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Diffstat (limited to 'core/src/main/java/org/spongycastle/pqc/math/ntru/polynomial/IntegerPolynomial.java')
| -rw-r--r-- | core/src/main/java/org/spongycastle/pqc/math/ntru/polynomial/IntegerPolynomial.java | 1358 |
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diff --git a/core/src/main/java/org/spongycastle/pqc/math/ntru/polynomial/IntegerPolynomial.java b/core/src/main/java/org/spongycastle/pqc/math/ntru/polynomial/IntegerPolynomial.java new file mode 100644 index 00000000..41e921bf --- /dev/null +++ b/core/src/main/java/org/spongycastle/pqc/math/ntru/polynomial/IntegerPolynomial.java @@ -0,0 +1,1358 @@ +package org.spongycastle.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.spongycastle.pqc.math.ntru.euclid.BigIntEuclidean; +import org.spongycastle.pqc.math.ntru.util.ArrayEncoder; +import org.spongycastle.pqc.math.ntru.util.Util; +import org.spongycastle.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.spongycastle.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); + } + } +} |