Exemple #1
0
            static SRP()
            {
                // initialize N
                {
                    NHex =
                        //512bit
                        //"D4C7F8A2B32C11B8FBA9581EC4BA4F1B04215642EF7355E37C0FC0443EF756EA2C6B8EEB755A1C723027663CAA265EF785B8FF6A9B35227A52D86633DBDFCA43";
                        //256bit
                        "EEAF0AB9ADB38DD69C33F80AFA8FC5E86072618775FF3C0B9EA2314C9C256576D674DF7496EA81D3383B4813D692C6E0E0D5D8E250B98BE48E495C1D6089DAD15DC7D7B46154D6B6CE8EF4AD69B15D4982559B297BCF1885C529F566660E57EC68EDBC3C05726CC02FD4CBF4976EAA9AFD5138FE8376435B9FC61D2FC0EB06E3".ToLowerInvariant();
                    N = new BigInteger(NHex, 16);
                    _nbits = N.bitCount();
                    Nminus1 = N - 1;

                //                    if (!N.isProbablePrime(80))
                //                    {
                //                        throw new Exception("Warning: N is not prime");
                //                    }
                //
                //                    if (!(Nminus1 / 2).isProbablePrime(80))
                //                    {
                //                        throw new Exception("Warning: (N-1)/2 is not prime");
                //                    }
                }

                // initialize g
                {
                    gHex = "2";
                    g = new BigInteger(gHex, 16);
                }

                // initialize k = H(N || g)
                {
                    BigInteger ktmp = new BigInteger(HHex(
                        (((NHex.Length & 1) == 0) ? "" : "0") + NHex +
                        new string('0', NHex.Length - gHex.Length) + gHex
                        ), 16);

                    k = (ktmp < N) ? ktmp : (ktmp % N);
                    kHex = k.ToString(16).ToLowerInvariant().TrimStart('0');
                }

                // initialize a, A
                {
                    a = new BigInteger();
                    a.genRandomBits(36);
                    A = g.modPow(a, N);

                    while (A.modInverse(N) == 0)
                    {
                        a = new BigInteger();
                        a.genRandomBits(36);
                        A = g.modPow(a, N);
                    }
                    Ahex = A.ToString(16).ToLowerInvariant().TrimStart('0');
                }
            }
        //***********************************************************************
        // Modulo Exponentiation
        //***********************************************************************

        public BigInteger modPow(BigInteger exp, BigInteger n)
        {
            if ((exp.data[maxLength - 1] & 0x80000000) != 0)
                throw (new ArithmeticException("Positive exponents only."));

            BigInteger resultNum = 1;
            BigInteger tempNum;
            bool thisNegative = false;

            if ((this.data[maxLength - 1] & 0x80000000) != 0)   // negative this
            {
                tempNum = -this % n;
                thisNegative = true;
            }
            else
                tempNum = this % n;  // ensures (tempNum * tempNum) < b^(2k)

            if ((n.data[maxLength - 1] & 0x80000000) != 0)   // negative n
                n = -n;

            // calculate constant = b^(2k) / m
            BigInteger constant = new BigInteger();

            int i = n.dataLength << 1;
            constant.data[i] = 0x00000001;
            constant.dataLength = i + 1;

            constant = constant / n;
            int totalBits = exp.bitCount();
            int count = 0;

            // perform squaring and multiply exponentiation
            for (int pos = 0; pos < exp.dataLength; pos++)
            {
                uint mask = 0x01;
                //Console.WriteLine("pos = " + pos);

                for (int index = 0; index < 32; index++)
                {
                    if ((exp.data[pos] & mask) != 0)
                        resultNum = BarrettReduction(resultNum * tempNum, n, constant);

                    mask <<= 1;

                    tempNum = BarrettReduction(tempNum * tempNum, n, constant);


                    if (tempNum.dataLength == 1 && tempNum.data[0] == 1)
                    {
                        if (thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
                            return -resultNum;
                        return resultNum;
                    }
                    count++;
                    if (count == totalBits)
                        break;
                }
            }

            if (thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
                return -resultNum;

            return resultNum;
        }
Exemple #3
0
            /// <summary>
            /// Respond to the server's challenge with a proof of password.
            /// </summary>
            /// <returns>The to challenge.</returns>
            /// <param name="password">Password.</param>
            /// <param name="identity">Identity.</param>
            /// <param name="salt">Salt.</param>
            /// <param name="Bhex">Bhex.</param>
            public static Meteor.ChallengeResponse RespondToChallenge(string password, string identity, string salt, string Bhex)
            {
                BigInteger B = new BigInteger (Bhex, 16);
                BigInteger u = new BigInteger (Hash (Ahex + Bhex), 16);
                BigInteger x = new BigInteger (Hash (salt + Hash (identity + ":" + password)),16);

                BigInteger kgx = k * (g.modPow (x, N));
                BigInteger aux = a + (u * x);
                S = (B - kgx).modPow (aux, N);
                string Shex = S.ToString(16).ToLowerInvariant().TrimStart('0');
                string M = Hash (Ahex + Bhex + Shex);
                HAMK = Hash (Ahex + M + Shex);

                return new Meteor.ChallengeResponse () {
                    M = M.ToLowerInvariant().TrimStart('0')
                };
            }
Exemple #4
0
            public static void AuthStep2(
                string vHex,
                string uHex,
                string AHex,
                string bHex,
                string BHex,
                out string m1serverHex,
                out string m2Hex)
            {
                BigInteger v = new BigInteger(vHex, 16);
                BigInteger u = new BigInteger(uHex, 16);
                BigInteger A = new BigInteger(AHex, 16);
                BigInteger b = new BigInteger(bHex, 16);
                   //BigInteger B = new BigInteger(BHex, 16); REMOVED WARNING JOE

                // S - common exponential value
                // S = (A * v^u) ^ b (mod N)
                BigInteger S = ((v.modPow(u, N) * A) % N).modPow(b, N);

                // K - the strong cryptographically session key
                // K = H(S)
                string KHex = HHex(S.ToHexString()).TrimStart('0');

                // m2 - expected client's proof as computed by the server
                m1serverHex = HHex(
                    AHex +
                    BHex +
                    KHex).TrimStart('0');

                // m2 - server's proof that it has the correct key
                m2Hex = HHex(
                    AHex +
                    m1serverHex +
                    KHex).TrimStart('0');
            }
        //***********************************************************************
        // Overloading of the NEGATE operator (2's complement)
        //***********************************************************************

        public static BigInteger operator -(BigInteger bi1)
        {
            // handle neg of zero separately since it'll cause an overflow
            // if we proceed.

            if (bi1.dataLength == 1 && bi1.data[0] == 0)
                return (new BigInteger());

            BigInteger result = new BigInteger(bi1);

            // 1's complement
            for (int i = 0; i < maxLength; i++)
                result.data[i] = (uint)(~(bi1.data[i]));

            // add one to result of 1's complement
            long val, carry = 1;
            int index = 0;

            while (carry != 0 && index < maxLength)
            {
                val = (long)(result.data[index]);
                val++;

                result.data[index] = (uint)(val & 0xFFFFFFFF);
                carry = val >> 32;

                index++;
            }

            if ((bi1.data[maxLength - 1] & 0x80000000) == (result.data[maxLength - 1] & 0x80000000))
                throw (new ArithmeticException("Overflow in negation.\n"));

            result.dataLength = maxLength;

            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;
            return result;
        }
        //***********************************************************************
        // Overloading of unary >> operators
        //***********************************************************************

        public static BigInteger operator >>(BigInteger bi1, int shiftVal)
        {
            BigInteger result = new BigInteger(bi1);
            result.dataLength = shiftRight(result.data, shiftVal);


            if ((bi1.data[maxLength - 1] & 0x80000000) != 0) // negative
            {
                for (int i = maxLength - 1; i >= result.dataLength; i--)
                    result.data[i] = 0xFFFFFFFF;

                uint mask = 0x80000000;
                for (int i = 0; i < 32; i++)
                {
                    if ((result.data[result.dataLength - 1] & mask) != 0)
                        break;

                    result.data[result.dataLength - 1] |= mask;
                    mask >>= 1;
                }
                result.dataLength = maxLength;
            }

            return result;
        }
        //***********************************************************************
        // Overloading of multiplication operator
        //***********************************************************************

        public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
        {
            int lastPos = maxLength - 1;
            bool bi1Neg = false, bi2Neg = false;

            // take the absolute value of the inputs
            try
            {
                if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
                {
                    bi1Neg = true; bi1 = -bi1;
                }
                if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
                {
                    bi2Neg = true; bi2 = -bi2;
                }
            }
            catch (Exception) { }

            BigInteger result = new BigInteger();

            // multiply the absolute values
            try
            {
                for (int i = 0; i < bi1.dataLength; i++)
                {
                    if (bi1.data[i] == 0) continue;

                    ulong mcarry = 0;
                    for (int j = 0, k = i; j < bi2.dataLength; j++, k++)
                    {
                        // k = i + j
                        ulong val = ((ulong)bi1.data[i] * (ulong)bi2.data[j]) +
                                     (ulong)result.data[k] + mcarry;

                        result.data[k] = (uint)(val & 0xFFFFFFFF);
                        mcarry = (val >> 32);
                    }

                    if (mcarry != 0)
                        result.data[i + bi2.dataLength] = (uint)mcarry;
                }
            }
            catch (Exception)
            {
                throw (new ArithmeticException("Multiplication overflow."));
            }


            result.dataLength = bi1.dataLength + bi2.dataLength;
            if (result.dataLength > maxLength)
                result.dataLength = maxLength;

            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;

            // overflow check (result is -ve)
            if ((result.data[lastPos] & 0x80000000) != 0)
            {
                if (bi1Neg != bi2Neg && result.data[lastPos] == 0x80000000)    // different sign
                {
                    // handle the special case where multiplication produces
                    // a max negative number in 2's complement.

                    if (result.dataLength == 1)
                        return result;
                    else
                    {
                        bool isMaxNeg = true;
                        for (int i = 0; i < result.dataLength - 1 && isMaxNeg; i++)
                        {
                            if (result.data[i] != 0)
                                isMaxNeg = false;
                        }

                        if (isMaxNeg)
                            return result;
                    }
                }

                throw (new ArithmeticException("Multiplication overflow."));
            }

            // if input has different signs, then result is -ve
            if (bi1Neg != bi2Neg)
                return -result;

            return result;
        }
        //***********************************************************************
        // Overloading of subtraction operator
        //***********************************************************************

        public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
        {
            BigInteger result = new BigInteger();

            result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

            long carryIn = 0;
            for (int i = 0; i < result.dataLength; i++)
            {
                long diff;

                diff = (long)bi1.data[i] - (long)bi2.data[i] - carryIn;
                result.data[i] = (uint)(diff & 0xFFFFFFFF);

                if (diff < 0)
                    carryIn = 1;
                else
                    carryIn = 0;
            }

            // roll over to negative
            if (carryIn != 0)
            {
                for (int i = result.dataLength; i < maxLength; i++)
                    result.data[i] = 0xFFFFFFFF;
                result.dataLength = maxLength;
            }

            // fixed in v1.03 to give correct datalength for a - (-b)
            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;

            // overflow check

            int lastPos = maxLength - 1;
            if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
               (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
            {
                throw (new ArithmeticException());
            }

            return result;
        }
        //***********************************************************************
        // Generates a positive BigInteger that is probably prime.
        //***********************************************************************

        public static BigInteger genPseudoPrime(int bits, int confidence)
        {
            BigInteger result = new BigInteger();
            bool done = false;

            while (!done)
            {
                result.genRandomBits(bits);
                result.data[0] |= 0x01;		// make it odd

                // prime test
                done = result.isProbablePrime(confidence);
            }
            return result;
        }
        //***********************************************************************
        // Computes the Jacobi Symbol for a and b.
        // Algorithm adapted from [3] and [4] with some optimizations
        //***********************************************************************

        public static int Jacobi(BigInteger a, BigInteger b)
        {
            // Jacobi defined only for odd integers
            if ((b.data[0] & 0x1) == 0)
                throw (new ArgumentException("Jacobi defined only for odd integers."));

            if (a >= b) a %= b;
            if (a.dataLength == 1 && a.data[0] == 0) return 0;  // a == 0
            if (a.dataLength == 1 && a.data[0] == 1) return 1;  // a == 1

            if (a < 0)
            {
                if ((((b - 1).data[0]) & 0x2) == 0)       //if( (((b-1) >> 1).data[0] & 0x1) == 0)
                    return Jacobi(-a, b);
                else
                    return -Jacobi(-a, b);
            }

            int e = 0;
            for (int index = 0; index < a.dataLength; index++)
            {
                uint mask = 0x01;

                for (int i = 0; i < 32; i++)
                {
                    if ((a.data[index] & mask) != 0)
                    {
                        index = a.dataLength;      // to break the outer loop
                        break;
                    }
                    mask <<= 1;
                    e++;
                }
            }

            BigInteger a1 = a >> e;

            int s = 1;
            if ((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5))
                s = -1;

            if ((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3)
                s = -s;

            if (a1.dataLength == 1 && a1.data[0] == 1)
                return s;
            else
                return (s * Jacobi(b % a1, a1));
        }
        private bool LucasStrongTestHelper(BigInteger thisVal)
        {
            // Do the test (selects D based on Selfridge)
            // Let D be the first element of the sequence
            // 5, -7, 9, -11, 13, ... for which J(D,n) = -1
            // Let P = 1, Q = (1-D) / 4

            long D = 5, sign = -1, dCount = 0;
            bool done = false;

            while (!done)
            {
                int Jresult = BigInteger.Jacobi(D, thisVal);

                if (Jresult == -1)
                    done = true;    // J(D, this) = 1
                else
                {
                    if (Jresult == 0 && Math.Abs(D) < thisVal)       // divisor found
                        return false;

                    if (dCount == 20)
                    {
                        // check for square
                        BigInteger root = thisVal.sqrt();
                        if (root * root == thisVal)
                            return false;
                    }

                    //Console.WriteLine(D);
                    D = (Math.Abs(D) + 2) * sign;
                    sign = -sign;
                }
                dCount++;
            }

            long Q = (1 - D) >> 2;

            /*
            Console.WriteLine("D = " + D);
            Console.WriteLine("Q = " + Q);
            Console.WriteLine("(n,D) = " + thisVal.gcd(D));
            Console.WriteLine("(n,Q) = " + thisVal.gcd(Q));
            Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal));
            */

            BigInteger p_add1 = thisVal + 1;
            int s = 0;

            for (int index = 0; index < p_add1.dataLength; index++)
            {
                uint mask = 0x01;

                for (int i = 0; i < 32; i++)
                {
                    if ((p_add1.data[index] & mask) != 0)
                    {
                        index = p_add1.dataLength;      // to break the outer loop
                        break;
                    }
                    mask <<= 1;
                    s++;
                }
            }

            BigInteger t = p_add1 >> s;

            // calculate constant = b^(2k) / m
            // for Barrett Reduction
            BigInteger constant = new BigInteger();

            int nLen = thisVal.dataLength << 1;
            constant.data[nLen] = 0x00000001;
            constant.dataLength = nLen + 1;

            constant = constant / thisVal;

            BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
            bool isPrime = false;

            if ((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
               (lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
            {
                // u(t) = 0 or V(t) = 0
                isPrime = true;
            }

            for (int i = 1; i < s; i++)
            {
                if (!isPrime)
                {
                    // doubling of index
                    lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant);
                    lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal;

                    //lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal;

                    if ((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
                        isPrime = true;
                }

                lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant);     //Q^k
            }


            if (isPrime)     // additional checks for composite numbers
            {
                // If n is prime and gcd(n, Q) == 1, then
                // Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n

                BigInteger g = thisVal.gcd(Q);
                if (g.dataLength == 1 && g.data[0] == 1)         // gcd(this, Q) == 1
                {
                    if ((lucas[2].data[maxLength - 1] & 0x80000000) != 0)
                        lucas[2] += thisVal;

                    BigInteger temp = (Q * BigInteger.Jacobi(Q, thisVal)) % thisVal;
                    if ((temp.data[maxLength - 1] & 0x80000000) != 0)
                        temp += thisVal;

                    if (lucas[2] != temp)
                        isPrime = false;
                }
            }

            return isPrime;
        }
        //***********************************************************************
        // Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
        //
        // p is probably prime if for any a < p (a is not multiple of p),
        // a^((p-1)/2) mod p = J(a, p)
        //
        // where J is the Jacobi symbol.
        //
        // Otherwise, p is composite.
        //
        // Returns
        // -------
        // True if "this" is a Euler pseudoprime to randomly chosen
        // bases.  The number of chosen bases is given by the "confidence"
        // parameter.
        //
        // False if "this" is definitely NOT prime.
        //
        //***********************************************************************

        public bool SolovayStrassenTest(int confidence)
        {
            BigInteger thisVal;
            if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
                thisVal = -this;
            else
                thisVal = this;

            if (thisVal.dataLength == 1)
            {
                // test small numbers
                if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
                    return false;
                else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
                    return true;
            }

            if ((thisVal.data[0] & 0x1) == 0)     // even numbers
                return false;


            int bits = thisVal.bitCount();
            BigInteger a = new BigInteger();
            BigInteger p_sub1 = thisVal - 1;
            BigInteger p_sub1_shift = p_sub1 >> 1;

            for (int round = 0; round < confidence; round++)
            {
                bool done = false;

                while (!done)		// generate a < n
                {
                    int testBits = 0;

                    // make sure "a" has at least 2 bits
                    while (testBits < 2)
						testBits = (int)(UnityEngine.Random.value * bits);

                    a.genRandomBits(testBits);

                    int byteLen = a.dataLength;

                    // make sure "a" is not 0
                    if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
                        done = true;
                }

                // check whether a factor exists (fix for version 1.03)
                BigInteger gcdTest = a.gcd(thisVal);
                if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                    return false;

                // calculate a^((p-1)/2) mod p

                BigInteger expResult = a.modPow(p_sub1_shift, thisVal);
                if (expResult == p_sub1)
                    expResult = -1;

                // calculate Jacobi symbol
                BigInteger jacob = Jacobi(a, thisVal);

                //Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
                //Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));

                // if they are different then it is not prime
                if (expResult != jacob)
                    return false;
            }

            return true;
        }
        //***********************************************************************
        // Probabilistic prime test based on Rabin-Miller's
        //
        // for any p > 0 with p - 1 = 2^s * t
        //
        // p is probably prime (strong pseudoprime) if for any a < p,
        // 1) a^t mod p = 1 or
        // 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
        //
        // Otherwise, p is composite.
        //
        // Returns
        // -------
        // True if "this" is a strong pseudoprime to randomly chosen
        // bases.  The number of chosen bases is given by the "confidence"
        // parameter.
        //
        // False if "this" is definitely NOT prime.
        //
        //***********************************************************************

        public bool RabinMillerTest(int confidence)
        {
            BigInteger thisVal;
            if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
                thisVal = -this;
            else
                thisVal = this;

            if (thisVal.dataLength == 1)
            {
                // test small numbers
                if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
                    return false;
                else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
                    return true;
            }

            if ((thisVal.data[0] & 0x1) == 0)     // even numbers
                return false;


            // calculate values of s and t
            BigInteger p_sub1 = thisVal - (new BigInteger(1));
            int s = 0;

            for (int index = 0; index < p_sub1.dataLength; index++)
            {
                uint mask = 0x01;

                for (int i = 0; i < 32; i++)
                {
                    if ((p_sub1.data[index] & mask) != 0)
                    {
                        index = p_sub1.dataLength;      // to break the outer loop
                        break;
                    }
                    mask <<= 1;
                    s++;
                }
            }

            BigInteger t = p_sub1 >> s;

            int bits = thisVal.bitCount();
            BigInteger a = new BigInteger();

			System.Random r = new System.Random ();

            for (int round = 0; round < confidence; round++)
            {
                bool done = false;

                while (!done)		// generate a < n
                {
                    int testBits = 0;

                    // make sure "a" has at least 2 bits
                    while (testBits < 2)
						testBits = (int)(r.NextDouble() * bits);

                    a.genRandomBits(testBits);

                    int byteLen = a.dataLength;

                    // make sure "a" is not 0
                    if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
                        done = true;
                }

                // check whether a factor exists (fix for version 1.03)
                BigInteger gcdTest = a.gcd(thisVal);
                if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
                    return false;

                BigInteger b = a.modPow(t, thisVal);

                /*
                Console.WriteLine("a = " + a.ToString(10));
                Console.WriteLine("b = " + b.ToString(10));
                Console.WriteLine("t = " + t.ToString(10));
                Console.WriteLine("s = " + s);
                */

                bool result = false;

                if (b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
                    result = true;

                for (int j = 0; result == false && j < s; j++)
                {
                    if (b == p_sub1)         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
                    {
                        result = true;
                        break;
                    }

                    b = (b * b) % thisVal;
                }

                if (result == false)
                    return false;
            }
            return true;
        }
        //***********************************************************************
        // Returns gcd(this, bi)
        //***********************************************************************

        public BigInteger gcd(BigInteger bi)
        {
            BigInteger x;
            BigInteger y;

            if ((data[maxLength - 1] & 0x80000000) != 0)     // negative
                x = -this;
            else
                x = this;

            if ((bi.data[maxLength - 1] & 0x80000000) != 0)     // negative
                y = -bi;
            else
                y = bi;

            BigInteger g = y;

            while (x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0))
            {
                g = x;
                x = y % x;
                y = g;
            }

            return g;
        }
        //***********************************************************************
        // Fast calculation of modular reduction using Barrett's reduction.
        // Requires x < b^(2k), where b is the base.  In this case, base is
        // 2^32 (uint).
        //
        // Reference [4]
        //***********************************************************************

        private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
        {
            int k = n.dataLength,
                kPlusOne = k + 1,
                kMinusOne = k - 1;

            BigInteger q1 = new BigInteger();

            // q1 = x / b^(k-1)
            for (int i = kMinusOne, j = 0; i < x.dataLength; i++, j++)
                q1.data[j] = x.data[i];
            q1.dataLength = x.dataLength - kMinusOne;
            if (q1.dataLength <= 0)
                q1.dataLength = 1;


            BigInteger q2 = q1 * constant;
            BigInteger q3 = new BigInteger();

            // q3 = q2 / b^(k+1)
            for (int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++)
                q3.data[j] = q2.data[i];
            q3.dataLength = q2.dataLength - kPlusOne;
            if (q3.dataLength <= 0)
                q3.dataLength = 1;


            // r1 = x mod b^(k+1)
            // i.e. keep the lowest (k+1) words
            BigInteger r1 = new BigInteger();
            int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength;
            for (int i = 0; i < lengthToCopy; i++)
                r1.data[i] = x.data[i];
            r1.dataLength = lengthToCopy;


            // r2 = (q3 * n) mod b^(k+1)
            // partial multiplication of q3 and n

            BigInteger r2 = new BigInteger();
            for (int i = 0; i < q3.dataLength; i++)
            {
                if (q3.data[i] == 0) continue;

                ulong mcarry = 0;
                int t = i;
                for (int j = 0; j < n.dataLength && t < kPlusOne; j++, t++)
                {
                    // t = i + j
                    ulong val = ((ulong)q3.data[i] * (ulong)n.data[j]) +
                                 (ulong)r2.data[t] + mcarry;

                    r2.data[t] = (uint)(val & 0xFFFFFFFF);
                    mcarry = (val >> 32);
                }

                if (t < kPlusOne)
                    r2.data[t] = (uint)mcarry;
            }
            r2.dataLength = kPlusOne;
            while (r2.dataLength > 1 && r2.data[r2.dataLength - 1] == 0)
                r2.dataLength--;

            r1 -= r2;
            if ((r1.data[maxLength - 1] & 0x80000000) != 0)        // negative
            {
                BigInteger val = new BigInteger();
                val.data[kPlusOne] = 0x00000001;
                val.dataLength = kPlusOne + 1;
                r1 += val;
            }

            while (r1 >= n)
                r1 -= n;

            return r1;
        }
        //***********************************************************************
        // Overloading of addition operator
        //***********************************************************************

        public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
        {
            BigInteger result = new BigInteger();

            result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

            long carry = 0;
            for (int i = 0; i < result.dataLength; i++)
            {
                long sum = (long)bi1.data[i] + (long)bi2.data[i] + carry;
                carry = sum >> 32;
                result.data[i] = (uint)(sum & 0xFFFFFFFF);
            }

            if (carry != 0 && result.dataLength < maxLength)
            {
                result.data[result.dataLength] = (uint)(carry);
                result.dataLength++;
            }

            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;


            // overflow check
            int lastPos = maxLength - 1;
            if ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
               (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
            {
                throw (new ArithmeticException());
            }

            return result;
        }
        //***********************************************************************
        // Overloading of the unary ++ operator
        //***********************************************************************

        public static BigInteger operator ++(BigInteger bi1)
        {
            BigInteger result = new BigInteger(bi1);

            long val, carry = 1;
            int index = 0;

            while (carry != 0 && index < maxLength)
            {
                val = (long)(result.data[index]);
                val++;

                result.data[index] = (uint)(val & 0xFFFFFFFF);
                carry = val >> 32;

                index++;
            }

            if (index > result.dataLength)
                result.dataLength = index;
            else
            {
                while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                    result.dataLength--;
            }

            // overflow check
            int lastPos = maxLength - 1;

            // overflow if initial value was +ve but ++ caused a sign
            // change to negative.

            if ((bi1.data[lastPos] & 0x80000000) == 0 &&
               (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
            {
                throw (new ArithmeticException("Overflow in ++."));
            }
            return result;
        }
        //***********************************************************************
        // Generates a random number with the specified number of bits such
        // that gcd(number, this) = 1
        //***********************************************************************

        public BigInteger genCoPrime(int bits)
        {
            bool done = false;
            BigInteger result = new BigInteger();

            while (!done)
            {
                result.genRandomBits(bits);
                //Console.WriteLine(result.ToString(16));

                // gcd test
                BigInteger g = result.gcd(this);
                if (g.dataLength == 1 && g.data[0] == 1)
                    done = true;
            }

            return result;
        }
        //***********************************************************************
        // Overloading of the unary -- operator
        //***********************************************************************

        public static BigInteger operator --(BigInteger bi1)
        {
            BigInteger result = new BigInteger(bi1);

            long val;
            bool carryIn = true;
            int index = 0;

            while (carryIn && index < maxLength)
            {
                val = (long)(result.data[index]);
                val--;

                result.data[index] = (uint)(val & 0xFFFFFFFF);

                if (val >= 0)
                    carryIn = false;

                index++;
            }

            if (index > result.dataLength)
                result.dataLength = index;

            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;

            // overflow check
            int lastPos = maxLength - 1;

            // overflow if initial value was -ve but -- caused a sign
            // change to positive.

            if ((bi1.data[lastPos] & 0x80000000) != 0 &&
               (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
            {
                throw (new ArithmeticException("Underflow in --."));
            }

            return result;
        }
        //***********************************************************************
        // Returns the modulo inverse of this.  Throws ArithmeticException if
        // the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
        //***********************************************************************

        public BigInteger modInverse(BigInteger modulus)
        {
            BigInteger[] p = { 0, 1 };
            BigInteger[] q = new BigInteger[2];    // quotients
            BigInteger[] r = { 0, 0 };             // remainders

            int step = 0;

            BigInteger a = modulus;
            BigInteger b = this;

            while (b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0))
            {
                BigInteger quotient = new BigInteger();
                BigInteger remainder = new BigInteger();

                if (step > 1)
                {
                    BigInteger pval = (p[0] - (p[1] * q[0])) % modulus;
                    p[0] = p[1];
                    p[1] = pval;
                }

                if (b.dataLength == 1)
                    singleByteDivide(a, b, quotient, remainder);
                else
                    multiByteDivide(a, b, quotient, remainder);

                /*
                Console.WriteLine(quotient.dataLength);
                Console.WriteLine("{0} = {1}({2}) + {3}  p = {4}", a.ToString(10),
                                  b.ToString(10), quotient.ToString(10), remainder.ToString(10),
                                  p[1].ToString(10));
                */

                q[0] = q[1];
                r[0] = r[1];
                q[1] = quotient; r[1] = remainder;

                a = b;
                b = remainder;

                step++;
            }

            if (r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1))
                throw (new ArithmeticException("No inverse!"));

            BigInteger result = ((p[0] - (p[1] * q[0])) % modulus);

            if ((result.data[maxLength - 1] & 0x80000000) != 0)
                result += modulus;  // get the least positive modulus

            return result;
        }
        //***********************************************************************
        // Overloading of unary << operators
        //***********************************************************************

        public static BigInteger operator <<(BigInteger bi1, int shiftVal)
        {
            BigInteger result = new BigInteger(bi1);
            result.dataLength = shiftLeft(result.data, shiftVal);

            return result;
        }
        //***********************************************************************
        // Constructor (Default value provided by BigInteger)
        //***********************************************************************

        public BigInteger(BigInteger bi)
        {
            data = new uint[maxLength];

            dataLength = bi.dataLength;

            for (int i = 0; i < dataLength; i++)
                data[i] = bi.data[i];
        }
        //***********************************************************************
        // Overloading of the NOT operator (1's complement)
        //***********************************************************************

        public static BigInteger operator ~(BigInteger bi1)
        {
            BigInteger result = new BigInteger(bi1);

            for (int i = 0; i < maxLength; i++)
                result.data[i] = (uint)(~(bi1.data[i]));

            result.dataLength = maxLength;

            while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
                result.dataLength--;

            return result;
        }
        //***********************************************************************
        // Returns a value that is equivalent to the integer square root
        // of the BigInteger.
        //
        // The integer square root of "this" is defined as the largest integer n
        // such that (n * n) <= this
        //
        //***********************************************************************

        public BigInteger sqrt()
        {
            uint numBits = (uint)this.bitCount();

            if ((numBits & 0x1) != 0)        // odd number of bits
                numBits = (numBits >> 1) + 1;
            else
                numBits = (numBits >> 1);

            uint bytePos = numBits >> 5;
            byte bitPos = (byte)(numBits & 0x1F);

            uint mask;

            BigInteger result = new BigInteger();
            if (bitPos == 0)
                mask = 0x80000000;
            else
            {
                mask = (uint)1 << bitPos;
                bytePos++;
            }
            result.dataLength = (int)bytePos;

            for (int i = (int)bytePos - 1; i >= 0; i--)
            {
                while (mask != 0)
                {
                    // guess
                    result.data[i] ^= mask;

                    // undo the guess if its square is larger than this
                    if ((result * result) > this)
                        result.data[i] ^= mask;

                    mask >>= 1;
                }
                mask = 0x80000000;
            }
            return result;
        }
Exemple #25
0
            public static void AuthStep1(
                string vHex,
                string AHex,
                out string bHex,
                out string BHex,
                out string uHex)
            {
                BigInteger v = new BigInteger(vHex, 16);
                //BigInteger A = new BigInteger(AHex, 16); REMOVED WARNING

                BigInteger b;
                // b - ephemeral private key
                // b = random between 2 and N-1
                {
                    b = new BigInteger();
                    //[TODO] perhaps here use a better random generator
                    b.genRandomBits(_nbits);

                    if (b >= N)
                    {
                        b = b % Nminus1;
                    }
                    if (b < 2)
                    {
                        b = 2;
                    }
                }
                bHex = b.ToHexString();

                // B = public key
                // B = kv + g^b (mod N)
                BigInteger B = (v * k + g.modPow(b, N)) % N;
                BHex = B.ToHexString();

                BigInteger u;
                // u - scrambling parameter
                // u = H (A || B)
                {
                    int nlen = 2 * ((_nbits + 7) >> 3);

                    BigInteger utmp = new BigInteger(HHex(
                        new string('0', nlen - AHex.Length) + AHex +
                        new string('0', nlen - BHex.Length) + BHex
                        ), 16);

                    u = (utmp < N) ? utmp : (utmp % Nminus1);
                }

                uHex = u.ToHexString();
            }
        //***********************************************************************
        // Returns the k_th number in the Lucas Sequence reduced modulo n.
        //
        // Uses index doubling to speed up the process.  For example, to calculate V(k),
        // we maintain two numbers in the sequence V(n) and V(n+1).
        //
        // To obtain V(2n), we use the identity
        //      V(2n) = (V(n) * V(n)) - (2 * Q^n)
        // To obtain V(2n+1), we first write it as
        //      V(2n+1) = V((n+1) + n)
        // and use the identity
        //      V(m+n) = V(m) * V(n) - Q * V(m-n)
        // Hence,
        //      V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n)
        //                   = V(n+1) * V(n) - Q^n * V(1)
        //                   = V(n+1) * V(n) - Q^n * P
        //
        // We use k in its binary expansion and perform index doubling for each
        // bit position.  For each bit position that is set, we perform an
        // index doubling followed by an index addition.  This means that for V(n),
        // we need to update it to V(2n+1).  For V(n+1), we need to update it to
        // V((2n+1)+1) = V(2*(n+1))
        //
        // This function returns
        // [0] = U(k)
        // [1] = V(k)
        // [2] = Q^n
        //
        // Where U(0) = 0 % n, U(1) = 1 % n
        //       V(0) = 2 % n, V(1) = P % n
        //***********************************************************************

        public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
                                                 BigInteger k, BigInteger n)
        {
            if (k.dataLength == 1 && k.data[0] == 0)
            {
                BigInteger[] result = new BigInteger[3];

                result[0] = 0; result[1] = 2 % n; result[2] = 1 % n;
                return result;
            }

            // calculate constant = b^(2k) / m
            // for Barrett Reduction
            BigInteger constant = new BigInteger();

            int nLen = n.dataLength << 1;
            constant.data[nLen] = 0x00000001;
            constant.dataLength = nLen + 1;

            constant = constant / n;

            // calculate values of s and t
            int s = 0;

            for (int index = 0; index < k.dataLength; index++)
            {
                uint mask = 0x01;

                for (int i = 0; i < 32; i++)
                {
                    if ((k.data[index] & mask) != 0)
                    {
                        index = k.dataLength;      // to break the outer loop
                        break;
                    }
                    mask <<= 1;
                    s++;
                }
            }

            BigInteger t = k >> s;

            //Console.WriteLine("s = " + s + " t = " + t);
            return LucasSequenceHelper(P, Q, t, n, constant, s);
        }
Exemple #27
0
            /// <summary>
            /// Generate a new SRP verifier. Password is the plaintext password.
            /// </summary>
            /// <returns>The verifier.</returns>
            /// <param name="password">Password.</param>
            //original: (didn't work with 3.5 .Net compiler and was annoying to keep switching to 4.0
            //public static Meteor.Verifier GenerateVerifier(string password, string identity = null, string salt = null)
            public static Meteor.Verifier GenerateVerifier(string password, string identity, string salt)
            {
                if (identity == null)
                {
                    BigInteger i = new BigInteger ();
                    i.genRandomBits (36);
                    identity = i.ToString(16).ToLowerInvariant().TrimStart('0');
                }

                if (salt == null)
                {
                    BigInteger s = new BigInteger ();
                    s.genRandomBits (36);
                    salt = s.ToString(16).ToLowerInvariant().TrimStart('0');
                }

                string x = Hash (salt + Hash (identity + ":" + password));

                BigInteger xi = new BigInteger (x, 16);
                BigInteger v = g.modPow (xi, N);

                return new Meteor.Verifier () {
                    identity = identity,
                    salt = salt,
                    verifier = v.ToString(16).ToLowerInvariant().TrimStart('0')
                };
            }
        //***********************************************************************
        // Performs the calculation of the kth term in the Lucas Sequence.
        // For details of the algorithm, see reference [9].
        //
        // k must be odd.  i.e LSB == 1
        //***********************************************************************

        private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q,
                                                        BigInteger k, BigInteger n,
                                                        BigInteger constant, int s)
        {
            BigInteger[] result = new BigInteger[3];

            if ((k.data[0] & 0x00000001) == 0)
                throw (new ArgumentException("Argument k must be odd."));

            int numbits = k.bitCount();
            uint mask = (uint)0x1 << ((numbits & 0x1F) - 1);

            // v = v0, v1 = v1, u1 = u1, Q_k = Q^0

            BigInteger v = 2 % n, Q_k = 1 % n,
                       v1 = P % n, u1 = Q_k;
            bool flag = true;

            for (int i = k.dataLength - 1; i >= 0; i--)     // iterate on the binary expansion of k
            {
                //Console.WriteLine("round");
                while (mask != 0)
                {
                    if (i == 0 && mask == 0x00000001)        // last bit
                        break;

                    if ((k.data[i] & mask) != 0)             // bit is set
                    {
                        // index doubling with addition

                        u1 = (u1 * v1) % n;

                        v = ((v * v1) - (P * Q_k)) % n;
                        v1 = n.BarrettReduction(v1 * v1, n, constant);
                        v1 = (v1 - ((Q_k * Q) << 1)) % n;

                        if (flag)
                            flag = false;
                        else
                            Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

                        Q_k = (Q_k * Q) % n;
                    }
                    else
                    {
                        // index doubling
                        u1 = ((u1 * v) - Q_k) % n;

                        v1 = ((v * v1) - (P * Q_k)) % n;
                        v = n.BarrettReduction(v * v, n, constant);
                        v = (v - (Q_k << 1)) % n;

                        if (flag)
                        {
                            Q_k = Q % n;
                            flag = false;
                        }
                        else
                            Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
                    }

                    mask >>= 1;
                }
                mask = 0x80000000;
            }

            // at this point u1 = u(n+1) and v = v(n)
            // since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1)

            u1 = ((u1 * v) - Q_k) % n;
            v = ((v * v1) - (P * Q_k)) % n;
            if (flag)
                flag = false;
            else
                Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

            Q_k = (Q_k * Q) % n;


            for (int i = 0; i < s; i++)
            {
                // index doubling
                u1 = (u1 * v) % n;
                v = ((v * v) - (Q_k << 1)) % n;

                if (flag)
                {
                    Q_k = Q % n;
                    flag = false;
                }
                else
                    Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
            }

            result[0] = u1;
            result[1] = v;
            result[2] = Q_k;

            return result;
        }
        //***********************************************************************
        // Constructor (Default value provided by a string of digits of the
        //              specified base)
        //
        // Example (base 10)
        // -----------------
        // To initialize "a" with the default value of 1234 in base 10
        //      BigInteger a = new BigInteger("1234", 10)
        //
        // To initialize "a" with the default value of -1234
        //      BigInteger a = new BigInteger("-1234", 10)
        //
        // Example (base 16)
        // -----------------
        // To initialize "a" with the default value of 0x1D4F in base 16
        //      BigInteger a = new BigInteger("1D4F", 16)
        //
        // To initialize "a" with the default value of -0x1D4F
        //      BigInteger a = new BigInteger("-1D4F", 16)
        //
        // Note that string values are specified in the <sign><magnitude>
        // format.
        //
        //***********************************************************************

        public BigInteger(string value, int radix)
        {
            BigInteger multiplier = new BigInteger(1);
            BigInteger result = new BigInteger();
            value = (value.ToUpperInvariant()).Trim();
            int limit = 0;

            if (value[0] == '-')
                limit = 1;

            for (int i = value.Length - 1; i >= limit; i--)
            {
                int posVal = (int)value[i];

                if (posVal >= '0' && posVal <= '9')
                    posVal -= '0';
                else if (posVal >= 'A' && posVal <= 'Z')
                    posVal = (posVal - 'A') + 10;
                else
                    posVal = 9999999;       // arbitrary large


                if (posVal >= radix)
                    throw (new ArithmeticException("Invalid string in constructor."));
                else
                {
                    if (value[0] == '-')
                        posVal = -posVal;

                    result = result + (multiplier * posVal);

                    if ((i - 1) >= limit)
                        multiplier = multiplier * radix;
                }
            }

            if (value[0] == '-')     // negative values
            {
                if ((result.data[maxLength - 1] & 0x80000000) == 0)
                    throw (new ArithmeticException("Negative underflow in constructor."));
            }
            else    // positive values
            {
                if ((result.data[maxLength - 1] & 0x80000000) != 0)
                    throw (new ArithmeticException("Positive overflow in constructor."));
            }

            data = new uint[maxLength];
            for (int i = 0; i < result.dataLength; i++)
                data[i] = result.data[i];

            dataLength = result.dataLength;
        }
        //***********************************************************************
        // Returns a string representing the BigInteger in sign-and-magnitude
        // format in the specified radix.
        //
        // Example
        // -------
        // If the value of BigInteger is -255 in base 10, then
        // ToString(16) returns "-FF"
        //
        //***********************************************************************

        public string ToString(int radix)
        {
            if (radix < 2 || radix > 36)
                throw (new ArgumentException("Radix must be >= 2 and <= 36"));

            string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
            string result = "";

            BigInteger a = this;

            bool negative = false;
            if ((a.data[maxLength - 1] & 0x80000000) != 0)
            {
                negative = true;
                try
                {
                    a = -a;
                }
                catch (Exception) { }
            }

            BigInteger quotient = new BigInteger();
            BigInteger remainder = new BigInteger();
            BigInteger biRadix = new BigInteger(radix);

            if (a.dataLength == 1 && a.data[0] == 0)
                result = "0";
            else
            {
                while (a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0))
                {
                    singleByteDivide(a, biRadix, quotient, remainder);

                    if (remainder.data[0] < 10)
                        result = remainder.data[0] + result;
                    else
                        result = charSet[(int)remainder.data[0] - 10] + result;

                    a = quotient;
                }
                if (negative)
                    result = "-" + result;
            }

            return result;
        }