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the secret @b@. + . + /Example in GHCi:/ + Suppose that you want to split the string \"my secret data\" into n=5 shares such that + at least m=3 shares are necessary to reconstruct the secret. + . + >> :m + Data.ByteString.Lazy.Char8 Crypto.SecretSharing + >> let secret = pack "my secret data" + >> shares <- encode 3 5 secret + >> mapM_ (Prelude.putStrLn . show) shares + > (1,"\134\168\154\SUBV\248\CAN:\250y<\GS\EOT*\t\222_\140") + > (2,"\225\206\241\136\SUBse\199r\169\162\131D4\179P\210x") + > (3,"~\238%\192\174\206\\\f\214\173\162\148\&3\139_\183\193\235") + > (4,"Z\b0\188\DC2\f\247\f,\136\&6S\209\&5\n\FS,\223") + > (5,"x\EM\CAN\DELI*<\193q7d\192!/\183v\DC3T") + >> let shares' = Prelude.drop 2 shares + >> decode shares' + > "my secret message!" + . + The mathematics behind the secret sharing scheme is described in: + \"/How to share a secret/.\" by Adi Shamir. + In Communications of the ACM 22 (11): 612–613, 1979. +license: LGPL-2.1 +license-file: LICENSE +author: Peter Robinson +maintainer: peter.robinson@monoid.at +copyright: Peter Robinson 2014 +category: Cryptography +build-type: Simple +cabal-version: >=1.8 +homepage: http://monoid.at/code +tested-with: GHC==7.8.3 +stability: experimental + +library + hs-source-dirs: src + exposed-modules: Crypto.SecretSharing + Crypto.SecretSharing.Internal + Crypto.SecretSharing.FiniteField + Crypto.SecretSharing.Prime + + build-depends: base ==4.6.*, + bytestring ==0.10.*, + dice-entropy-conduit >= 1.0.0.0, + binary >=0.5.1.1, + vector >=0.10.11.0, + finite-field >=0.8.0, + polynomial >= 0.7.1 + 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 + hs-source-dirs: src, tests + main-is: Tests.hs + diff --git a/src/Crypto/SecretSharing.hs b/src/Crypto/SecretSharing.hs new file mode 100644 --- /dev/null +++ b/src/Crypto/SecretSharing.hs @@ -0,0 +1,36 @@ +----------------------------------------------------------------------------- +-- | +-- Module : Crypto.SecretSharing +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : stable +-- Portability : portable +-- +-- Implementation of an (@m@,@n@)-threshold secret sharing scheme. +-- A given ByteString @b@ (the secret) is split into @n@ shares, +-- and any @m@ shares are sufficient to reconstruct @b@. +-- The scheme preserves perfect secrecy in the sense that the knowledge of up +-- to @m-1@ shares does not reveal any information about the secret @b@. +-- +-- Typically, there are @n@ parties and we would like to give the @i@-th party +-- the @i@-share of each byte. +-- For example, to encode a bytestring @secret@ as @10@ shares, any @5@ of which +-- are sufficient for reconstruction we could write: +-- +-- > shares <- encode 5 10 secret +-- +-- Note that each byte is encoded separately using a fresh set of random +-- coefficients. +-- +-- The mathematics behind the secret sharing scheme is described in: +-- \"How to share a secret.\" by Shamir, Adi. +-- In Communications of the ACM 22 (11): 612–613, 1979. +-- +-- +----------------------------------------------------------------------------- + +module Crypto.SecretSharing( encode, decode, Share ) +where +import Crypto.SecretSharing.Internal diff --git a/src/Crypto/SecretSharing/FiniteField.hs b/src/Crypto/SecretSharing/FiniteField.hs new file mode 100644 --- /dev/null +++ b/src/Crypto/SecretSharing/FiniteField.hs @@ -0,0 +1,34 @@ +{-# LANGUAGE DeriveDataTypeable, DeriveGeneric, GeneralizedNewtypeDeriving, TemplateHaskell #-} +----------------------------------------------------------------------------- +-- | +-- Module : Crypto.SecretSharing.FiniteField +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : stable +-- Portability : portable +-- +----------------------------------------------------------------------------- + +module Crypto.SecretSharing.FiniteField +where + +import Data.Typeable +import GHC.Generics +import Data.FiniteField.PrimeField as PF +import Crypto.SecretSharing.Prime + + +-- | A finite prime field. All computations are performed in this field. +newtype FField = FField { number :: $(primeField $ fromIntegral prime) } + deriving(Show,Read,Ord,Eq,Num,Fractional,Generic,Typeable) + + +-- | A polynomial over the finite field given as a list of coefficients. +type Polyn = [FField] + +-- | Evaluates the polynomial at a given point. +evalPolynomial :: Polyn -> FField -> FField +evalPolynomial coeffs x + = foldr (\c res -> c + (x * res)) 0 coeffs diff --git a/src/Crypto/SecretSharing/Internal.hs b/src/Crypto/SecretSharing/Internal.hs new file mode 100644 --- /dev/null +++ b/src/Crypto/SecretSharing/Internal.hs @@ -0,0 +1,127 @@ +{-# LANGUAGE DeriveDataTypeable, DeriveGeneric, GeneralizedNewtypeDeriving #-} +----------------------------------------------------------------------------- +-- | +-- Module : Crypto.SecretSharing.Internal +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : stable +-- Portability : portable +-- +----------------------------------------------------------------------------- + +module Crypto.SecretSharing.Internal +where +import Math.Polynomial.Interpolation + +import Data.ByteString.Lazy( ByteString ) +import qualified Data.ByteString.Lazy as BL +import qualified Data.ByteString.Lazy.Char8 as BLC +import qualified Data.List as L +import Data.Maybe +import Data.Char +import Data.Vector( Vector ) +import qualified Data.Vector as V +import Data.Typeable +import Control.Exception +import Control.Monad +import Data.Binary( Binary ) +import GHC.Generics +import Data.FiniteField.PrimeField as PF + +import Crypto.SecretSharing.FiniteField +import Crypto.SecretSharing.Prime +import System.Random.Dice + + + +-- | A share of an encoded byte. +data ByteShare = ByteShare + { shareId :: !Int -- ^ the index of this share + , reconstructionThreshold :: !Int -- ^ number of shares required for + -- reconstruction + , shareValue :: !Int -- ^ the value of p(shareId) where p(x) is the + -- generated (secret) polynomial + } + deriving(Typeable,Eq,Generic) + +instance Show ByteShare where + show = show . shareValue + +-- | A share of the encoded secret. +data Share = Share + { theShare :: ![ByteShare] } + deriving(Typeable,Eq,Generic) + +instance Show Share where + show s = show (shareId $ head $ theShare s,BLC.pack $ map (chr . shareValue) $ theShare s) + +instance Binary ByteShare +instance Binary Share + +-- | Encodes a 'ByteString' as a list of n shares, m of which are required for +-- reconstruction. +-- Lives in the 'IO' to access a random source. +encode :: Int -- ^ m + -> Int -- ^ n + -> ByteString -- ^ the secret that we want to share + -> IO [Share] -- a list of n-shares (per byte) +encode m n bstr + | n >= prime || m > n = throw $ AssertionFailed $ + "encode: require n < " ++ show prime ++ " and m<=n." + | BL.null bstr = return [] + | otherwise = do + let bytes = map fromIntegral $ BL.unpack bstr + let len = max 1 ((length bytes) * (m-1)) + coeffs <- (groupInto (m-1) . map fromIntegral . take len ) + `liftM` (getDiceRolls prime len) + let byteVecs = zipWith (encodeByte m n) coeffs bytes + return [ Share $ map (V.! (i-1)) byteVecs | i <- [1..n] ] + + +-- | Reconstructs a (secret) bytestring from a list of (at least @m@) shares. +-- Throws 'AssertionFailed' if the number of shares is too small. +decode :: [Share] -- ^ list of at least @m@ shares + -> ByteString -- ^ reconstructed secret +decode [] = BL.pack [] +decode shares@((Share s):_) + | length shares < reconstructionThreshold (head s) = throw $ AssertionFailed + "decode: not enough shares for reconstruction." + | otherwise = + let origLength = length s in + let byteVecs = map (V.fromList . theShare) shares in + let byteShares = [ map ((V.! (i-1))) byteVecs | i <- [1..origLength] ] in + BL.pack . map (fromInteger . PF.toInteger . number) + . catMaybes . map decodeByte $ byteShares + + +encodeByte :: Int -> Int -> Polyn -> FField -> Vector ByteShare +encodeByte m n coeffs secret = + V.fromList[ ByteShare i m $ fromInteger . PF.toInteger . number $ + evalPolynomial (secret:coeffs) (fromIntegral i::FField) + | i <- [1..n] + ] + + +decodeByte :: [ByteShare] -> Maybe FField +decodeByte ss = + let m = reconstructionThreshold $ head ss in + if length ss < m + then Nothing + else + let shares = take m ss in + let pts = map (\s -> (fromIntegral $ shareId s,fromIntegral $ shareValue s)) shares in + Just $ polyInterp pts 0 + + +-- | Groups a list into blocks of certain size. Running time: /O(n)/ +groupInto :: Int -> [a] -> [[a]] +groupInto num as + | num < 0 = throw $ AssertionFailed "groupInto: Need positive number as argument." + | otherwise = + let (fs,ss) = L.splitAt num as in + if L.null ss + then [fs] + else fs : groupInto num ss + diff --git a/src/Crypto/SecretSharing/Prime.hs b/src/Crypto/SecretSharing/Prime.hs new file mode 100644 --- /dev/null +++ b/src/Crypto/SecretSharing/Prime.hs @@ -0,0 +1,17 @@ +----------------------------------------------------------------------------- +-- | +-- Module : Crypto.SecretSharing.Prime +-- Copyright : Peter Robinson 2014 +-- License : LGPL +-- +-- Maintainer : Peter Robinson +-- Stability : stable +-- Portability : portable +-- +----------------------------------------------------------------------------- + +module Crypto.SecretSharing.Prime( prime ) +where +-- | Determines the size of the finite field and the maximum number of shares. +prime :: Int +prime = 1021 diff --git a/tests/Tests.hs b/tests/Tests.hs new file mode 100644 --- /dev/null +++ b/tests/Tests.hs @@ -0,0 +1,41 @@ +module Main +where +import Data.Monoid +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import Data.Maybe +import qualified Data.List as L +import Control.Monad + +import Data.ByteString.Lazy( ByteString ) +import qualified Data.ByteString.Lazy as B + +import Crypto.SecretSharing.Internal +import Crypto.SecretSharing.FiniteField +import Crypto.SecretSharing.Prime + +instance Arbitrary ByteString where + arbitrary = fmap B.pack arbitrary + +main :: IO () +main = defaultMainWithOpts + [ testProperty "encodingDecoding" propEncodingDecoding + ] mempty + +data ShareIdxs = ShareIdxs { shareIdxs :: [Int] } + deriving (Show,Eq,Ord) + +instance Arbitrary ShareIdxs where + arbitrary = + liftM (ShareIdxs . L.nub) $ sequence [ choose (1,prime-1) + | i <- [1..(prime-1) `div` 2]] + +propEncodingDecoding bstr idxs = ioProperty $ do + shares <- encode 20 (prime-1) bstr + if null shares && B.null bstr + then return True + else do + let chosen = [ shares !! (i-1) | i <- shareIdxs idxs ] + return (bstr == (decode chosen)) +