lower (getCol 1 $ pmatrix * colVector bV) + xV = backwardSub upper yV + in colVector xV + in + foldr (<|>) (fromLists [[]]) columnsOfInverse + + +-- | Construct a Vandermonde matrix. The i-th list element is the i-th seed of +-- the geometric progression of the i-th row. +vandermonde :: Int -> [FField] -> FMatrix +vandermonde m is = matrix (length is) m $ \(i,j) -> (fromIntegral i)^(fromIntegral $ j-1) + +-- | Create an nxm Vandermonde matrix. /O(n m)/. +vmatrix :: Int -> Int -> FMatrix +vmatrix numFragments m = vandermonde m [1..fromIntegral numFragments] + + diff --git a/src/Data/IDA/Internal.hs b/src/Data/IDA/Internal.hs new file mode 100644 --- /dev/null +++ b/src/Data/IDA/Internal.hs @@ -0,0 +1,137 @@ +----------------------------------------------------------------------------- +-- | +-- Module : Data.IDA.Internal +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : experimental +-- Portability : portable +-- +----------------------------------------------------------------------------- +{-# LANGUAGE DeriveDataTypeable, ScopedTypeVariables, DeriveGeneric #-} +module Data.IDA.Internal +where +import Data.ByteString( ByteString ) +import qualified Data.ByteString as B +import Data.Vector(Vector,(!)) +import qualified Data.Vector as V +import Data.Array( array ) +import qualified Data.Array as A +import Control.Exception +import Data.Typeable +import Data.Binary( Binary ) +import GHC.Generics +import qualified Data.Matrix as M + +import Data.IDA.FiniteField +import qualified Data.FiniteField.PrimeField as PF + + + + +-- | A fragment of the original data. +data Fragment = Fragment + { fragmentId :: !Int -- ^ index of this fragment + , trailLength :: !Int -- ^ number of symbols added to the original message + , reconstructionThreshold :: !Int -- ^ number of fragments required for reconstruction + , theContent :: ![FField] -- ^ the encoded content of the fragment + , msgLength :: !Int -- ^ length of the original message + } + deriving(Typeable,Show,Eq,Generic) + +instance Binary Fragment + + +-- | Takes a message (a bytestring) and yields 'n' fragments such that any 'm' of +-- them are sufficient for reconstructing the original message. +encode :: Int -- ^ m: we need ≥ 'm' fragments to reconstruct the original message + -> Int -- ^ n: total number of fragments into which we are going to + -- split the message; 'n' ≥ 'm' + -> ByteString -- ^ the original message + -> [Fragment] -- ^ 'n' fragments +encode m numFragments msg + | numFragments >= 1021 && numFragments <1 = + throw $ AssertionFailed "encode: invalid number of fragments." + | otherwise = + let (intseq,trailLen) = toIntVec m msg + len = V.length intseq + blocks = V.fromList $ groupInto m intseq + vm = vmatrix numFragments m + c i k = dotProduct (M.getRow i vm) (blocks ! (k-1)) -- product M.! (i,k) + content i = [ c i j + | j <- [ 1 .. ceiling $ fromIntegral len / fromIntegral m ] + ] in + [ Fragment { fragmentId = i + , trailLength = trailLen + , reconstructionThreshold = m + , theContent = content i + , msgLength = len + } + | i <- [1 .. numFragments] + ] + +-- | Takes a list of at least m fragments (where 'm' is the reconstruction +-- threshold used for 'encode') and tries to reconstruct the original message. +-- Throws an 'AssertionFailed' exception if there are less than m fragments +-- or if the fragments belong to a different message. +decode :: [Fragment] + -> ByteString +decode [] = throw $ AssertionFailed + "decode: need at least m fragments for reconstruction." +decode pss@(p:_) + | length pss < reconstructionThreshold p = throw $ AssertionFailed + "decode: need at least m fragments for reconstruction." + | otherwise = + let m = reconstructionThreshold p + idxs = map fragmentId (take m pss) + n = maximum idxs + fragments :: [Vector FField] + fragments = map (V.fromList . theContent) (take m pss) + idxVec = V.fromList idxs + vecA = M.matrix m m $ \(i,j) -> + vmatrix n m M.! (idxVec!(i-1),j) + matrixBInv = inverse vecA + colVecR :: Vector FMatrix + colVecR = V.fromList + [ M.transpose $ M.fromLists [ map (! (k-1)) fragments ] + | k <- [1..V.length (head fragments)] + ] + idxList = [(j,k) | k <- [1..V.length (head fragments)], j <- [1..m]] + matrixBInvTimesColVecr = array ((1,1),(V.length (head fragments),m)) + [ ((k,j),head $ + array (1,m) (zip [1..] $ M.toLists $ matrixBInv + * + (colVecR ! (k-1))) A.! j) + | k <- [1..V.length (head fragments)], j <- [1..m]] + mCont :: [FField] + mCont = map (\(j,k) -> matrixBInvTimesColVecr A.! (k,j)) idxList in + fromIntVec (msgLength p - trailLength p) $ V.fromList mCont + + +-- | Takes an integer m and a bytestring and converts the bytestring into a +-- 'Vector Word8', appending 0s at the end such that the length is dividable by +-- m. +toIntVec :: Int -> ByteString -> (Vector FField,Int) +toIntVec m bStr = + let len = B.length bStr in + let trailLen = if (len `mod` m) == 0 then 0 + else ((len `div` m)+1)*m - len in + let bStrApp = bStr `B.append` B.pack (replicate trailLen 0) in + (V.fromList $ map fromIntegral $ B.unpack bStrApp,trailLen) + + +-- | Converts a bytestring to a 'Vector Word8', removing the trailing 0s. +fromIntVec :: Int -> Vector FField -> ByteString +fromIntVec originalLength intVec = + B.pack $ map (fromInteger . PF.toInteger . number) $ V.toList $ V.slice 0 originalLength intVec + + +-- | Splits a vector into lists of the given size. O(vector-length \/ size). +groupInto :: Int -> Vector a -> [Vector a] +groupInto size as = + let (fs,ss) = V.splitAt size as in + if V.null ss + then [fs] + else fs : groupInto size ss + diff --git a/tests/Tests.hs b/tests/Tests.hs new file mode 100644 --- /dev/null +++ b/tests/Tests.hs @@ -0,0 +1,52 @@ +module Main +where +import Data.Monoid +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import Data.Maybe +import GHC.Conc + +import qualified Data.ByteString as BS +import qualified Data.ByteString.Lazy as BL +import qualified Data.IDA as IDA +import qualified Crypto.IDA as SecretIDA + +instance Arbitrary BS.ByteString where + arbitrary = fmap BS.pack arbitrary +instance Arbitrary BL.ByteString where + arbitrary = fmap BL.pack arbitrary + +main :: IO () +main = do + defaultMainWithOpts + [ testProperty "IDAencodingDecoding" propIDAEncodingDecoding + , testProperty "SecretIDAencodingDecoding" propSecretIDAEncodingDecoding + ] mempty + + + +-- | Input: bytestring b, list of booleans. +-- First, encode b, then randomly select sufficiently many fragments according to +-- the boolean list, finally decode these fragments and check if the original +-- bytestring is matched. +propIDAEncodingDecoding bstr blist = + let frags = IDA.encode 10 40 bstr in + let chosen = fst $ unzip $ filter snd $ zip frags blist in + if length chosen < 10 + then True + else bstr == IDA.decode chosen + + +-- | Input: bytestring b, list of booleans. +-- First, encode b, then randomly select sufficiently many fragments according to +-- the boolean list, finally decode these fragments and check if the original +-- bytestring is matched. +propSecretIDAEncodingDecoding bstr blist = ioProperty $ do + frags <- SecretIDA.encode 5 30 Nothing bstr + let chosen = fst $ unzip $ filter snd $ zip frags blist + if length chosen < 5 + then return True + else + return $ bstr == SecretIDA.decode chosen +