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>.<br/>
*
* @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);
}
}
}