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package org.bouncycastle.pqc.math.linearalgebra;
import java.security.SecureRandom;
/**
* This class describes operations with polynomials from the ring R =
* GF(2^m)[X], where 2 <= m <=31.
*
* @see GF2mField
* @see PolynomialRingGF2m
*/
public class PolynomialGF2mSmallM
{
/**
* the finite field GF(2^m)
*/
private GF2mField field;
/**
* the degree of this polynomial
*/
private int degree;
/**
* For the polynomial representation the map f: R->Z*,
* poly(X) -> [coef_0, coef_1, ...] is used, where
* coef_i is the ith coefficient of the polynomial
* represented as int (see {@link GF2mField}). The polynomials are stored
* as int arrays.
*/
private int[] coefficients;
/*
* some types of polynomials
*/
/**
* Constant used for polynomial construction (see constructor
* {@link #PolynomialGF2mSmallM(GF2mField, int, char, SecureRandom)}).
*/
public static final char RANDOM_IRREDUCIBLE_POLYNOMIAL = 'I';
/**
* Construct the zero polynomial over the finite field GF(2^m).
*
* @param field the finite field GF(2^m)
*/
public PolynomialGF2mSmallM(GF2mField field)
{
this.field = field;
degree = -1;
coefficients = new int[1];
}
/**
* Construct a polynomial over the finite field GF(2^m).
*
* @param field the finite field GF(2^m)
* @param deg degree of polynomial
* @param typeOfPolynomial type of polynomial
* @param sr PRNG
*/
public PolynomialGF2mSmallM(GF2mField field, int deg,
char typeOfPolynomial, SecureRandom sr)
{
this.field = field;
switch (typeOfPolynomial)
{
case PolynomialGF2mSmallM.RANDOM_IRREDUCIBLE_POLYNOMIAL:
coefficients = createRandomIrreduciblePolynomial(deg, sr);
break;
default:
throw new IllegalArgumentException(" Error: type "
+ typeOfPolynomial
+ " is not defined for GF2smallmPolynomial");
}
computeDegree();
}
/**
* Create an irreducible polynomial with the given degree over the field
* GF(2^m).
*
* @param deg polynomial degree
* @param sr source of randomness
* @return the generated irreducible polynomial
*/
private int[] createRandomIrreduciblePolynomial(int deg, SecureRandom sr)
{
int[] resCoeff = new int[deg + 1];
resCoeff[deg] = 1;
resCoeff[0] = field.getRandomNonZeroElement(sr);
for (int i = 1; i < deg; i++)
{
resCoeff[i] = field.getRandomElement(sr);
}
while (!isIrreducible(resCoeff))
{
int n = RandUtils.nextInt(sr, deg);
if (n == 0)
{
resCoeff[0] = field.getRandomNonZeroElement(sr);
}
else
{
resCoeff[n] = field.getRandomElement(sr);
}
}
return resCoeff;
}
/**
* Construct a monomial of the given degree over the finite field GF(2^m).
*
* @param field the finite field GF(2^m)
* @param degree the degree of the monomial
*/
public PolynomialGF2mSmallM(GF2mField field, int degree)
{
this.field = field;
this.degree = degree;
coefficients = new int[degree + 1];
coefficients[degree] = 1;
}
/**
* Construct the polynomial over the given finite field GF(2^m) from the
* given coefficient vector.
*
* @param field finite field GF2m
* @param coeffs the coefficient vector
*/
public PolynomialGF2mSmallM(GF2mField field, int[] coeffs)
{
this.field = field;
coefficients = normalForm(coeffs);
computeDegree();
}
/**
* Create a polynomial over the finite field GF(2^m).
*
* @param field the finite field GF(2^m)
* @param enc byte[] polynomial in byte array form
*/
public PolynomialGF2mSmallM(GF2mField field, byte[] enc)
{
this.field = field;
// decodes polynomial
int d = 8;
int count = 1;
while (field.getDegree() > d)
{
count++;
d += 8;
}
if ((enc.length % count) != 0)
{
throw new IllegalArgumentException(
" Error: byte array is not encoded polynomial over given finite field GF2m");
}
coefficients = new int[enc.length / count];
count = 0;
for (int i = 0; i < coefficients.length; i++)
{
for (int j = 0; j < d; j += 8)
{
coefficients[i] ^= (enc[count++] & 0x000000ff) << j;
}
if (!this.field.isElementOfThisField(coefficients[i]))
{
throw new IllegalArgumentException(
" Error: byte array is not encoded polynomial over given finite field GF2m");
}
}
// if HC = 0 for non-zero polynomial, returns error
if ((coefficients.length != 1)
&& (coefficients[coefficients.length - 1] == 0))
{
throw new IllegalArgumentException(
" Error: byte array is not encoded polynomial over given finite field GF2m");
}
computeDegree();
}
/**
* Copy constructor.
*
* @param other another {@link PolynomialGF2mSmallM}
*/
public PolynomialGF2mSmallM(PolynomialGF2mSmallM other)
{
// field needs not to be cloned since it is immutable
field = other.field;
degree = other.degree;
coefficients = IntUtils.clone(other.coefficients);
}
/**
* Create a polynomial over the finite field GF(2^m) out of the given
* coefficient vector. The finite field is also obtained from the
* {@link GF2mVector}.
*
* @param vect the coefficient vector
*/
public PolynomialGF2mSmallM(GF2mVector vect)
{
this(vect.getField(), vect.getIntArrayForm());
}
/*
* ------------------------
*/
/**
* Return the degree of this polynomial
*
* @return int degree of this polynomial if this is zero polynomial return
* -1
*/
public int getDegree()
{
int d = coefficients.length - 1;
if (coefficients[d] == 0)
{
return -1;
}
return d;
}
/**
* @return the head coefficient of this polynomial
*/
public int getHeadCoefficient()
{
if (degree == -1)
{
return 0;
}
return coefficients[degree];
}
/**
* Return the head coefficient of a polynomial.
*
* @param a the polynomial
* @return the head coefficient of a
*/
private static int headCoefficient(int[] a)
{
int degree = computeDegree(a);
if (degree == -1)
{
return 0;
}
return a[degree];
}
/**
* Return the coefficient with the given index.
*
* @param index the index
* @return the coefficient with the given index
*/
public int getCoefficient(int index)
{
if ((index < 0) || (index > degree))
{
return 0;
}
return coefficients[index];
}
/**
* Returns encoded polynomial, i.e., this polynomial in byte array form
*
* @return the encoded polynomial
*/
public byte[] getEncoded()
{
int d = 8;
int count = 1;
while (field.getDegree() > d)
{
count++;
d += 8;
}
byte[] res = new byte[coefficients.length * count];
count = 0;
for (int i = 0; i < coefficients.length; i++)
{
for (int j = 0; j < d; j += 8)
{
res[count++] = (byte)(coefficients[i] >>> j);
}
}
return res;
}
/**
* Evaluate this polynomial p at a value e (in
* GF(2^m)) with the Horner scheme.
*
* @param e the element of the finite field GF(2^m)
* @return this(e)
*/
public int evaluateAt(int e)
{
int result = coefficients[degree];
for (int i = degree - 1; i >= 0; i--)
{
result = field.mult(result, e) ^ coefficients[i];
}
return result;
}
/**
* Compute the sum of this polynomial and the given polynomial.
*
* @param addend the addend
* @return this + a (newly created)
*/
public PolynomialGF2mSmallM add(PolynomialGF2mSmallM addend)
{
int[] resultCoeff = add(coefficients, addend.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Add the given polynomial to this polynomial (overwrite this).
*
* @param addend the addend
*/
public void addToThis(PolynomialGF2mSmallM addend)
{
coefficients = add(coefficients, addend.coefficients);
computeDegree();
}
/**
* Compute the sum of two polynomials a and b over the finite field
* GF(2^m).
*
* @param a the first polynomial
* @param b the second polynomial
* @return a + b
*/
private int[] add(int[] a, int[] b)
{
int[] result, addend;
if (a.length < b.length)
{
result = new int[b.length];
System.arraycopy(b, 0, result, 0, b.length);
addend = a;
}
else
{
result = new int[a.length];
System.arraycopy(a, 0, result, 0, a.length);
addend = b;
}
for (int i = addend.length - 1; i >= 0; i--)
{
result[i] = field.add(result[i], addend[i]);
}
return result;
}
/**
* Compute the sum of this polynomial and the monomial of the given degree.
*
* @param degree the degree of the monomial
* @return this + X^k
*/
public PolynomialGF2mSmallM addMonomial(int degree)
{
int[] monomial = new int[degree + 1];
monomial[degree] = 1;
int[] resultCoeff = add(coefficients, monomial);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the product of this polynomial with an element from GF(2^m).
*
* @param element an element of the finite field GF(2^m)
* @return this * element (newly created)
* @throws ArithmeticException if element is not an element of the finite
* field this polynomial is defined over.
*/
public PolynomialGF2mSmallM multWithElement(int element)
{
if (!field.isElementOfThisField(element))
{
throw new ArithmeticException(
"Not an element of the finite field this polynomial is defined over.");
}
int[] resultCoeff = multWithElement(coefficients, element);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Multiply this polynomial with an element from GF(2^m).
*
* @param element an element of the finite field GF(2^m)
* @throws ArithmeticException if element is not an element of the finite
* field this polynomial is defined over.
*/
public void multThisWithElement(int element)
{
if (!field.isElementOfThisField(element))
{
throw new ArithmeticException(
"Not an element of the finite field this polynomial is defined over.");
}
coefficients = multWithElement(coefficients, element);
computeDegree();
}
/**
* Compute the product of a polynomial a with an element from the finite
* field GF(2^m).
*
* @param a the polynomial
* @param element an element of the finite field GF(2^m)
* @return a * element
*/
private int[] multWithElement(int[] a, int element)
{
int degree = computeDegree(a);
if (degree == -1 || element == 0)
{
return new int[1];
}
if (element == 1)
{
return IntUtils.clone(a);
}
int[] result = new int[degree + 1];
for (int i = degree; i >= 0; i--)
{
result[i] = field.mult(a[i], element);
}
return result;
}
/**
* Compute the product of this polynomial with a monomial X^k.
*
* @param k the degree of the monomial
* @return this * X^k
*/
public PolynomialGF2mSmallM multWithMonomial(int k)
{
int[] resultCoeff = multWithMonomial(coefficients, k);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the product of a polynomial with a monomial X^k.
*
* @param a the polynomial
* @param k the degree of the monomial
* @return a * X^k
*/
private static int[] multWithMonomial(int[] a, int k)
{
int d = computeDegree(a);
if (d == -1)
{
return new int[1];
}
int[] result = new int[d + k + 1];
System.arraycopy(a, 0, result, k, d + 1);
return result;
}
/**
* Divide this polynomial by the given polynomial.
*
* @param f a polynomial
* @return polynomial pair = {q,r} where this = q*f+r and deg(r) <
* deg(f);
*/
public PolynomialGF2mSmallM[] div(PolynomialGF2mSmallM f)
{
int[][] resultCoeffs = div(coefficients, f.coefficients);
return new PolynomialGF2mSmallM[]{
new PolynomialGF2mSmallM(field, resultCoeffs[0]),
new PolynomialGF2mSmallM(field, resultCoeffs[1])};
}
/**
* Compute the result of the division of two polynomials over the field
* GF(2^m).
*
* @param a the first polynomial
* @param f the second polynomial
* @return int[][] {q,r}, where a = q*f+r and deg(r) < deg(f);
*/
private int[][] div(int[] a, int[] f)
{
int df = computeDegree(f);
int da = computeDegree(a) + 1;
if (df == -1)
{
throw new ArithmeticException("Division by zero.");
}
int[][] result = new int[2][];
result[0] = new int[1];
result[1] = new int[da];
int hc = headCoefficient(f);
hc = field.inverse(hc);
result[0][0] = 0;
System.arraycopy(a, 0, result[1], 0, result[1].length);
while (df <= computeDegree(result[1]))
{
int[] q;
int[] coeff = new int[1];
coeff[0] = field.mult(headCoefficient(result[1]), hc);
q = multWithElement(f, coeff[0]);
int n = computeDegree(result[1]) - df;
q = multWithMonomial(q, n);
coeff = multWithMonomial(coeff, n);
result[0] = add(coeff, result[0]);
result[1] = add(q, result[1]);
}
return result;
}
/**
* Return the greatest common divisor of this and a polynomial f
*
* @param f polynomial
* @return GCD(this, f)
*/
public PolynomialGF2mSmallM gcd(PolynomialGF2mSmallM f)
{
int[] resultCoeff = gcd(coefficients, f.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Return the greatest common divisor of two polynomials over the field
* GF(2^m).
*
* @param f the first polynomial
* @param g the second polynomial
* @return gcd(f, g)
*/
private int[] gcd(int[] f, int[] g)
{
int[] a = f;
int[] b = g;
if (computeDegree(a) == -1)
{
return b;
}
while (computeDegree(b) != -1)
{
int[] c = mod(a, b);
a = new int[b.length];
System.arraycopy(b, 0, a, 0, a.length);
b = new int[c.length];
System.arraycopy(c, 0, b, 0, b.length);
}
int coeff = field.inverse(headCoefficient(a));
return multWithElement(a, coeff);
}
/**
* Compute the product of this polynomial and the given factor using a
* Karatzuba like scheme.
*
* @param factor the polynomial
* @return this * factor
*/
public PolynomialGF2mSmallM multiply(PolynomialGF2mSmallM factor)
{
int[] resultCoeff = multiply(coefficients, factor.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the product of two polynomials over the field GF(2^m)
* using a Karatzuba like multiplication.
*
* @param a the first polynomial
* @param b the second polynomial
* @return a * b
*/
private int[] multiply(int[] a, int[] b)
{
int[] mult1, mult2;
if (computeDegree(a) < computeDegree(b))
{
mult1 = b;
mult2 = a;
}
else
{
mult1 = a;
mult2 = b;
}
mult1 = normalForm(mult1);
mult2 = normalForm(mult2);
if (mult2.length == 1)
{
return multWithElement(mult1, mult2[0]);
}
int d1 = mult1.length;
int d2 = mult2.length;
int[] result = new int[d1 + d2 - 1];
if (d2 != d1)
{
int[] res1 = new int[d2];
int[] res2 = new int[d1 - d2];
System.arraycopy(mult1, 0, res1, 0, res1.length);
System.arraycopy(mult1, d2, res2, 0, res2.length);
res1 = multiply(res1, mult2);
res2 = multiply(res2, mult2);
res2 = multWithMonomial(res2, d2);
result = add(res1, res2);
}
else
{
d2 = (d1 + 1) >>> 1;
int d = d1 - d2;
int[] firstPartMult1 = new int[d2];
int[] firstPartMult2 = new int[d2];
int[] secondPartMult1 = new int[d];
int[] secondPartMult2 = new int[d];
System
.arraycopy(mult1, 0, firstPartMult1, 0,
firstPartMult1.length);
System.arraycopy(mult1, d2, secondPartMult1, 0,
secondPartMult1.length);
System
.arraycopy(mult2, 0, firstPartMult2, 0,
firstPartMult2.length);
System.arraycopy(mult2, d2, secondPartMult2, 0,
secondPartMult2.length);
int[] helpPoly1 = add(firstPartMult1, secondPartMult1);
int[] helpPoly2 = add(firstPartMult2, secondPartMult2);
int[] res1 = multiply(firstPartMult1, firstPartMult2);
int[] res2 = multiply(helpPoly1, helpPoly2);
int[] res3 = multiply(secondPartMult1, secondPartMult2);
res2 = add(res2, res1);
res2 = add(res2, res3);
res3 = multWithMonomial(res3, d2);
result = add(res2, res3);
result = multWithMonomial(result, d2);
result = add(result, res1);
}
return result;
}
/*
* ---------------- PART II ----------------
*
*/
/**
* Check a polynomial for irreducibility over the field GF(2^m).
*
* @param a the polynomial to check
* @return true if a is irreducible, false otherwise
*/
private boolean isIrreducible(int[] a)
{
if (a[0] == 0)
{
return false;
}
int d = computeDegree(a) >> 1;
int[] u = {0, 1};
final int[] Y = {0, 1};
int fieldDegree = field.getDegree();
for (int i = 0; i < d; i++)
{
for (int j = fieldDegree - 1; j >= 0; j--)
{
u = modMultiply(u, u, a);
}
u = normalForm(u);
int[] g = gcd(add(u, Y), a);
if (computeDegree(g) != 0)
{
return false;
}
}
return true;
}
/**
* Reduce this polynomial modulo another polynomial.
*
* @param f the reduction polynomial
* @return this mod f
*/
public PolynomialGF2mSmallM mod(PolynomialGF2mSmallM f)
{
int[] resultCoeff = mod(coefficients, f.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Reduce a polynomial modulo another polynomial.
*
* @param a the polynomial
* @param f the reduction polynomial
* @return a mod f
*/
private int[] mod(int[] a, int[] f)
{
int df = computeDegree(f);
if (df == -1)
{
throw new ArithmeticException("Division by zero");
}
int[] result = new int[a.length];
int hc = headCoefficient(f);
hc = field.inverse(hc);
System.arraycopy(a, 0, result, 0, result.length);
while (df <= computeDegree(result))
{
int[] q;
int coeff = field.mult(headCoefficient(result), hc);
q = multWithMonomial(f, computeDegree(result) - df);
q = multWithElement(q, coeff);
result = add(q, result);
}
return result;
}
/**
* Compute the product of this polynomial and another polynomial modulo a
* third polynomial.
*
* @param a another polynomial
* @param b the reduction polynomial
* @return this * a mod b
*/
public PolynomialGF2mSmallM modMultiply(PolynomialGF2mSmallM a,
PolynomialGF2mSmallM b)
{
int[] resultCoeff = modMultiply(coefficients, a.coefficients,
b.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Square this polynomial using a squaring matrix.
*
* @param matrix the squaring matrix
* @return this^2 modulo the reduction polynomial implicitly
* given via the squaring matrix
*/
public PolynomialGF2mSmallM modSquareMatrix(PolynomialGF2mSmallM[] matrix)
{
int length = matrix.length;
int[] resultCoeff = new int[length];
int[] thisSquare = new int[length];
// square each entry of this polynomial
for (int i = 0; i < coefficients.length; i++)
{
thisSquare[i] = field.mult(coefficients[i], coefficients[i]);
}
// do matrix-vector multiplication
for (int i = 0; i < length; i++)
{
// compute scalar product of i-th row and coefficient vector
for (int j = 0; j < length; j++)
{
if (i >= matrix[j].coefficients.length)
{
continue;
}
int scalarTerm = field.mult(matrix[j].coefficients[i],
thisSquare[j]);
resultCoeff[i] = field.add(resultCoeff[i], scalarTerm);
}
}
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the product of two polynomials modulo a third polynomial over the
* finite field GF(2^m).
*
* @param a the first polynomial
* @param b the second polynomial
* @param g the reduction polynomial
* @return a * b mod g
*/
private int[] modMultiply(int[] a, int[] b, int[] g)
{
return mod(multiply(a, b), g);
}
/**
* Compute the square root of this polynomial modulo the given polynomial.
*
* @param a the reduction polynomial
* @return this^(1/2) mod a
*/
public PolynomialGF2mSmallM modSquareRoot(PolynomialGF2mSmallM a)
{
int[] resultCoeff = IntUtils.clone(coefficients);
int[] help = modMultiply(resultCoeff, resultCoeff, a.coefficients);
while (!isEqual(help, coefficients))
{
resultCoeff = normalForm(help);
help = modMultiply(resultCoeff, resultCoeff, a.coefficients);
}
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the square root of this polynomial using a square root matrix.
*
* @param matrix the matrix for computing square roots in
* (GF(2^m))^t the polynomial ring defining the
* square root matrix
* @return this^(1/2) modulo the reduction polynomial implicitly
* given via the square root matrix
*/
public PolynomialGF2mSmallM modSquareRootMatrix(
PolynomialGF2mSmallM[] matrix)
{
int length = matrix.length;
int[] resultCoeff = new int[length];
// do matrix multiplication
for (int i = 0; i < length; i++)
{
// compute scalar product of i-th row and j-th column
for (int j = 0; j < length; j++)
{
if (i >= matrix[j].coefficients.length)
{
continue;
}
if (j < coefficients.length)
{
int scalarTerm = field.mult(matrix[j].coefficients[i],
coefficients[j]);
resultCoeff[i] = field.add(resultCoeff[i], scalarTerm);
}
}
}
// compute the square root of each entry of the result coefficients
for (int i = 0; i < length; i++)
{
resultCoeff[i] = field.sqRoot(resultCoeff[i]);
}
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the result of the division of this polynomial by another
* polynomial modulo a third polynomial.
*
* @param divisor the divisor
* @param modulus the reduction polynomial
* @return this * divisor^(-1) mod modulus
*/
public PolynomialGF2mSmallM modDiv(PolynomialGF2mSmallM divisor,
PolynomialGF2mSmallM modulus)
{
int[] resultCoeff = modDiv(coefficients, divisor.coefficients,
modulus.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute the result of the division of two polynomials modulo a third
* polynomial over the field GF(2^m).
*
* @param a the first polynomial
* @param b the second polynomial
* @param g the reduction polynomial
* @return a * b^(-1) mod g
*/
private int[] modDiv(int[] a, int[] b, int[] g)
{
int[] r0 = normalForm(g);
int[] r1 = mod(b, g);
int[] s0 = {0};
int[] s1 = mod(a, g);
int[] s2;
int[][] q;
while (computeDegree(r1) != -1)
{
q = div(r0, r1);
r0 = normalForm(r1);
r1 = normalForm(q[1]);
s2 = add(s0, modMultiply(q[0], s1, g));
s0 = normalForm(s1);
s1 = normalForm(s2);
}
int hc = headCoefficient(r0);
s0 = multWithElement(s0, field.inverse(hc));
return s0;
}
/**
* Compute the inverse of this polynomial modulo the given polynomial.
*
* @param a the reduction polynomial
* @return this^(-1) mod a
*/
public PolynomialGF2mSmallM modInverse(PolynomialGF2mSmallM a)
{
int[] unit = {1};
int[] resultCoeff = modDiv(unit, coefficients, a.coefficients);
return new PolynomialGF2mSmallM(field, resultCoeff);
}
/**
* Compute a polynomial pair (a,b) from this polynomial and the given
* polynomial g with the property b*this = a mod g and deg(a)<=deg(g)/2.
*
* @param g the reduction polynomial
* @return PolynomialGF2mSmallM[] {a,b} with b*this = a mod g and deg(a)<=
* deg(g)/2
*/
public PolynomialGF2mSmallM[] modPolynomialToFracton(PolynomialGF2mSmallM g)
{
int dg = g.degree >> 1;
int[] a0 = normalForm(g.coefficients);
int[] a1 = mod(coefficients, g.coefficients);
int[] b0 = {0};
int[] b1 = {1};
while (computeDegree(a1) > dg)
{
int[][] q = div(a0, a1);
a0 = a1;
a1 = q[1];
int[] b2 = add(b0, modMultiply(q[0], b1, g.coefficients));
b0 = b1;
b1 = b2;
}
return new PolynomialGF2mSmallM[]{
new PolynomialGF2mSmallM(field, a1),
new PolynomialGF2mSmallM(field, b1)};
}
/**
* checks if given object is equal to this polynomial.
*
* The method returns false whenever the given object is not polynomial over
* GF(2^m).
*
* @param other object
* @return true or false
*/
public boolean equals(Object other)
{
if (other == null || !(other instanceof PolynomialGF2mSmallM))
{
return false;
}
PolynomialGF2mSmallM p = (PolynomialGF2mSmallM)other;
if ((field.equals(p.field)) && (degree == p.degree)
&& (isEqual(coefficients, p.coefficients)))
{
return true;
}
return false;
}
/**
* Compare two polynomials given as int arrays.
*
* @param a the first polynomial
* @param b the second polynomial
* @return true if a and b represent the
* same polynomials, false otherwise
*/
private static boolean isEqual(int[] a, int[] b)
{
int da = computeDegree(a);
int db = computeDegree(b);
if (da != db)
{
return false;
}
for (int i = 0; i <= da; i++)
{
if (a[i] != b[i])
{
return false;
}
}
return true;
}
/**
* @return the hash code of this polynomial
*/
public int hashCode()
{
int hash = field.hashCode();
for (int j = 0; j < coefficients.length; j++)
{
hash = hash * 31 + coefficients[j];
}
return hash;
}
/**
* Returns a human readable form of the polynomial.
*
* @return a human readable form of the polynomial.
*/
public String toString()
{
String str = " Polynomial over " + field.toString() + ": \n";
for (int i = 0; i < coefficients.length; i++)
{
str = str + field.elementToStr(coefficients[i]) + "Y^" + i + "+";
}
str = str + ";";
return str;
}
/**
* Compute the degree of this polynomial. If this is the zero polynomial,
* the degree is -1.
*/
private void computeDegree()
{
for (degree = coefficients.length - 1; degree >= 0
&& coefficients[degree] == 0; degree--)
{
;
}
}
/**
* Compute the degree of a polynomial.
*
* @param a the polynomial
* @return the degree of the polynomial a. If a is
* the zero polynomial, return -1.
*/
private static int computeDegree(int[] a)
{
int degree;
for (degree = a.length - 1; degree >= 0 && a[degree] == 0; degree--)
{
;
}
return degree;
}
/**
* Strip leading zero coefficients from the given polynomial.
*
* @param a the polynomial
* @return the reduced polynomial
*/
private static int[] normalForm(int[] a)
{
int d = computeDegree(a);
// if a is the zero polynomial
if (d == -1)
{
// return new zero polynomial
return new int[1];
}
// if a already is in normal form
if (a.length == d + 1)
{
// return a clone of a
return IntUtils.clone(a);
}
// else, reduce a
int[] result = new int[d + 1];
System.arraycopy(a, 0, result, 0, d + 1);
return result;
}
}