Exemplo n.º 1
0
        /// <summary>
        /// Modulo Exponentiation
        /// </summary>
        /// <param name="exp"></param>
        /// <param name="n"></param>
        /// <returns></returns>
        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 ((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
            var 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;
        }
Exemplo n.º 2
0
        /// <summary>
        /// Overloading of unary operators
        /// </summary>
        /// <param name="bi1"></param>
        /// <param name="shiftVal"></param>
        /// <returns></returns>
        public static BigInteger operator <<(BigInteger bi1, int shiftVal)
        {
            var result = new BigInteger(bi1);
            result.dataLength = shiftLeft(result.data, shiftVal);

            return result;
        }
Exemplo n.º 3
0
        /// <summary>
        ///  Overloading of the NOT operator (1's complement)
        /// </summary>
        /// <param name="bi1"></param>
        /// <returns></returns>
        public static BigInteger operator ~(BigInteger bi1)
        {
            var result = new BigInteger(bi1);

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

            result.dataLength = maxLength;

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

            return result;
        }
Exemplo n.º 4
0
        /// <summary>
        /// Overloading of the unary ++ operator
        /// </summary>
        /// <param name="bi1"></param>
        /// <returns></returns>
        public static BigInteger operator ++(BigInteger bi1)
        {
            var result = new BigInteger(bi1);

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

            while (carry != 0 && index < maxLength)
            {
                val = (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;
        }
Exemplo n.º 5
0
        /// <summary>
        ///  Overloading of the unary -- operator
        /// </summary>
        /// <param name="bi1"></param>
        /// <returns></returns>
        public static BigInteger operator --(BigInteger bi1)
        {
            var result = new BigInteger(bi1);

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

            while (carryIn && index < maxLength)
            {
                val = (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;
        }
Exemplo n.º 6
0
        /// <summary>
        ///  Tests the correct implementation of the modulo exponential function
        /// using RSA encryption and decryption (using pre-computed encryption and
        /// decryption keys).
        /// </summary>
        /// <param name="rounds"></param>
        public static void RSATest(int rounds)
        {
            var rand = new Random(1);
            var val = new byte[64];

            // private and public key
            var bi_e =
                new BigInteger(
                    "a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7",
                    16);
            var bi_d =
                new BigInteger(
                    "4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7",
                    16);
            var bi_n =
                new BigInteger(
                    "e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f",
                    16);

            Console.WriteLine("e =\n" + bi_e.ToString(10));
            Console.WriteLine("\nd =\n" + bi_d.ToString(10));
            Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

            for (int count = 0; count < rounds; count++)
            {
                // generate data of random length
                int t1 = 0;
                while (t1 == 0)
                    t1 = (int) (rand.NextDouble()*65);

                bool done = false;
                while (!done)
                {
                    for (int i = 0; i < 64; i++)
                    {
                        if (i < t1)
                            val[i] = (byte) (rand.NextDouble()*256);
                        else
                            val[i] = 0;

                        if (val[i] != 0)
                            done = true;
                    }
                }

                while (val[0] == 0)
                    val[0] = (byte) (rand.NextDouble()*256);

                Console.Write("Round = " + count);

                // encrypt and decrypt data
                var bi_data = new BigInteger(val, t1);
                BigInteger bi_encrypted = bi_data.ModPow(bi_e, bi_n);
                BigInteger bi_decrypted = bi_encrypted.ModPow(bi_d, bi_n);

                // compare
                if (bi_decrypted != bi_data)
                {
                    Console.WriteLine("\nError at round " + count);
                    Console.WriteLine(bi_data + "\n");
                    return;
                }
                Console.WriteLine(" <PASSED>.");
            }
        }
Exemplo n.º 7
0
        /// <summary>
        /// Tests the correct implementation of sqrt() method.
        /// </summary>
        /// <param name="rounds"></param>
        public static void SqrtTest(int rounds)
        {
            var rand = new Random();
            for (int count = 0; count < rounds; count++)
            {
                // generate data of random length
                int t1 = 0;
                while (t1 == 0)
                    t1 = (int) (rand.NextDouble()*1024);

                Console.Write("Round = " + count);

                var a = new BigInteger();
                a.genRandomBits(t1, rand);

                BigInteger b = a.sqrt();
                BigInteger c = (b + 1)*(b + 1);

                // check that b is the largest integer such that b*b <= a
                if (c <= a)
                {
                    Console.WriteLine("\nError at round " + count);
                    Console.WriteLine(a + "\n");
                    return;
                }
                Console.WriteLine(" <PASSED>.");
            }
        }
Exemplo n.º 8
0
        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 = 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
            var 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*Jacobi(Q, thisVal))%thisVal;
                    if ((temp.data[maxLength - 1] & 0x80000000) != 0)
                        temp += thisVal;

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

            return isPrime;
        }
Exemplo n.º 9
0
        /// <summary>
        /// Constructor (Default value provided by BigInteger)
        /// </summary>
        /// <param name="bi"></param>
        public BigInteger(BigInteger bi)
        {
            data = new uint[maxLength];

            dataLength = bi.dataLength;

            for (int i = 0; i < dataLength; i++)
                data[i] = bi.data[i];
        }
Exemplo n.º 10
0
        /// <summary>
        /// Probabilistic prime test based on Rabin-Miller's
        /// <![CDATA[
        /// 
        /// 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.
        /// ]]>
        /// </summary>
        /// <param name="confidence"></param>
        /// <returns></returns>
        public bool RabinMillerTest(int confidence)
        {
            BigInteger thisVal;
            if ((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();
            var a = new BigInteger();
            var rand = new 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) (rand.NextDouble()*bits);

                    a.genRandomBits(testBits, rand);

                    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;
        }
Exemplo n.º 11
0
        /// <summary>
        /// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
        /// <remarks>
        /// <![CDATA[
        /// 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.
        /// ]]></remarks>
        /// </summary>
        /// <param name="confidence"></param>
        /// <returns></returns>
        public bool SolovayStrassenTest(int confidence)
        {
            BigInteger thisVal;
            if ((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();
            var a = new BigInteger();
            BigInteger p_sub1 = thisVal - 1;
            BigInteger p_sub1_shift = p_sub1 >> 1;

            var rand = new 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) (rand.NextDouble()*bits);

                    a.genRandomBits(testBits, rand);

                    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;
        }
Exemplo n.º 12
0
        //***********************************************************************
        // Probabilistic prime test based on Fermat's little theorem
        //

        //
        //***********************************************************************
        /// <summary>
        /// Probabilistic prime test based on Fermat's little theorem
        /// <example>
        /// <![CDATA[
        /// 
        /// for any a < p (p does not divide a) if
        ///      a^(p-1) mod p != 1 then p is not prime.
        ///
        /// Otherwise, p is probably prime (pseudoprime to the chosen base).
        ///
        /// Returns
        /// -------
        /// True if "this" is a pseudoprime to randomly chosen
        /// bases.  The number of chosen bases is given by the "confidence"
        /// parameter.
        ///
        /// False if "this" is definitely NOT prime.
        ///
        /// Note - this method is fast but fails for Carmichael numbers except
        /// when the randomly chosen base is a factor of the number.
        /// 
        /// ]]>
        /// </example>
        /// </summary>
        /// <param name="confidence"></param>
        /// <returns></returns>
        public bool FermatLittleTest(int confidence)
        {
            BigInteger thisVal;
            if ((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();
            var a = new BigInteger();
            BigInteger p_sub1 = thisVal - (new BigInteger(1));
            var rand = new 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) (rand.NextDouble()*bits);

                    a.genRandomBits(testBits, rand);

                    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) mod p
                BigInteger expResult = a.ModPow(p_sub1, thisVal);

                int resultLen = expResult.dataLength;

                // is NOT prime is a^(p-1) mod p != 1

                if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
                {
                    //Console.WriteLine("a = " + a.ToString());
                    return false;
                }
            }

            return true;
        }
Exemplo n.º 13
0
        /// <summary>
        /// Returns gcd(this, bi)
        /// </summary>
        /// <param name="bi"></param>
        /// <returns></returns>
        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;
        }
Exemplo n.º 14
0
        //***********************************************************************
        // 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;

            var 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;
            var 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
            var 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

            var 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 = (q3.data[i]*(ulong) n.data[j]) +
                                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
            {
                var val = new BigInteger();
                val.data[kPlusOne] = 0x00000001;
                val.dataLength = kPlusOne + 1;
                r1 += val;
            }

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

            return r1;
        }
Exemplo n.º 15
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)
        {
            var 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;
        }
Exemplo n.º 16
0
        /// <summary>
        /// Computes the Jacobi Symbol for a and b.
        /// Algorithm adapted from [3] and [4] with some optimizations
        /// </summary>
        /// <param name="a"></param>
        /// <param name="b"></param>
        /// <returns></returns>
        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));
        }
Exemplo n.º 17
0
        /// <summary>
        /// Tests the correct implementation of the /, %, * and + operators
        /// </summary>
        /// <param name="rounds"></param>
        public static void MulDivTest(int rounds)
        {
            var rand = new Random();
            var val = new byte[64];
            var val2 = new byte[64];

            for (int count = 0; count < rounds; count++)
            {
                // generate 2 numbers of random length
                int t1 = 0;
                while (t1 == 0)
                    t1 = (int) (rand.NextDouble()*65);

                int t2 = 0;
                while (t2 == 0)
                    t2 = (int) (rand.NextDouble()*65);

                bool done = false;
                while (!done)
                {
                    for (int i = 0; i < 64; i++)
                    {
                        if (i < t1)
                            val[i] = (byte) (rand.NextDouble()*256);
                        else
                            val[i] = 0;

                        if (val[i] != 0)
                            done = true;
                    }
                }

                done = false;
                while (!done)
                {
                    for (int i = 0; i < 64; i++)
                    {
                        if (i < t2)
                            val2[i] = (byte) (rand.NextDouble()*256);
                        else
                            val2[i] = 0;

                        if (val2[i] != 0)
                            done = true;
                    }
                }

                while (val[0] == 0)
                    val[0] = (byte) (rand.NextDouble()*256);
                while (val2[0] == 0)
                    val2[0] = (byte) (rand.NextDouble()*256);

                Console.WriteLine(count);
                var bn1 = new BigInteger(val, t1);
                var bn2 = new BigInteger(val2, t2);


                // Determine the quotient and remainder by dividing
                // the first number by the second.

                BigInteger bn3 = bn1/bn2;
                BigInteger bn4 = bn1%bn2;

                // Recalculate the number
                BigInteger bn5 = (bn3*bn2) + bn4;

                // Make sure they're the same
                if (bn5 != bn1)
                {
                    Console.WriteLine("Error at " + count);
                    Console.WriteLine(bn1 + "\n");
                    Console.WriteLine(bn2 + "\n");
                    Console.WriteLine(bn3 + "\n");
                    Console.WriteLine(bn4 + "\n");
                    Console.WriteLine(bn5 + "\n");
                    return;
                }
            }
        }
Exemplo n.º 18
0
        /// <summary>
        ///  Generates a positive BigInteger that is probably prime.
        /// </summary>
        /// <param name="bits"></param>
        /// <param name="confidence"></param>
        /// <param name="rand"></param>
        /// <returns></returns>
        public static BigInteger genPseudoPrime(int bits, int confidence, Random rand)
        {
            var result = new BigInteger();
            bool done = false;

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

                // prime test
                done = result.isProbablePrime(confidence);
            }
            return result;
        }
Exemplo n.º 19
0
        /// <summary>
        /// Tests the correct implementation of the modulo exponential and
        /// inverse modulo functions using RSA encryption and decryption.  The two
        /// pseudoprimes p and q are fixed, but the two RSA keys are generated
        /// for each round of testing.
        /// </summary>
        /// <param name="rounds"></param>
        public static void RSATest2(int rounds)
        {
            var rand = new Random();
            var val = new byte[64];

            byte[] pseudoPrime1 =
                {
                    0x85, 0x84, 0x64, 0xFD, 0x70, 0x6A,
                    0x9F, 0xF0, 0x94, 0x0C, 0x3E, 0x2C,
                    0x74, 0x34, 0x05, 0xC9, 0x55, 0xB3,
                    0x85, 0x32, 0x98, 0x71, 0xF9, 0x41,
                    0x21, 0x5F, 0x02, 0x9E, 0xEA, 0x56,
                    0x8D, 0x8C, 0x44, 0xCC, 0xEE, 0xEE,
                    0x3D, 0x2C, 0x9D, 0x2C, 0x12, 0x41,
                    0x1E, 0xF1, 0xC5, 0x32, 0xC3, 0xAA,
                    0x31, 0x4A, 0x52, 0xD8, 0xE8, 0xAF,
                    0x42, 0xF4, 0x72, 0xA1, 0x2A, 0x0D,
                    0x97, 0xB1, 0x31, 0xB3,
                };

            byte[] pseudoPrime2 =
                {
                    0x99, 0x98, 0xCA, 0xB8, 0x5E, 0xD7,
                    0xE5, 0xDC, 0x28, 0x5C, 0x6F, 0x0E,
                    0x15, 0x09, 0x59, 0x6E, 0x84, 0xF3,
                    0x81, 0xCD, 0xDE, 0x42, 0xDC, 0x93,
                    0xC2, 0x7A, 0x62, 0xAC, 0x6C, 0xAF,
                    0xDE, 0x74, 0xE3, 0xCB, 0x60, 0x20,
                    0x38, 0x9C, 0x21, 0xC3, 0xDC, 0xC8,
                    0xA2, 0x4D, 0xC6, 0x2A, 0x35, 0x7F,
                    0xF3, 0xA9, 0xE8, 0x1D, 0x7B, 0x2C,
                    0x78, 0xFA, 0xB8, 0x02, 0x55, 0x80,
                    0x9B, 0xC2, 0xA5, 0xCB,
                };


            var bi_p = new BigInteger(pseudoPrime1);
            var bi_q = new BigInteger(pseudoPrime2);
            BigInteger bi_pq = (bi_p - 1)*(bi_q - 1);
            BigInteger bi_n = bi_p*bi_q;

            for (int count = 0; count < rounds; count++)
            {
                // generate private and public key
                BigInteger bi_e = bi_pq.genCoPrime(512, rand);
                BigInteger bi_d = bi_e.modInverse(bi_pq);

                Console.WriteLine("\ne =\n" + bi_e.ToString(10));
                Console.WriteLine("\nd =\n" + bi_d.ToString(10));
                Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

                // generate data of random length
                int t1 = 0;
                while (t1 == 0)
                    t1 = (int) (rand.NextDouble()*65);

                bool done = false;
                while (!done)
                {
                    for (int i = 0; i < 64; i++)
                    {
                        if (i < t1)
                            val[i] = (byte) (rand.NextDouble()*256);
                        else
                            val[i] = 0;

                        if (val[i] != 0)
                            done = true;
                    }
                }

                while (val[0] == 0)
                    val[0] = (byte) (rand.NextDouble()*256);

                Console.Write("Round = " + count);

                // encrypt and decrypt data
                var bi_data = new BigInteger(val, t1);
                BigInteger bi_encrypted = bi_data.ModPow(bi_e, bi_n);
                BigInteger bi_decrypted = bi_encrypted.ModPow(bi_d, bi_n);

                // compare
                if (bi_decrypted != bi_data)
                {
                    Console.WriteLine("\nError at round " + count);
                    Console.WriteLine(bi_data + "\n");
                    return;
                }
                Console.WriteLine(" <PASSED>.");
            }
        }
Exemplo n.º 20
0
        /// <summary>
        /// Generates a random number with the specified number of bits such
        /// that gcd(number, this) = 1
        /// </summary>
        /// <param name="bits"></param>
        /// <param name="rand"></param>
        /// <returns></returns>
        public BigInteger genCoPrime(int bits, Random rand)
        {
            bool done = false;
            var result = new BigInteger();

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

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

            return result;
        }
Exemplo n.º 21
0
        /// <summary>
        /// Overloading of addition operator
        /// </summary>
        /// <param name="bi1"></param>
        /// <param name="bi2"></param>
        /// <returns></returns>
        public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
        {
            var 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 = 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;
        }
Exemplo n.º 22
0
        /// <summary>
        ///  Returns the modulo inverse of this.  Throws ArithmeticException if 
        /// the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
        /// </summary>
        /// <param name="modulus"></param>
        /// <returns></returns>
        public BigInteger modInverse(BigInteger modulus)
        {
            BigInteger[] p = {0, 1};
            var 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))
            {
                var quotient = new BigInteger();
                var 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;
        }
Exemplo n.º 23
0
        /// <summary>
        /// Overloading of subtraction operator
        /// </summary>
        /// <param name="bi1"></param>
        /// <param name="bi2"></param>
        /// <returns></returns>
        public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
        {
            var 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 = 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;
        }
Exemplo n.º 24
0
        /// <summary>
        /// Returns a value that is equivalent to the integer square root
        /// of the BigInteger.
        /// <remarks>
        /// <![CDATA[
        ///  The integer square root of "this" is defined as the largest integer n
        /// such that (n * n) <= this
        /// 
        /// ]]>
        /// </remarks>
        /// </summary>
        /// <returns></returns>
        public BigInteger sqrt()
        {
            var numBits = (uint) bitCount();

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

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

            uint mask;

            var 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;
        }
Exemplo n.º 25
0
        /// <summary>
        /// Overloading of multiplication operator
        /// </summary>
        /// <param name="bi1"></param>
        /// <param name="bi2"></param>
        /// <returns></returns>
        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)
            {
            }

            var 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 = (bi1.data[i]*(ulong) bi2.data[j]) +
                                    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;
        }
Exemplo n.º 26
0
        /// <summary>
        /// Constructor (Default value provided by a string of digits of the specified base)
        /// <example>
        /// 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.
        /// </example>
        /// </summary>
        /// <param name="value"></param>
        /// <param name="radix"></param>
        public BigInteger(string value, int radix)
        {
            var multiplier = new BigInteger(1);
            var result = new BigInteger();
            value = (value.ToUpper()).Trim();
            int limit = 0;

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

            for (int i = value.Length - 1; i >= limit; i--)
            {
                int posVal = 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;
        }
Exemplo n.º 27
0
        /// <summary>
        /// Overloading of unary >> operators
        /// </summary>
        /// <param name="bi1"></param>
        /// <param name="shiftVal"></param>
        /// <returns></returns>
        public static BigInteger operator >>(BigInteger bi1, int shiftVal)
        {
            var 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;
        }
Exemplo n.º 28
0
        /// <summary>
        /// <![CDATA[
        /// 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
        /// ]]>
        /// </summary>
        /// <param name="P"></param>
        /// <param name="Q"></param>
        /// <param name="k"></param>
        /// <param name="n"></param>
        /// <returns></returns>
        public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
                                                 BigInteger k, BigInteger n)
        {
            if (k.dataLength == 1 && k.data[0] == 0)
            {
                var 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
            var 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);
        }
Exemplo n.º 29
0
        /// <summary>
        /// Overloading of the NEGATE operator (2's complement)
        /// </summary>
        /// <param name="bi1"></param>
        /// <returns></returns>
        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());

            var result = new BigInteger(bi1);

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

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

            while (carry != 0 && index < maxLength)
            {
                val = (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;
        }
Exemplo n.º 30
0
        //***********************************************************************
        // 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"
        //
        //***********************************************************************
        /// <summary>
        /// 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"
        /// </example>
        /// </summary>
        /// <param name="radix"></param>
        /// <returns></returns>
        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)
                {
                }
            }

            var quotient = new BigInteger();
            var remainder = new BigInteger();
            var 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;
        }