BarrettReduction() приватный Метод

private BarrettReduction ( BigInteger x, BigInteger n, BigInteger constant ) : BigInteger
x BigInteger
n BigInteger
constant BigInteger
Результат BigInteger
Пример #1
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;
        }
Пример #2
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 = 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;
        }