package org.bouncycastle.pqc.math.ntru.polynomial; import java.io.IOException; import java.io.InputStream; import java.math.BigInteger; import java.util.ArrayList; import java.util.Iterator; import java.util.LinkedList; import java.util.List; import java.util.concurrent.Callable; import java.util.concurrent.ExecutorService; import java.util.concurrent.Executors; import java.util.concurrent.Future; import java.util.concurrent.LinkedBlockingQueue; import org.bouncycastle.pqc.math.ntru.euclid.BigIntEuclidean; import org.bouncycastle.pqc.math.ntru.util.ArrayEncoder; import org.bouncycastle.pqc.math.ntru.util.Util; import org.bouncycastle.util.Arrays; /** * A polynomial with <code>int</code> coefficients.<br> * Some methods (like <code>add</code>) change the polynomial, others (like <code>mult</code>) do * not but return the result as a new polynomial. */ public class IntegerPolynomial implements Polynomial { private static final int NUM_EQUAL_RESULTANTS = 3; /** * Prime numbers > 4500 for resultant computation. Starting them below ~4400 causes incorrect results occasionally. * Fortunately, 4500 is about the optimum number for performance.<br/> * This array contains enough prime numbers so primes never have to be computed on-line for any standard {@link org.bouncycastle.pqc.crypto.ntru.NTRUSigningParameters}. */ private static final int[] PRIMES = new int[]{ 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973}; private static final List BIGINT_PRIMES; static { BIGINT_PRIMES = new ArrayList(); for (int i = 0; i != PRIMES.length; i++) { BIGINT_PRIMES.add(BigInteger.valueOf(PRIMES[i])); } } public int[] coeffs; /** * Constructs a new polynomial with <code>N</code> coefficients initialized to 0. * * @param N the number of coefficients */ public IntegerPolynomial(int N) { coeffs = new int[N]; } /** * Constructs a new polynomial with a given set of coefficients. * * @param coeffs the coefficients */ public IntegerPolynomial(int[] coeffs) { this.coeffs = coeffs; } /** * Constructs a <code>IntegerPolynomial</code> from a <code>BigIntPolynomial</code>. The two polynomials are independent of each other. * * @param p the original polynomial */ public IntegerPolynomial(BigIntPolynomial p) { coeffs = new int[p.coeffs.length]; for (int i = 0; i < p.coeffs.length; i++) { coeffs[i] = p.coeffs[i].intValue(); } } /** * Decodes a byte array to a polynomial with <code>N</code> ternary coefficients<br> * Ignores any excess bytes. * * @param data an encoded ternary polynomial * @param N number of coefficients * @return the decoded polynomial */ public static IntegerPolynomial fromBinary3Sves(byte[] data, int N) { return new IntegerPolynomial(ArrayEncoder.decodeMod3Sves(data, N)); } /** * Converts a byte array produced by {@link #toBinary3Tight()} to a polynomial. * * @param b a byte array * @param N number of coefficients * @return the decoded polynomial */ public static IntegerPolynomial fromBinary3Tight(byte[] b, int N) { return new IntegerPolynomial(ArrayEncoder.decodeMod3Tight(b, N)); } /** * Reads data produced by {@link #toBinary3Tight()} from an input stream and converts it to a polynomial. * * @param is an input stream * @param N number of coefficients * @return the decoded polynomial */ public static IntegerPolynomial fromBinary3Tight(InputStream is, int N) throws IOException { return new IntegerPolynomial(ArrayEncoder.decodeMod3Tight(is, N)); } /** * Returns a polynomial with N coefficients between <code>0</code> and <code>q-1</code>.<br> * <code>q</code> must be a power of 2.<br> * Ignores any excess bytes. * * @param data an encoded ternary polynomial * @param N number of coefficients * @param q * @return the decoded polynomial */ public static IntegerPolynomial fromBinary(byte[] data, int N, int q) { return new IntegerPolynomial(ArrayEncoder.decodeModQ(data, N, q)); } /** * Returns a polynomial with N coefficients between <code>0</code> and <code>q-1</code>.<br> * <code>q</code> must be a power of 2.<br> * Ignores any excess bytes. * * @param is an encoded ternary polynomial * @param N number of coefficients * @param q * @return the decoded polynomial */ public static IntegerPolynomial fromBinary(InputStream is, int N, int q) throws IOException { return new IntegerPolynomial(ArrayEncoder.decodeModQ(is, N, q)); } /** * Encodes a polynomial with ternary coefficients to binary. * <code>coeffs[2*i]</code> and <code>coeffs[2*i+1]</code> must not both equal -1 for any integer <code>i</code>, * so this method is only safe to use with polynomials produced by <code>fromBinary3Sves()</code>. * * @return the encoded polynomial */ public byte[] toBinary3Sves() { return ArrayEncoder.encodeMod3Sves(coeffs); } /** * Converts a polynomial with ternary coefficients to binary. * * @return the encoded polynomial */ public byte[] toBinary3Tight() { BigInteger sum = Constants.BIGINT_ZERO; for (int i = coeffs.length - 1; i >= 0; i--) { sum = sum.multiply(BigInteger.valueOf(3)); sum = sum.add(BigInteger.valueOf(coeffs[i] + 1)); } int size = (BigInteger.valueOf(3).pow(coeffs.length).bitLength() + 7) / 8; byte[] arr = sum.toByteArray(); if (arr.length < size) { // pad with leading zeros so arr.length==size byte[] arr2 = new byte[size]; System.arraycopy(arr, 0, arr2, size - arr.length, arr.length); return arr2; } if (arr.length > size) // drop sign bit { arr = Arrays.copyOfRange(arr, 1, arr.length); } return arr; } /** * Encodes a polynomial whose coefficients are between 0 and q, to binary. q must be a power of 2. * * @param q * @return the encoded polynomial */ public byte[] toBinary(int q) { return ArrayEncoder.encodeModQ(coeffs, q); } /** * Multiplies the polynomial with another, taking the values mod modulus and the indices mod N */ public IntegerPolynomial mult(IntegerPolynomial poly2, int modulus) { IntegerPolynomial c = mult(poly2); c.mod(modulus); return c; } /** * Multiplies the polynomial with another, taking the indices mod N */ public IntegerPolynomial mult(IntegerPolynomial poly2) { int N = coeffs.length; if (poly2.coeffs.length != N) { throw new IllegalArgumentException("Number of coefficients must be the same"); } IntegerPolynomial c = multRecursive(poly2); if (c.coeffs.length > N) { for (int k = N; k < c.coeffs.length; k++) { c.coeffs[k - N] += c.coeffs[k]; } c.coeffs = Arrays.copyOf(c.coeffs, N); } return c; } public BigIntPolynomial mult(BigIntPolynomial poly2) { return new BigIntPolynomial(this).mult(poly2); } /** * Karazuba multiplication */ private IntegerPolynomial multRecursive(IntegerPolynomial poly2) { int[] a = coeffs; int[] b = poly2.coeffs; int n = poly2.coeffs.length; if (n <= 32) { int cn = 2 * n - 1; IntegerPolynomial c = new IntegerPolynomial(new int[cn]); for (int k = 0; k < cn; k++) { for (int i = Math.max(0, k - n + 1); i <= Math.min(k, n - 1); i++) { c.coeffs[k] += b[i] * a[k - i]; } } return c; } else { int n1 = n / 2; IntegerPolynomial a1 = new IntegerPolynomial(Arrays.copyOf(a, n1)); IntegerPolynomial a2 = new IntegerPolynomial(Arrays.copyOfRange(a, n1, n)); IntegerPolynomial b1 = new IntegerPolynomial(Arrays.copyOf(b, n1)); IntegerPolynomial b2 = new IntegerPolynomial(Arrays.copyOfRange(b, n1, n)); IntegerPolynomial A = (IntegerPolynomial)a1.clone(); A.add(a2); IntegerPolynomial B = (IntegerPolynomial)b1.clone(); B.add(b2); IntegerPolynomial c1 = a1.multRecursive(b1); IntegerPolynomial c2 = a2.multRecursive(b2); IntegerPolynomial c3 = A.multRecursive(B); c3.sub(c1); c3.sub(c2); IntegerPolynomial c = new IntegerPolynomial(2 * n - 1); for (int i = 0; i < c1.coeffs.length; i++) { c.coeffs[i] = c1.coeffs[i]; } for (int i = 0; i < c3.coeffs.length; i++) { c.coeffs[n1 + i] += c3.coeffs[i]; } for (int i = 0; i < c2.coeffs.length; i++) { c.coeffs[2 * n1 + i] += c2.coeffs[i]; } return c; } } /** * Computes the inverse mod <code>q; q</code> must be a power of 2.<br> * Returns <code>null</code> if the polynomial is not invertible. * * @param q the modulus * @return a new polynomial */ public IntegerPolynomial invertFq(int q) { int N = coeffs.length; int k = 0; IntegerPolynomial b = new IntegerPolynomial(N + 1); b.coeffs[0] = 1; IntegerPolynomial c = new IntegerPolynomial(N + 1); IntegerPolynomial f = new IntegerPolynomial(N + 1); f.coeffs = Arrays.copyOf(coeffs, N + 1); f.modPositive(2); // set g(x) = x^N − 1 IntegerPolynomial g = new IntegerPolynomial(N + 1); g.coeffs[0] = 1; g.coeffs[N] = 1; while (true) { while (f.coeffs[0] == 0) { for (int i = 1; i <= N; i++) { f.coeffs[i - 1] = f.coeffs[i]; // f(x) = f(x) / x c.coeffs[N + 1 - i] = c.coeffs[N - i]; // c(x) = c(x) * x } f.coeffs[N] = 0; c.coeffs[0] = 0; k++; if (f.equalsZero()) { return null; // not invertible } } if (f.equalsOne()) { break; } if (f.degree() < g.degree()) { // exchange f and g IntegerPolynomial temp = f; f = g; g = temp; // exchange b and c temp = b; b = c; c = temp; } f.add(g, 2); b.add(c, 2); } if (b.coeffs[N] != 0) { return null; } // Fq(x) = x^(N-k) * b(x) IntegerPolynomial Fq = new IntegerPolynomial(N); int j = 0; k %= N; for (int i = N - 1; i >= 0; i--) { j = i - k; if (j < 0) { j += N; } Fq.coeffs[j] = b.coeffs[i]; } return mod2ToModq(Fq, q); } /** * Computes the inverse mod q from the inverse mod 2 * * @param Fq * @param q * @return The inverse of this polynomial mod q */ private IntegerPolynomial mod2ToModq(IntegerPolynomial Fq, int q) { if (Util.is64BitJVM() && q == 2048) { LongPolynomial2 thisLong = new LongPolynomial2(this); LongPolynomial2 FqLong = new LongPolynomial2(Fq); int v = 2; while (v < q) { v *= 2; LongPolynomial2 temp = (LongPolynomial2)FqLong.clone(); temp.mult2And(v - 1); FqLong = thisLong.mult(FqLong).mult(FqLong); temp.subAnd(FqLong, v - 1); FqLong = temp; } return FqLong.toIntegerPolynomial(); } else { int v = 2; while (v < q) { v *= 2; IntegerPolynomial temp = new IntegerPolynomial(Arrays.copyOf(Fq.coeffs, Fq.coeffs.length)); temp.mult2(v); Fq = mult(Fq, v).mult(Fq, v); temp.sub(Fq, v); Fq = temp; } return Fq; } } /** * Computes the inverse mod 3. * Returns <code>null</code> if the polynomial is not invertible. * * @return a new polynomial */ public IntegerPolynomial invertF3() { int N = coeffs.length; int k = 0; IntegerPolynomial b = new IntegerPolynomial(N + 1); b.coeffs[0] = 1; IntegerPolynomial c = new IntegerPolynomial(N + 1); IntegerPolynomial f = new IntegerPolynomial(N + 1); f.coeffs = Arrays.copyOf(coeffs, N + 1); f.modPositive(3); // set g(x) = x^N − 1 IntegerPolynomial g = new IntegerPolynomial(N + 1); g.coeffs[0] = -1; g.coeffs[N] = 1; while (true) { while (f.coeffs[0] == 0) { for (int i = 1; i <= N; i++) { f.coeffs[i - 1] = f.coeffs[i]; // f(x) = f(x) / x c.coeffs[N + 1 - i] = c.coeffs[N - i]; // c(x) = c(x) * x } f.coeffs[N] = 0; c.coeffs[0] = 0; k++; if (f.equalsZero()) { return null; // not invertible } } if (f.equalsAbsOne()) { break; } if (f.degree() < g.degree()) { // exchange f and g IntegerPolynomial temp = f; f = g; g = temp; // exchange b and c temp = b; b = c; c = temp; } if (f.coeffs[0] == g.coeffs[0]) { f.sub(g, 3); b.sub(c, 3); } else { f.add(g, 3); b.add(c, 3); } } if (b.coeffs[N] != 0) { return null; } // Fp(x) = [+-] x^(N-k) * b(x) IntegerPolynomial Fp = new IntegerPolynomial(N); int j = 0; k %= N; for (int i = N - 1; i >= 0; i--) { j = i - k; if (j < 0) { j += N; } Fp.coeffs[j] = f.coeffs[0] * b.coeffs[i]; } Fp.ensurePositive(3); return Fp; } /** * Resultant of this polynomial with <code>x^n-1</code> using a probabilistic algorithm. * <p> * Unlike EESS, this implementation does not compute all resultants modulo primes * such that their product exceeds the maximum possible resultant, but rather stops * when <code>NUM_EQUAL_RESULTANTS</code> consecutive modular resultants are equal.<br> * This means the return value may be incorrect. Experiments show this happens in * about 1 out of 100 cases when <code>N=439</code> and <code>NUM_EQUAL_RESULTANTS=2</code>, * so the likelyhood of leaving the loop too early is <code>(1/100)^(NUM_EQUAL_RESULTANTS-1)</code>. * <p> * Because of the above, callers must verify the output and try a different polynomial if necessary. * * @return <code>(rho, res)</code> satisfying <code>res = rho*this + t*(x^n-1)</code> for some integer <code>t</code>. */ public Resultant resultant() { int N = coeffs.length; // Compute resultants modulo prime numbers. Continue until NUM_EQUAL_RESULTANTS consecutive modular resultants are equal. LinkedList<ModularResultant> modResultants = new LinkedList<ModularResultant>(); BigInteger prime = null; BigInteger pProd = Constants.BIGINT_ONE; BigInteger res = Constants.BIGINT_ONE; int numEqual = 1; // number of consecutive modular resultants equal to each other Iterator<BigInteger> primes = BIGINT_PRIMES.iterator(); while (true) { prime = primes.hasNext() ? primes.next() : prime.nextProbablePrime(); ModularResultant crr = resultant(prime.intValue()); modResultants.add(crr); BigInteger temp = pProd.multiply(prime); BigIntEuclidean er = BigIntEuclidean.calculate(prime, pProd); BigInteger resPrev = res; res = res.multiply(er.x.multiply(prime)); BigInteger res2 = crr.res.multiply(er.y.multiply(pProd)); res = res.add(res2).mod(temp); pProd = temp; BigInteger pProd2 = pProd.divide(BigInteger.valueOf(2)); BigInteger pProd2n = pProd2.negate(); if (res.compareTo(pProd2) > 0) { res = res.subtract(pProd); } else if (res.compareTo(pProd2n) < 0) { res = res.add(pProd); } if (res.equals(resPrev)) { numEqual++; if (numEqual >= NUM_EQUAL_RESULTANTS) { break; } } else { numEqual = 1; } } // Combine modular rho's to obtain the final rho. // For efficiency, first combine all pairs of small resultants to bigger resultants, // then combine pairs of those, etc. until only one is left. while (modResultants.size() > 1) { ModularResultant modRes1 = modResultants.removeFirst(); ModularResultant modRes2 = modResultants.removeFirst(); ModularResultant modRes3 = ModularResultant.combineRho(modRes1, modRes2); modResultants.addLast(modRes3); } BigIntPolynomial rhoP = modResultants.getFirst().rho; BigInteger pProd2 = pProd.divide(BigInteger.valueOf(2)); BigInteger pProd2n = pProd2.negate(); if (res.compareTo(pProd2) > 0) { res = res.subtract(pProd); } if (res.compareTo(pProd2n) < 0) { res = res.add(pProd); } for (int i = 0; i < N; i++) { BigInteger c = rhoP.coeffs[i]; if (c.compareTo(pProd2) > 0) { rhoP.coeffs[i] = c.subtract(pProd); } if (c.compareTo(pProd2n) < 0) { rhoP.coeffs[i] = c.add(pProd); } } return new Resultant(rhoP, res); } /** * Multithreaded version of {@link #resultant()}. * * @return <code>(rho, res)</code> satisfying <code>res = rho*this + t*(x^n-1)</code> for some integer <code>t</code>. */ public Resultant resultantMultiThread() { int N = coeffs.length; // upper bound for resultant(f, g) = ||f, 2||^deg(g) * ||g, 2||^deg(f) = squaresum(f)^(N/2) * 2^(deg(f)/2) because g(x)=x^N-1 // see http://jondalon.mathematik.uni-osnabrueck.de/staff/phpages/brunsw/CompAlg.pdf chapter 3 BigInteger max = squareSum().pow((N + 1) / 2); max = max.multiply(BigInteger.valueOf(2).pow((degree() + 1) / 2)); BigInteger max2 = max.multiply(BigInteger.valueOf(2)); // compute resultants modulo prime numbers BigInteger prime = BigInteger.valueOf(10000); BigInteger pProd = Constants.BIGINT_ONE; LinkedBlockingQueue<Future<ModularResultant>> resultantTasks = new LinkedBlockingQueue<Future<ModularResultant>>(); Iterator<BigInteger> primes = BIGINT_PRIMES.iterator(); ExecutorService executor = Executors.newFixedThreadPool(Runtime.getRuntime().availableProcessors()); while (pProd.compareTo(max2) < 0) { if (primes.hasNext()) { prime = primes.next(); } else { prime = prime.nextProbablePrime(); } Future<ModularResultant> task = executor.submit(new ModResultantTask(prime.intValue())); resultantTasks.add(task); pProd = pProd.multiply(prime); } // Combine modular resultants to obtain the resultant. // For efficiency, first combine all pairs of small resultants to bigger resultants, // then combine pairs of those, etc. until only one is left. ModularResultant overallResultant = null; while (!resultantTasks.isEmpty()) { try { Future<ModularResultant> modRes1 = resultantTasks.take(); Future<ModularResultant> modRes2 = resultantTasks.poll(); if (modRes2 == null) { // modRes1 is the only one left overallResultant = modRes1.get(); break; } Future<ModularResultant> newTask = executor.submit(new CombineTask(modRes1.get(), modRes2.get())); resultantTasks.add(newTask); } catch (Exception e) { throw new IllegalStateException(e.toString()); } } executor.shutdown(); BigInteger res = overallResultant.res; BigIntPolynomial rhoP = overallResultant.rho; BigInteger pProd2 = pProd.divide(BigInteger.valueOf(2)); BigInteger pProd2n = pProd2.negate(); if (res.compareTo(pProd2) > 0) { res = res.subtract(pProd); } if (res.compareTo(pProd2n) < 0) { res = res.add(pProd); } for (int i = 0; i < N; i++) { BigInteger c = rhoP.coeffs[i]; if (c.compareTo(pProd2) > 0) { rhoP.coeffs[i] = c.subtract(pProd); } if (c.compareTo(pProd2n) < 0) { rhoP.coeffs[i] = c.add(pProd); } } return new Resultant(rhoP, res); } /** * Resultant of this polynomial with <code>x^n-1 mod p</code>. * * @return <code>(rho, res)</code> satisfying <code>res = rho*this + t*(x^n-1) mod p</code> for some integer <code>t</code>. */ public ModularResultant resultant(int p) { // Add a coefficient as the following operations involve polynomials of degree deg(f)+1 int[] fcoeffs = Arrays.copyOf(coeffs, coeffs.length + 1); IntegerPolynomial f = new IntegerPolynomial(fcoeffs); int N = fcoeffs.length; IntegerPolynomial a = new IntegerPolynomial(N); a.coeffs[0] = -1; a.coeffs[N - 1] = 1; IntegerPolynomial b = new IntegerPolynomial(f.coeffs); IntegerPolynomial v1 = new IntegerPolynomial(N); IntegerPolynomial v2 = new IntegerPolynomial(N); v2.coeffs[0] = 1; int da = N - 1; int db = b.degree(); int ta = da; int c = 0; int r = 1; while (db > 0) { c = Util.invert(b.coeffs[db], p); c = (c * a.coeffs[da]) % p; a.multShiftSub(b, c, da - db, p); v1.multShiftSub(v2, c, da - db, p); da = a.degree(); if (da < db) { r *= Util.pow(b.coeffs[db], ta - da, p); r %= p; if (ta % 2 == 1 && db % 2 == 1) { r = (-r) % p; } IntegerPolynomial temp = a; a = b; b = temp; int tempdeg = da; da = db; temp = v1; v1 = v2; v2 = temp; ta = db; db = tempdeg; } } r *= Util.pow(b.coeffs[0], da, p); r %= p; c = Util.invert(b.coeffs[0], p); v2.mult(c); v2.mod(p); v2.mult(r); v2.mod(p); // drop the highest coefficient so #coeffs matches the original input v2.coeffs = Arrays.copyOf(v2.coeffs, v2.coeffs.length - 1); return new ModularResultant(new BigIntPolynomial(v2), BigInteger.valueOf(r), BigInteger.valueOf(p)); } /** * Computes <code>this-b*c*(x^k) mod p</code> and stores the result in this polynomial.<br/> * See steps 4a,4b in EESS algorithm 2.2.7.1. * * @param b * @param c * @param k * @param p */ private void multShiftSub(IntegerPolynomial b, int c, int k, int p) { int N = coeffs.length; for (int i = k; i < N; i++) { coeffs[i] = (coeffs[i] - b.coeffs[i - k] * c) % p; } } /** * Adds the squares of all coefficients. * * @return the sum of squares */ private BigInteger squareSum() { BigInteger sum = Constants.BIGINT_ZERO; for (int i = 0; i < coeffs.length; i++) { sum = sum.add(BigInteger.valueOf(coeffs[i] * coeffs[i])); } return sum; } /** * Returns the degree of the polynomial * * @return the degree */ int degree() { int degree = coeffs.length - 1; while (degree > 0 && coeffs[degree] == 0) { degree--; } return degree; } /** * Adds another polynomial which can have a different number of coefficients, * and takes the coefficient values mod <code>modulus</code>. * * @param b another polynomial */ public void add(IntegerPolynomial b, int modulus) { add(b); mod(modulus); } /** * Adds another polynomial which can have a different number of coefficients. * * @param b another polynomial */ public void add(IntegerPolynomial b) { if (b.coeffs.length > coeffs.length) { coeffs = Arrays.copyOf(coeffs, b.coeffs.length); } for (int i = 0; i < b.coeffs.length; i++) { coeffs[i] += b.coeffs[i]; } } /** * Subtracts another polynomial which can have a different number of coefficients, * and takes the coefficient values mod <code>modulus</code>. * * @param b another polynomial */ public void sub(IntegerPolynomial b, int modulus) { sub(b); mod(modulus); } /** * Subtracts another polynomial which can have a different number of coefficients. * * @param b another polynomial */ public void sub(IntegerPolynomial b) { if (b.coeffs.length > coeffs.length) { coeffs = Arrays.copyOf(coeffs, b.coeffs.length); } for (int i = 0; i < b.coeffs.length; i++) { coeffs[i] -= b.coeffs[i]; } } /** * Subtracts a <code>int</code> from each coefficient. Does not return a new polynomial but modifies this polynomial. * * @param b */ void sub(int b) { for (int i = 0; i < coeffs.length; i++) { coeffs[i] -= b; } } /** * Multiplies each coefficient by a <code>int</code>. Does not return a new polynomial but modifies this polynomial. * * @param factor */ public void mult(int factor) { for (int i = 0; i < coeffs.length; i++) { coeffs[i] *= factor; } } /** * Multiplies each coefficient by a 2 and applies a modulus. Does not return a new polynomial but modifies this polynomial. * * @param modulus a modulus */ private void mult2(int modulus) { for (int i = 0; i < coeffs.length; i++) { coeffs[i] *= 2; coeffs[i] %= modulus; } } /** * Multiplies each coefficient by a 2 and applies a modulus. Does not return a new polynomial but modifies this polynomial. * * @param modulus a modulus */ public void mult3(int modulus) { for (int i = 0; i < coeffs.length; i++) { coeffs[i] *= 3; coeffs[i] %= modulus; } } /** * Divides each coefficient by <code>k</code> and rounds to the nearest integer. Does not return a new polynomial but modifies this polynomial. * * @param k the divisor */ public void div(int k) { int k2 = (k + 1) / 2; for (int i = 0; i < coeffs.length; i++) { coeffs[i] += coeffs[i] > 0 ? k2 : -k2; coeffs[i] /= k; } } /** * Takes each coefficient modulo 3 such that all coefficients are ternary. */ public void mod3() { for (int i = 0; i < coeffs.length; i++) { coeffs[i] %= 3; if (coeffs[i] > 1) { coeffs[i] -= 3; } if (coeffs[i] < -1) { coeffs[i] += 3; } } } /** * Ensures all coefficients are between 0 and <code>modulus-1</code> * * @param modulus a modulus */ public void modPositive(int modulus) { mod(modulus); ensurePositive(modulus); } /** * Reduces all coefficients to the interval [-modulus/2, modulus/2) */ void modCenter(int modulus) { mod(modulus); for (int j = 0; j < coeffs.length; j++) { while (coeffs[j] < modulus / 2) { coeffs[j] += modulus; } while (coeffs[j] >= modulus / 2) { coeffs[j] -= modulus; } } } /** * Takes each coefficient modulo <code>modulus</code>. */ public void mod(int modulus) { for (int i = 0; i < coeffs.length; i++) { coeffs[i] %= modulus; } } /** * Adds <code>modulus</code> until all coefficients are above 0. * * @param modulus a modulus */ public void ensurePositive(int modulus) { for (int i = 0; i < coeffs.length; i++) { while (coeffs[i] < 0) { coeffs[i] += modulus; } } } /** * Computes the centered euclidean norm of the polynomial. * * @param q a modulus * @return the centered norm */ public long centeredNormSq(int q) { int N = coeffs.length; IntegerPolynomial p = (IntegerPolynomial)clone(); p.shiftGap(q); long sum = 0; long sqSum = 0; for (int i = 0; i != p.coeffs.length; i++) { int c = p.coeffs[i]; sum += c; sqSum += c * c; } long centeredNormSq = sqSum - sum * sum / N; return centeredNormSq; } /** * Shifts all coefficients so the largest gap is centered around <code>-q/2</code>. * * @param q a modulus */ void shiftGap(int q) { modCenter(q); int[] sorted = Arrays.clone(coeffs); sort(sorted); int maxrange = 0; int maxrangeStart = 0; for (int i = 0; i < sorted.length - 1; i++) { int range = sorted[i + 1] - sorted[i]; if (range > maxrange) { maxrange = range; maxrangeStart = sorted[i]; } } int pmin = sorted[0]; int pmax = sorted[sorted.length - 1]; int j = q - pmax + pmin; int shift; if (j > maxrange) { shift = (pmax + pmin) / 2; } else { shift = maxrangeStart + maxrange / 2 + q / 2; } sub(shift); } private void sort(int[] ints) { boolean swap = true; while (swap) { swap = false; for (int i = 0; i != ints.length - 1; i++) { if (ints[i] > ints[i+1]) { int tmp = ints[i]; ints[i] = ints[i+1]; ints[i+1] = tmp; swap = true; } } } } /** * Shifts the values of all coefficients to the interval <code>[-q/2, q/2]</code>. * * @param q a modulus */ public void center0(int q) { for (int i = 0; i < coeffs.length; i++) { while (coeffs[i] < -q / 2) { coeffs[i] += q; } while (coeffs[i] > q / 2) { coeffs[i] -= q; } } } /** * Returns the sum of all coefficients, i.e. evaluates the polynomial at 0. * * @return the sum of all coefficients */ public int sumCoeffs() { int sum = 0; for (int i = 0; i < coeffs.length; i++) { sum += coeffs[i]; } return sum; } /** * Tests if <code>p(x) = 0</code>. * * @return true iff all coefficients are zeros */ private boolean equalsZero() { for (int i = 0; i < coeffs.length; i++) { if (coeffs[i] != 0) { return false; } } return true; } /** * Tests if <code>p(x) = 1</code>. * * @return true iff all coefficients are equal to zero, except for the lowest coefficient which must equal 1 */ public boolean equalsOne() { for (int i = 1; i < coeffs.length; i++) { if (coeffs[i] != 0) { return false; } } return coeffs[0] == 1; } /** * Tests if <code>|p(x)| = 1</code>. * * @return true iff all coefficients are equal to zero, except for the lowest coefficient which must equal 1 or -1 */ private boolean equalsAbsOne() { for (int i = 1; i < coeffs.length; i++) { if (coeffs[i] != 0) { return false; } } return Math.abs(coeffs[0]) == 1; } /** * Counts the number of coefficients equal to an integer * * @param value an integer * @return the number of coefficients equal to <code>value</code> */ public int count(int value) { int count = 0; for (int i = 0; i != coeffs.length; i++) { if (coeffs[i] == value) { count++; } } return count; } /** * Multiplication by <code>X</code> in <code>Z[X]/Z[X^n-1]</code>. */ public void rotate1() { int clast = coeffs[coeffs.length - 1]; for (int i = coeffs.length - 1; i > 0; i--) { coeffs[i] = coeffs[i - 1]; } coeffs[0] = clast; } public void clear() { for (int i = 0; i < coeffs.length; i++) { coeffs[i] = 0; } } public IntegerPolynomial toIntegerPolynomial() { return (IntegerPolynomial)clone(); } public Object clone() { return new IntegerPolynomial(coeffs.clone()); } public boolean equals(Object obj) { if (obj instanceof IntegerPolynomial) { return Arrays.areEqual(coeffs, ((IntegerPolynomial)obj).coeffs); } else { return false; } } /** * Calls {@link IntegerPolynomial#resultant(int) */ private class ModResultantTask implements Callable<ModularResultant> { private int modulus; private ModResultantTask(int modulus) { this.modulus = modulus; } public ModularResultant call() { return resultant(modulus); } } /** * Calls {@link ModularResultant#combineRho(ModularResultant, ModularResultant) */ private class CombineTask implements Callable<ModularResultant> { private ModularResultant modRes1; private ModularResultant modRes2; private CombineTask(ModularResultant modRes1, ModularResultant modRes2) { this.modRes1 = modRes1; this.modRes2 = modRes2; } public ModularResultant call() { return ModularResultant.combineRho(modRes1, modRes2); } } }