/// <summary>
/// Decode a binary vector of length n and weight t into a number between 0 and (n|t) (binomial coefficient).
/// <para>The result is given as a byte array of length floor[(s+7)/8], where s = floor[log(n|t)].</para>
/// </summary>
///
/// <param name="N">The "n" integer</param>
/// <param name="T">The "t" integer</param>
/// <param name="GVector">The binary vector</param>
///
/// <returns>The decoded vector as a byte array</returns>
public static byte[] Decode(int N, int T, GF2Vector GVector)
{
if ((GVector.Length != N) || (GVector.HammingWeight() != T))
throw new ArgumentException("vector has wrong length or hamming weight");
int[] vecArray = GVector.VectorArray;
BigInteger bc = BigMath.Binomial(N, T);
BigInteger d = ZERO;
int nn = N;
int tt = T;
for (int i = 0; i < N; i++)
{
bc = bc.Multiply(BigInteger.ValueOf(nn - tt)).Divide(BigInteger.ValueOf(nn));
nn--;
int q = i >> 5;
int e = vecArray[q] & (1 << (i & 0x1f));
if (e != 0)
{
d = d.Add(bc);
tt--;
if (nn == tt)
bc = ONE;
else
bc = bc.Multiply(BigInteger.ValueOf(tt + 1)).Divide(BigInteger.ValueOf(nn - tt));
}
}
return BigIntUtils.ToMinimalByteArray(d);
}
/// <summary>
/// Return a new vector consisting of the first <c>K</c> elements of this vector
/// </summary>
///
/// <param name="K">The number of elements to extract</param>
///
/// <returns>Returns a new GF2Vector consisting of the first <c>K</c> elements of this vector</returns>
public GF2Vector ExtractLeftVector(int K)
{
if (K > Length)
{
throw new ArithmeticException("invalid length");
}
if (K == Length)
{
return(new GF2Vector(this));
}
GF2Vector result = new GF2Vector(K);
int q = K >> 5;
int r = K & 0x1f;
Array.Copy(_elements, 0, result._elements, 0, q);
if (r != 0)
{
result._elements[q] = _elements[q] & ((1 << r) - 1);
}
return(result);
}
/// <summary>
/// Find an error vector <c>E</c> over <c>GF(2)</c> from an input syndrome <c>S</c> over <c>GF(2^M)</c>
/// </summary>
///
/// <param name="SyndVec">The syndrome</param>
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="SqRootMatrix">The matrix for computing square roots in <c>(GF(2M))<sup>T</sup></c></param>
///
/// <returns>The error vector</returns>
public static GF2Vector SyndromeDecode(GF2Vector SyndVec, GF2mField Field, PolynomialGF2mSmallM Gp, PolynomialGF2mSmallM[] SqRootMatrix)
{
int n = 1 << Field.Degree;
// the error vector
GF2Vector errors = new GF2Vector(n);
// if the syndrome vector is zero, the error vector is also zero
if (!SyndVec.IsZero())
{
// convert syndrome vector to polynomial over GF(2^m)
PolynomialGF2mSmallM syndrome = new PolynomialGF2mSmallM(SyndVec.ToExtensionFieldVector(Field));
// compute T = syndrome^-1 mod gp
PolynomialGF2mSmallM t = syndrome.ModInverse(Gp);
// compute tau = sqRoot(T + X) mod gp
PolynomialGF2mSmallM tau = t.AddMonomial(1);
tau = tau.ModSquareRootMatrix(SqRootMatrix);
// compute polynomials a and b satisfying a + b*tau = 0 mod gp
PolynomialGF2mSmallM[] ab = tau.ModPolynomialToFracton(Gp);
// compute the polynomial a^2 + X*b^2
PolynomialGF2mSmallM a2 = ab[0].Multiply(ab[0]);
PolynomialGF2mSmallM b2 = ab[1].Multiply(ab[1]);
PolynomialGF2mSmallM xb2 = b2.MultWithMonomial(1);
PolynomialGF2mSmallM a2plusXb2 = a2.Add(xb2);
// normalize a^2 + X*b^2 to obtain the error locator polynomial
int headCoeff = a2plusXb2.Head;
int invHeadCoeff = Field.Inverse(headCoeff);
PolynomialGF2mSmallM elp = a2plusXb2.MultWithElement(invHeadCoeff);
// for all elements i of GF(2^m)
for (int i = 0; i < n; i++)
{
// evaluate the error locator polynomial at i
int z = elp.EvaluateAt(i);
// if polynomial evaluates to zero
if (z == 0)
{
// set the i-th coefficient of the error vector
errors.SetBit(i);
}
}
}
return(errors);
}
/// <summary>
/// Decides whether the given object <c>other</c> is the same as this field
/// </summary>
///
/// <param name="Obj">The object for comparison</param>
///
/// <returns>Returns <c>(this == other)</c></returns>
public override bool Equals(Object Obj)
{
if (!(Obj is GF2Vector))
{
return(false);
}
GF2Vector otherVec = (GF2Vector)Obj;
if (Length != otherVec.Length)
{
return(false);
}
if (!Compare.AreEqual(_elements, otherVec._elements))
{
return(false);
}
return(true);
}
/// <summary>
/// Return a new vector consisting of the last <c>k</c> elements of this vector
/// </summary>
///
/// <param name="K">The number of elements to extract</param>
///
/// <returns>Returns a new GF2Vector consisting of the last <c>K</c> elements of this vector</returns>
public GF2Vector ExtractRightVector(int K)
{
if (K > base.Length)
{
throw new ArithmeticException("invalid length");
}
if (K == base.Length)
{
return(new GF2Vector(this));
}
GF2Vector result = new GF2Vector(K);
int q = (base.Length - K) >> 5;
int r = (base.Length - K) & 0x1f;
int length = (K + 31) >> 5;
int ind = q;
// if words have to be shifted
if (r != 0)
{
// process all but last word
for (int i = 0; i < length - 1; i++)
{
result._elements[i] = (IntUtils.URShift(_elements[ind++], r)) | (_elements[ind] << (32 - r));
}
// process last word
result._elements[length - 1] = IntUtils.URShift(_elements[ind++], r);
if (ind < _elements.Length)
{
result._elements[length - 1] |= _elements[ind] << (32 - r);
}
}
else
{
// no shift necessary
Array.Copy(_elements, q, result._elements, 0, length);
}
return(result);
}
/// <summary>
/// Multiply this vector with a permutation
/// </summary>
///
/// <param name="P">The permutation</param>
///
/// <returns>Returns <c>this*p = p*this</c></returns>
public override Vector Multiply(Permutation P)
{
int[] pVec = P.GetVector();
if (Length != pVec.Length)
{
throw new ArithmeticException("GF2Vector: Length mismatch!");
}
GF2Vector result = new GF2Vector(Length);
for (int i = 0; i < pVec.Length; i++)
{
int e = _elements[pVec[i] >> 5] & (1 << (pVec[i] & 0x1f));
if (e != 0)
{
result._elements[i >> 5] |= 1 << (i & 0x1f);
}
}
return(result);
}
/// <summary>
/// Return a new vector consisting of the elements of this vector with the indices given by the set <c>SetJ</c>
/// </summary>
///
/// <param name="SetJ">The set of indices of elements to extract</param>
///
/// <returns>Return the new GF2Vector <c>[SetJ[0], SetJ[1], ..., SetJ[#SetJ-1]]</c></returns>
public GF2Vector ExtractVector(int[] SetJ)
{
int k = SetJ.Length;
if (SetJ[k - 1] > Length)
{
throw new ArithmeticException("invalid index set");
}
GF2Vector result = new GF2Vector(k);
for (int i = 0; i < k; i++)
{
int e = _elements[SetJ[i] >> 5] & (1 << (SetJ[i] & 0x1f));
if (e != 0)
{
result._elements[i >> 5] |= 1 << (i & 0x1f);
}
}
return(result);
}
/// <summary>
/// Adds another GF2Vector to this vector
/// </summary>
///
/// <param name="V">The GF2Vector to add</param>
///
/// <returns>Returns <c>this + V</c></returns>
public override Vector Add(Vector V)
{
if (!(V is GF2Vector))
{
throw new ArithmeticException("GF2Vector: Vector is not defined over GF(2)!");
}
GF2Vector otherVec = (GF2Vector)V;
if (Length != otherVec.Length)
{
throw new ArithmeticException("GF2Vector: Length mismatch!");
}
int[] vec = IntUtils.DeepCopy(((GF2Vector)V)._elements);
for (int i = vec.Length - 1; i >= 0; i--)
{
vec[i] ^= _elements[i];
}
return(new GF2Vector(Length, vec));
}
/// <summary>
/// Return a new vector consisting of the first <c>K</c> elements of this vector
/// </summary>
///
/// <param name="K">The number of elements to extract</param>
///
/// <returns>Returns a new GF2Vector consisting of the first <c>K</c> elements of this vector</returns>
public GF2Vector ExtractLeftVector(int K)
{
if (K > Length)
throw new ArithmeticException("invalid length");
if (K == Length)
return new GF2Vector(this);
GF2Vector result = new GF2Vector(K);
int q = K >> 5;
int r = K & 0x1f;
Array.Copy(_elements, 0, result._elements, 0, q);
if (r != 0)
result._elements[q] = _elements[q] & ((1 << r) - 1);
return result;
}
/// <summary>
/// Encrypt a plain text message
/// </summary>
///
/// <param name="Input">The plain text</param>
///
/// <returns>The cipher text</returns>
public byte[] Encrypt(byte[] Input)
{
// generate random vector r of length k bits
GF2Vector r = new GF2Vector(_K, _secRnd);
// convert r to byte array
byte[] rBytes = r.GetEncoded();
// compute (r||input)
byte[] rm = ByteUtils.Concatenate(rBytes, Input);
// compute H(r||input)
_dgtEngine.BlockUpdate(rm, 0, rm.Length);
byte[] hrm = new byte[_dgtEngine.DigestSize];
_dgtEngine.DoFinal(hrm, 0);
// convert H(r||input) to error vector z
GF2Vector z = CCA2Conversions.Encode(_N, _T, hrm);
// compute c1 = E(r, z)
byte[] c1 = CCA2Primitives.Encrypt((MPKCPublicKey)_keyPair.PublicKey, r, z).GetEncoded();
byte[] c2;
// get PRNG object
using (KDF2Drbg sr0 = new KDF2Drbg(GetDigest(_cipherParams.Digest)))
{
// seed PRNG with r'
sr0.Initialize(rBytes);
// generate random c2
c2 = new byte[Input.Length];
sr0.Generate(c2);
}
// XOR with input
for (int i = 0; i < Input.Length; i++)
c2[i] ^= Input[i];
// return (c1||c2)
return ByteUtils.Concatenate(c1, c2);
}
/// <summary>
/// Find an error vector <c>E</c> over <c>GF(2)</c> from an input syndrome <c>S</c> over <c>GF(2^M)</c>
/// </summary>
///
/// <param name="SyndVec">The syndrome</param>
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="SqRootMatrix">The matrix for computing square roots in <c>(GF(2M))<sup>T</sup></c></param>
///
/// <returns>The error vector</returns>
public static GF2Vector SyndromeDecode(GF2Vector SyndVec, GF2mField Field, PolynomialGF2mSmallM Gp, PolynomialGF2mSmallM[] SqRootMatrix)
{
int n = 1 << Field.Degree;
// the error vector
GF2Vector errors = new GF2Vector(n);
// if the syndrome vector is zero, the error vector is also zero
if (!SyndVec.IsZero())
{
// convert syndrome vector to polynomial over GF(2^m)
PolynomialGF2mSmallM syndrome = new PolynomialGF2mSmallM(SyndVec.ToExtensionFieldVector(Field));
// compute T = syndrome^-1 mod gp
PolynomialGF2mSmallM t = syndrome.ModInverse(Gp);
// compute tau = sqRoot(T + X) mod gp
PolynomialGF2mSmallM tau = t.AddMonomial(1);
tau = tau.ModSquareRootMatrix(SqRootMatrix);
// compute polynomials a and b satisfying a + b*tau = 0 mod gp
PolynomialGF2mSmallM[] ab = tau.ModPolynomialToFracton(Gp);
// compute the polynomial a^2 + X*b^2
PolynomialGF2mSmallM a2 = ab[0].Multiply(ab[0]);
PolynomialGF2mSmallM b2 = ab[1].Multiply(ab[1]);
PolynomialGF2mSmallM xb2 = b2.MultWithMonomial(1);
PolynomialGF2mSmallM a2plusXb2 = a2.Add(xb2);
// normalize a^2 + X*b^2 to obtain the error locator polynomial
int headCoeff = a2plusXb2.Head;
int invHeadCoeff = Field.Inverse(headCoeff);
PolynomialGF2mSmallM elp = a2plusXb2.MultWithElement(invHeadCoeff);
// for all elements i of GF(2^m)
for (int i = 0; i < n; i++)
{
// evaluate the error locator polynomial at i
int z = elp.EvaluateAt(i);
// if polynomial evaluates to zero
if (z == 0)
{
// set the i-th coefficient of the error vector
errors.SetBit(i);
}
}
}
return errors;
}
/// <summary>
/// Multiply this vector with a permutation
/// </summary>
///
/// <param name="P">The permutation</param>
///
/// <returns>Returns <c>this*p = p*this</c></returns>
public override Vector Multiply(Permutation P)
{
int[] pVec = P.GetVector();
if (Length != pVec.Length)
throw new ArithmeticException("GF2Vector: Length mismatch!");
GF2Vector result = new GF2Vector(Length);
for (int i = 0; i < pVec.Length; i++)
{
int e = _elements[pVec[i] >> 5] & (1 << (pVec[i] & 0x1f));
if (e != 0)
result._elements[i >> 5] |= 1 << (i & 0x1f);
}
return result;
}
/// <summary>
/// Return a new vector consisting of the elements of this vector with the indices given by the set <c>SetJ</c>
/// </summary>
///
/// <param name="SetJ">The set of indices of elements to extract</param>
///
/// <returns>Return the new GF2Vector <c>[SetJ[0], SetJ[1], ..., SetJ[#SetJ-1]]</c></returns>
public GF2Vector ExtractVector(int[] SetJ)
{
int k = SetJ.Length;
if (SetJ[k - 1] > Length)
throw new ArithmeticException("invalid index set");
GF2Vector result = new GF2Vector(k);
for (int i = 0; i < k; i++)
{
int e = _elements[SetJ[i] >> 5] & (1 << (SetJ[i] & 0x1f));
if (e != 0)
result._elements[i >> 5] |= 1 << (i & 0x1f);
}
return result;
}
/// <summary>
/// Return a new vector consisting of the last <c>k</c> elements of this vector
/// </summary>
///
/// <param name="K">The number of elements to extract</param>
///
/// <returns>Returns a new GF2Vector consisting of the last <c>K</c> elements of this vector</returns>
public GF2Vector ExtractRightVector(int K)
{
if (K > base.Length)
throw new ArithmeticException("invalid length");
if (K == base.Length)
return new GF2Vector(this);
GF2Vector result = new GF2Vector(K);
int q = (base.Length - K) >> 5;
int r = (base.Length - K) & 0x1f;
int length = (K + 31) >> 5;
int ind = q;
// if words have to be shifted
if (r != 0)
{
// process all but last word
for (int i = 0; i < length - 1; i++)
result._elements[i] = (IntUtils.URShift(_elements[ind++], r)) | (_elements[ind] << (32 - r));
// process last word
result._elements[length - 1] = IntUtils.URShift(_elements[ind++], r);
if (ind < _elements.Length)
result._elements[length - 1] |= _elements[ind] << (32 - r);
}
else
{
// no shift necessary
Array.Copy(_elements, q, result._elements, 0, length);
}
return result;
}
/// <summary>
/// Copy constructor
/// </summary>
///
/// <param name="G">The GF2Vector to copy</param>
public GF2Vector(GF2Vector G)
{
this.Length = G.Length;
this._elements = IntUtils.DeepCopy(G._elements);
}
/// <summary>
/// Encode a number between 0 and (n|t) (binomial coefficient) into a binary vector of length n with weight t.
/// <para>The number is given as a byte array. Only the first s bits are used, where s = floor[log(n|t)].</para>
/// </summary>
///
/// <param name="N">The "n" integer</param>
/// <param name="T">The "t" integer</param>
/// <param name="M">The message as a byte array</param>
///
/// <returns>The encoded message as GF2Vector</returns>
public static GF2Vector Encode(int N, int T, byte[] M)
{
if (N < T)
throw new ArgumentException("n < t");
// compute the binomial c = (n|t)
BigInteger c = BigMath.Binomial(N, T);
// get the number encoded in m
BigInteger i = new BigInteger(1, M);
// compare
if (i.CompareTo(c) >= 0)
throw new ArgumentException("Encoded number too large.");
GF2Vector result = new GF2Vector(N);
int nn = N;
int tt = T;
for (int j = 0; j < N; j++)
{
c = c.Multiply(BigInteger.ValueOf(nn - tt)).Divide(BigInteger.ValueOf(nn));
nn--;
if (c.CompareTo(i) <= 0)
{
result.SetBit(j);
i = i.Subtract(c);
tt--;
if (nn == tt)
c = ONE;
else
c = (c.Multiply(BigInteger.ValueOf(tt + 1))).Divide(BigInteger.ValueOf(nn - tt));
}
}
return result;
}
/// <summary>
/// Copy constructor
/// </summary>
///
/// <param name="G">The GF2Vector to copy</param>
public GF2Vector(GF2Vector G)
{
this.Length = G.Length;
this._elements = IntUtils.DeepCopy(G._elements);
}