Bouncy Castle Cryptography Library 1.80 · org.bouncycastle.pqc.legacy.math.linearalgebra.PolynomialGF2mSmallM

Modifier and Type	Field and Description
private int[]	coefficients For the polynomial representation the map f: R->Z*, `poly(X) -> [coef_0, coef_1, ...]` is used, where `coef_i` is the `i`th coefficient of the polynomial represented as int (see `GF2mField`).
private int	degree the degree of this polynomial
private GF2mField	field the finite field GF(2^m)
public static final char	RANDOM_IRREDUCIBLE_POLYNOMIAL Constant used for polynomial construction (see constructor `PolynomialGF2mSmallM(GF2mField, int, char, SecureRandom)`).

Access	Constructor and Description
public	PolynomialGF2mSmallM(GF2mField the finite field GF(2^m) field) Construct the zero polynomial over the finite field GF(2^m).
public	PolynomialGF2mSmallM(GF2mField the finite field GF(2^m) field, int degree of polynomial deg, char type of polynomial typeOfPolynomial, SecureRandom PRNG sr) Construct a polynomial over the finite field GF(2^m).
public	PolynomialGF2mSmallM(GF2mField the finite field GF(2^m) field, int the degree of the monomial degree) Construct a monomial of the given degree over the finite field GF(2^m).
public	PolynomialGF2mSmallM(GF2mField finite field GF2m field, int[] the coefficient vector coeffs) Construct the polynomial over the given finite field GF(2^m) from the given coefficient vector.
public	PolynomialGF2mSmallM(GF2mField the finite field GF(2^m) field, byte[] byte[] polynomial in byte array form enc) Create a polynomial over the finite field GF(2^m).
public	PolynomialGF2mSmallM(PolynomialGF2mSmallM another `PolynomialGF2mSmallM` other) Copy constructor.
public	PolynomialGF2mSmallM(GF2mVector the coefficient vector vect) Create a polynomial over the finite field GF(2^m) out of the given coefficient vector.

Method Summary

Modifier and Type	Method and Description
public PolynomialGF2mSmallM	Returns: `this + a` (newly created) add(PolynomialGF2mSmallM the addend addend) Compute the sum of this polynomial and the given polynomial.
private int[]	Returns: a + b add(int[] the first polynomial a, int[] the second polynomial b) Compute the sum of two polynomials a and b over the finite field `GF(2^m)`.
public PolynomialGF2mSmallM	Returns: `this + X^k` addMonomial(int the degree of the monomial degree) Compute the sum of this polynomial and the monomial of the given degree.
public void	addToThis(PolynomialGF2mSmallM the addend addend) Add the given polynomial to this polynomial (overwrite this).
private void	computeDegree() Compute the degree of this polynomial.
private static int	Returns: the degree of the polynomial `a`. If `a` is the zero polynomial, return -1. computeDegree(int[] the polynomial a) Compute the degree of a polynomial.
private int[]	Returns: the generated irreducible polynomial createRandomIrreduciblePolynomial(int polynomial degree deg, SecureRandom source of randomness sr) Create an irreducible polynomial with the given degree over the field `GF(2^m)`.
public PolynomialGF2mSmallM[]	Returns: polynomial pair = {q,r} where this = q*f+r and deg(r) < deg(f); div(PolynomialGF2mSmallM a polynomial f) Divide this polynomial by the given polynomial.
private int[][]	Returns: int[][] {q,r}, where a = q*f+r and deg(r) < deg(f); div(int[] the first polynomial a, int[] the second polynomial f) Compute the result of the division of two polynomials over the field `GF(2^m)`.
public boolean	Returns: true or false equals(Object object other) Overrides java.lang.Object.equals. checks if given object is equal to this polynomial.
public int	Returns: `this(e)` evaluateAt(int the element of the finite field GF(2^m) e) Evaluate this polynomial `p` at a value `e` (in `GF(2^m)`) with the Horner scheme.
public PolynomialGF2mSmallM	Returns: GCD(this, f) gcd(PolynomialGF2mSmallM polynomial f) Return the greatest common divisor of this and a polynomial f
private int[]	Returns: `gcd(f, g)` gcd(int[] the first polynomial f, int[] the second polynomial g) Return the greatest common divisor of two polynomials over the field `GF(2^m)`.
public int	Returns: the coefficient with the given index getCoefficient(int the index index) Return the coefficient with the given index.
public int	Returns: int degree of this polynomial if this is zero polynomial return -1 getDegree() Return the degree of this polynomial
public byte[]	Returns: the encoded polynomial getEncoded() Returns encoded polynomial, i.e., this polynomial in byte array form
public int	Returns: the head coefficient of this polynomial getHeadCoefficient()
public int	Returns: the hash code of this polynomial hashCode() Overrides java.lang.Object.hashCode. Returns a hash code value for this object.
private static int	Returns: the head coefficient of `a` headCoefficient(int[] the polynomial a) Return the head coefficient of a polynomial.
private static boolean	Returns: `true` if `a` and `b` represent the same polynomials, `false` otherwise isEqual(int[] the first polynomial a, int[] the second polynomial b) Compare two polynomials given as int arrays.
private boolean	Returns: true if a is irreducible, false otherwise isIrreducible(int[] the polynomial to check a) Check a polynomial for irreducibility over the field `GF(2^m)`.
public PolynomialGF2mSmallM	Returns: `this mod f` mod(PolynomialGF2mSmallM the reduction polynomial f) Reduce this polynomial modulo another polynomial.
private int[]	Returns: `a mod f` mod(int[] the polynomial a, int[] the reduction polynomial f) Reduce a polynomial modulo another polynomial.
public PolynomialGF2mSmallM	Returns: `this * divisor^(-1) mod modulus` modDiv(PolynomialGF2mSmallM the divisor divisor, PolynomialGF2mSmallM the reduction polynomial modulus) Compute the result of the division of this polynomial by another polynomial modulo a third polynomial.
private int[]	Returns: `a * b^(-1) mod g` modDiv(int[] the first polynomial a, int[] the second polynomial b, int[] the reduction polynomial g) Compute the result of the division of two polynomials modulo a third polynomial over the field `GF(2^m)`.
public PolynomialGF2mSmallM	Returns: `this^(-1) mod a` modInverse(PolynomialGF2mSmallM the reduction polynomial a) Compute the inverse of this polynomial modulo the given polynomial.
public PolynomialGF2mSmallM	Returns: `this * a mod b` modMultiply(PolynomialGF2mSmallM another polynomial a, PolynomialGF2mSmallM the reduction polynomial b) Compute the product of this polynomial and another polynomial modulo a third polynomial.
private int[]	Returns: `a * b mod g` modMultiply(int[] the first polynomial a, int[] the second polynomial b, int[] the reduction polynomial g) Compute the product of two polynomials modulo a third polynomial over the finite field `GF(2^m)`.
public PolynomialGF2mSmallM[]	Returns: PolynomialGF2mSmallM[] {a,b} with bthis = a mod g and deg(a)<= deg(g)/2 modPolynomialToFracton(PolynomialGF2mSmallM the reduction polynomial g) Compute a polynomial pair (a,b) from this polynomial and the given polynomial g with the property bthis = a mod g and deg(a)<=deg(g)/2.
public PolynomialGF2mSmallM	Returns: `this^2` modulo the reduction polynomial implicitly given via the squaring matrix modSquareMatrix(PolynomialGF2mSmallM[] the squaring matrix matrix) Square this polynomial using a squaring matrix.
public PolynomialGF2mSmallM	Returns: `this^(1/2) mod a` modSquareRoot(PolynomialGF2mSmallM the reduction polynomial a) Compute the square root of this polynomial modulo the given polynomial.
public PolynomialGF2mSmallM	Returns: `this^(1/2)` modulo the reduction polynomial implicitly given via the square root matrix modSquareRootMatrix(PolynomialGF2mSmallM[] the matrix for computing square roots in `(GF(2^m))^t` the polynomial ring defining the square root matrix matrix) Compute the square root of this polynomial using a square root matrix.
public PolynomialGF2mSmallM	Returns: `this * factor` multiply(PolynomialGF2mSmallM the polynomial factor) Compute the product of this polynomial and the given factor using a Karatzuba like scheme.
private int[]	Returns: a * b multiply(int[] the first polynomial a, int[] the second polynomial b) Compute the product of two polynomials over the field `GF(2^m)` using a Karatzuba like multiplication.
public void	multThisWithElement(int an element of the finite field GF(2^m) element) Multiply this polynomial with an element from GF(2^m).
public PolynomialGF2mSmallM	Returns: `this * element` (newly created) multWithElement(int an element of the finite field GF(2^m) element) Compute the product of this polynomial with an element from GF(2^m).
private int[]	Returns: `a * element` multWithElement(int[] the polynomial a, int an element of the finite field GF(2^m) element) Compute the product of a polynomial a with an element from the finite field `GF(2^m)`.
public PolynomialGF2mSmallM	Returns: `this * X^k` multWithMonomial(int the degree of the monomial k) Compute the product of this polynomial with a monomial X^k.
private static int[]	Returns: `a * X^k` multWithMonomial(int[] the polynomial a, int the degree of the monomial k) Compute the product of a polynomial with a monomial X^k.
private static int[]	Returns: the reduced polynomial normalForm(int[] the polynomial a) Strip leading zero coefficients from the given polynomial.
public String	Returns: a human readable form of the polynomial. toString() Overrides java.lang.Object.toString. Returns a human readable form of the polynomial.

Inherited from java.lang.Object:: clone finalize getClass notify notifyAll wait wait wait

coefficients	back to summary
private int[] coefficients For the polynomial representation the map f: R->Z*, `poly(X) -> [coef_0, coef_1, ...]` is used, where `coef_i` is the `i`th coefficient of the polynomial represented as int (see `GF2mField`). The polynomials are stored as int arrays.

degree	back to summary
private int degree the degree of this polynomial

field	back to summary
private GF2mField field the finite field GF(2^m)

RANDOM_IRREDUCIBLE_POLYNOMIAL	back to summary
public static final char RANDOM_IRREDUCIBLE_POLYNOMIAL Constant used for polynomial construction (see constructor `PolynomialGF2mSmallM(GF2mField, int, char, SecureRandom)`).

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mField field) Construct the zero polynomial over the finite field GF(2^m). Parameters field:GF2mField the finite field GF(2^m)

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mField field, int deg, char typeOfPolynomial, SecureRandom sr) Construct a polynomial over the finite field GF(2^m). Parameters field:GF2mField the finite field GF(2^m) deg:int degree of polynomial typeOfPolynomial:char type of polynomial sr:SecureRandom PRNG

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mField field, int degree) Construct a monomial of the given degree over the finite field GF(2^m). Parameters field:GF2mField the finite field GF(2^m) degree:int the degree of the monomial

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mField field, int[] coeffs) Construct the polynomial over the given finite field GF(2^m) from the given coefficient vector. Parameters field:GF2mField finite field GF2m coeffs:int[] the coefficient vector

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mField field, byte[] enc) Create a polynomial over the finite field GF(2^m). Parameters field:GF2mField the finite field GF(2^m) enc:byte[] byte[] polynomial in byte array form

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(PolynomialGF2mSmallM other) Copy constructor. Parameters other:PolynomialGF2mSmallM another `PolynomialGF2mSmallM`

PolynomialGF2mSmallM	back to summary
public PolynomialGF2mSmallM(GF2mVector vect) Create a polynomial over the finite field GF(2^m) out of the given coefficient vector. The finite field is also obtained from the `GF2mVector`. Parameters vect:GF2mVector the coefficient vector

Method Detail

add back to summary

add	back to summary
public PolynomialGF2mSmallM add(PolynomialGF2mSmallM addend) Compute the sum of this polynomial and the given polynomial. Parameters addend:PolynomialGF2mSmallM the addend Returns:PolynomialGF2mSmallM `this + a` (newly created)

public PolynomialGF2mSmallM add(PolynomialGF2mSmallM addend)

Compute the sum of this polynomial and the given polynomial.

Parameters

addend:PolynomialGF2mSmallM: the addend

Returns:PolynomialGF2mSmallM

this + a (newly created)

add back to summary

add	back to summary
private int[] add(int[] a, int[] b) Compute the sum of two polynomials a and b over the finite field `GF(2^m)`. Parameters a:int[] the first polynomial b:int[] the second polynomial Returns:int[] a + b

private int[] add(int[] a, int[] b)

Compute the sum of two polynomials a and b over the finite field GF(2^m).

Parameters

a:int[]: the first polynomial
b:int[]: the second polynomial

Returns:int[]

a + b

addMonomial back to summary

addMonomial	back to summary
public PolynomialGF2mSmallM addMonomial(int degree) Compute the sum of this polynomial and the monomial of the given degree. Parameters degree:int the degree of the monomial Returns:PolynomialGF2mSmallM `this + X^k`

public PolynomialGF2mSmallM addMonomial(int degree)

Compute the sum of this polynomial and the monomial of the given degree.

Parameters

degree:int: the degree of the monomial

Returns:PolynomialGF2mSmallM

this + X^k

addToThis	back to summary
public void addToThis(PolynomialGF2mSmallM addend) Add the given polynomial to this polynomial (overwrite this). Parameters addend:PolynomialGF2mSmallM the addend

addToThis

back to summary

public void addToThis(PolynomialGF2mSmallM addend)

Add the given polynomial to this polynomial (overwrite this).

Parameters

addend:PolynomialGF2mSmallM: the addend

computeDegree	back to summary
private void computeDegree() Compute the degree of this polynomial. If this is the zero polynomial, the degree is -1.

computeDegree back to summary

computeDegree	back to summary
private static int computeDegree(int[] a) Compute the degree of a polynomial. Parameters a:int[] the polynomial Returns:int the degree of the polynomial `a`. If `a` is the zero polynomial, return -1.

private static int computeDegree(int[] a)

Compute the degree of a polynomial.

Parameters

a:int[]: the polynomial

Returns:int

the degree of the polynomial a. If a is the zero polynomial, return -1.

createRandomIrreduciblePolynomial back to summary

createRandomIrreduciblePolynomial	back to summary
private int[] createRandomIrreduciblePolynomial(int deg, SecureRandom sr) Create an irreducible polynomial with the given degree over the field `GF(2^m)`. Parameters deg:int polynomial degree sr:SecureRandom source of randomness Returns:int[] the generated irreducible polynomial

private int[] createRandomIrreduciblePolynomial(int deg, SecureRandom sr)

Create an irreducible polynomial with the given degree over the field GF(2^m).

Parameters

deg:int: polynomial degree
sr:SecureRandom: source of randomness

Returns:int[]

the generated irreducible polynomial

div	back to summary
public PolynomialGF2mSmallM[] div(PolynomialGF2mSmallM f) Divide this polynomial by the given polynomial. Parameters f:PolynomialGF2mSmallM a polynomial Returns:PolynomialGF2mSmallM[] polynomial pair = {q,r} where this = q*f+r and deg(r) < deg(f);

div

back to summary

public PolynomialGF2mSmallM[] div(PolynomialGF2mSmallM f)

Divide this polynomial by the given polynomial.

Parameters

f:PolynomialGF2mSmallM: a polynomial

Returns:PolynomialGF2mSmallM[]

polynomial pair = {q,r} where this = q*f+r and deg(r) < deg(f);

div back to summary

div	back to summary
private int[][] div(int[] a, int[] f) Compute the result of the division of two polynomials over the field `GF(2^m)`. Parameters a:int[] the first polynomial f:int[] the second polynomial Returns:int[][] int[][] {q,r}, where a = q*f+r and deg(r) < deg(f);

private int[][] div(int[] a, int[] f)

Compute the result of the division of two polynomials over the field GF(2^m).

Parameters

a:int[]: the first polynomial
f:int[]: the second polynomial

Returns:int[][]

int[][] {q,r}, where a = q*f+r and deg(r) < deg(f);

equals	back to summary
public boolean equals(Object other) Overrides java.lang.Object.equals. checks if given object is equal to this polynomial. The method returns false whenever the given object is not polynomial over GF(2^m). Parameters other:Object object Returns:boolean true or false

equals

back to summary

public boolean equals(Object other)

Overrides java.lang.Object.equals.

checks if given object is equal to this polynomial.

The method returns false whenever the given object is not polynomial over GF(2^m).

Parameters

other:Object: object

Returns:boolean

true or false

evaluateAt back to summary

evaluateAt	back to summary
public int evaluateAt(int e) Evaluate this polynomial `p` at a value `e` (in `GF(2^m)`) with the Horner scheme. Parameters e:int the element of the finite field GF(2^m) Returns:int `this(e)`

public int evaluateAt(int e)

Evaluate this polynomial p at a value e (in GF(2^m)) with the Horner scheme.

Parameters

e:int: the element of the finite field GF(2^m)

Returns:int

this(e)

gcd	back to summary
public PolynomialGF2mSmallM gcd(PolynomialGF2mSmallM f) Return the greatest common divisor of this and a polynomial f Parameters f:PolynomialGF2mSmallM polynomial Returns:PolynomialGF2mSmallM GCD(this, f)

gcd

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public PolynomialGF2mSmallM gcd(PolynomialGF2mSmallM f)

Return the greatest common divisor of this and a polynomial f

Parameters

f:PolynomialGF2mSmallM: polynomial

Returns:PolynomialGF2mSmallM

GCD(this, f)

gcd back to summary

gcd	back to summary
private int[] gcd(int[] f, int[] g) Return the greatest common divisor of two polynomials over the field `GF(2^m)`. Parameters f:int[] the first polynomial g:int[] the second polynomial Returns:int[] `gcd(f, g)`

private int[] gcd(int[] f, int[] g)

Return the greatest common divisor of two polynomials over the field GF(2^m).

Parameters

f:int[]: the first polynomial
g:int[]: the second polynomial

Returns:int[]

gcd(f, g)

getCoefficient	back to summary
public int getCoefficient(int index) Return the coefficient with the given index. Parameters index:int the index Returns:int the coefficient with the given index

getCoefficient

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public int getCoefficient(int index)

Return the coefficient with the given index.

Parameters

index:int: the index

Returns:int

the coefficient with the given index

getDegree	back to summary
public int getDegree() Return the degree of this polynomial Returns:int int degree of this polynomial if this is zero polynomial return -1

getDegree

back to summary

public int getDegree()

Return the degree of this polynomial

Returns:int: int degree of this polynomial if this is zero polynomial return -1

getEncoded	back to summary
public byte[] getEncoded() Returns encoded polynomial, i.e., this polynomial in byte array form Returns:byte[] the encoded polynomial

getEncoded

back to summary

public byte[] getEncoded()

Returns encoded polynomial, i.e., this polynomial in byte array form

Returns:byte[]: the encoded polynomial

getHeadCoefficient	back to summary
public int getHeadCoefficient() Returns:int the head coefficient of this polynomial

hashCode back to summary

hashCode	back to summary
public int hashCode() Overrides java.lang.Object.hashCode. Doc from java.lang.Object.hashCode. Returns a hash code value for this object. This method is supported for the benefit of hash tables such as those provided by `java.util.HashMap`. The general contract of `hashCode` is: Whenever it is invoked on the same object more than once during an execution of a Java application, the `hashCode` method must consistently return the same integer, provided no information used in `equals` comparisons on the object is modified. This integer need not remain consistent from one execution of an application to another execution of the same application. If two objects are equal according to the `equals` method, then calling the `hashCode` method on each of the two objects must produce the same integer result. It is not required that if two objects are unequal according to the `equals` method, then calling the `hashCode` method on each of the two objects must produce distinct integer results. However, the programmer should be aware that producing distinct integer results for unequal objects may improve the performance of hash tables. Returns:int the hash code of this polynomial

public int hashCode()

Overrides java.lang.Object.hashCode.

Doc from java.lang.Object.hashCode.

Returns a hash code value for this object. This method is supported for the benefit of hash tables such as those provided by java.util.HashMap.

The general contract of hashCode is:

Whenever it is invoked on the same object more than once during an execution of a Java application, the hashCode method must consistently return the same integer, provided no information used in equals comparisons on the object is modified. This integer need not remain consistent from one execution of an application to another execution of the same application.
If two objects are equal according to the equals method, then calling the hashCode method on each of the two objects must produce the same integer result.
It is not required that if two objects are unequal according to the equals method, then calling the hashCode method on each of the two objects must produce distinct integer results. However, the programmer should be aware that producing distinct integer results for unequal objects may improve the performance of hash tables.

Returns:int: the hash code of this polynomial

headCoefficient back to summary

headCoefficient	back to summary
private static int headCoefficient(int[] a) Return the head coefficient of a polynomial. Parameters a:int[] the polynomial Returns:int the head coefficient of `a`

private static int headCoefficient(int[] a)

Return the head coefficient of a polynomial.

Parameters

a:int[]: the polynomial

Returns:int

the head coefficient of a

isEqual back to summary

isEqual	back to summary
private static boolean isEqual(int[] a, int[] b) Compare two polynomials given as int arrays. Parameters a:int[] the first polynomial b:int[] the second polynomial Returns:boolean `true` if `a` and `b` represent the same polynomials, `false` otherwise

private static boolean isEqual(int[] a, int[] b)

Compare two polynomials given as int arrays.

Parameters

a:int[]: the first polynomial
b:int[]: the second polynomial

Returns:boolean

true if a and b represent the same polynomials, false otherwise

isIrreducible back to summary

isIrreducible	back to summary
private boolean isIrreducible(int[] a) Check a polynomial for irreducibility over the field `GF(2^m)`. Parameters a:int[] the polynomial to check Returns:boolean true if a is irreducible, false otherwise

private boolean isIrreducible(int[] a)

Check a polynomial for irreducibility over the field GF(2^m).

Parameters

a:int[]: the polynomial to check

Returns:boolean

true if a is irreducible, false otherwise

mod back to summary

mod	back to summary
public PolynomialGF2mSmallM mod(PolynomialGF2mSmallM f) Reduce this polynomial modulo another polynomial. Parameters f:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM `this mod f`

public PolynomialGF2mSmallM mod(PolynomialGF2mSmallM f)

Reduce this polynomial modulo another polynomial.

Parameters

f:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM

this mod f

mod back to summary

mod	back to summary
private int[] mod(int[] a, int[] f) Reduce a polynomial modulo another polynomial. Parameters a:int[] the polynomial f:int[] the reduction polynomial Returns:int[] `a mod f`

private int[] mod(int[] a, int[] f)

Reduce a polynomial modulo another polynomial.

Parameters

a:int[]: the polynomial
f:int[]: the reduction polynomial

Returns:int[]

a mod f

modDiv back to summary

modDiv	back to summary
public PolynomialGF2mSmallM modDiv(PolynomialGF2mSmallM divisor, PolynomialGF2mSmallM modulus) Compute the result of the division of this polynomial by another polynomial modulo a third polynomial. Parameters divisor:PolynomialGF2mSmallM the divisor modulus:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM `this * divisor^(-1) mod modulus`

public PolynomialGF2mSmallM modDiv(PolynomialGF2mSmallM divisor, PolynomialGF2mSmallM modulus)

Compute the result of the division of this polynomial by another polynomial modulo a third polynomial.

Parameters

divisor:PolynomialGF2mSmallM: the divisor
modulus:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM

this * divisor^(-1) mod modulus

modDiv back to summary

modDiv	back to summary
private int[] modDiv(int[] a, int[] b, int[] g) Compute the result of the division of two polynomials modulo a third polynomial over the field `GF(2^m)`. Parameters a:int[] the first polynomial b:int[] the second polynomial g:int[] the reduction polynomial Returns:int[] `a * b^(-1) mod g`

private int[] modDiv(int[] a, int[] b, int[] g)

Compute the result of the division of two polynomials modulo a third polynomial over the field GF(2^m).

Parameters

a:int[]: the first polynomial
b:int[]: the second polynomial
g:int[]: the reduction polynomial

Returns:int[]

a * b^(-1) mod g

modInverse back to summary

modInverse	back to summary
public PolynomialGF2mSmallM modInverse(PolynomialGF2mSmallM a) Compute the inverse of this polynomial modulo the given polynomial. Parameters a:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM `this^(-1) mod a`

public PolynomialGF2mSmallM modInverse(PolynomialGF2mSmallM a)

Compute the inverse of this polynomial modulo the given polynomial.

Parameters

a:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM

this^(-1) mod a

modMultiply back to summary

modMultiply	back to summary
public PolynomialGF2mSmallM modMultiply(PolynomialGF2mSmallM a, PolynomialGF2mSmallM b) Compute the product of this polynomial and another polynomial modulo a third polynomial. Parameters a:PolynomialGF2mSmallM another polynomial b:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM `this * a mod b`

public PolynomialGF2mSmallM modMultiply(PolynomialGF2mSmallM a, PolynomialGF2mSmallM b)

Compute the product of this polynomial and another polynomial modulo a third polynomial.

Parameters

a:PolynomialGF2mSmallM: another polynomial
b:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM

this * a mod b

modMultiply back to summary

modMultiply	back to summary
private int[] modMultiply(int[] a, int[] b, int[] g) Compute the product of two polynomials modulo a third polynomial over the finite field `GF(2^m)`. Parameters a:int[] the first polynomial b:int[] the second polynomial g:int[] the reduction polynomial Returns:int[] `a * b mod g`

private int[] modMultiply(int[] a, int[] b, int[] g)

Compute the product of two polynomials modulo a third polynomial over the finite field GF(2^m).

Parameters

a:int[]: the first polynomial
b:int[]: the second polynomial
g:int[]: the reduction polynomial

Returns:int[]

a * b mod g

modPolynomialToFracton	back to summary
public PolynomialGF2mSmallM[] modPolynomialToFracton(PolynomialGF2mSmallM g) Compute a polynomial pair (a,b) from this polynomial and the given polynomial g with the property bthis = a mod g and deg(a)<=deg(g)/2. Parameters g:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM[] PolynomialGF2mSmallM[] {a,b} with bthis = a mod g and deg(a)<= deg(g)/2

modPolynomialToFracton

back to summary

public PolynomialGF2mSmallM[] modPolynomialToFracton(PolynomialGF2mSmallM g)

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.

Parameters

g:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM[]

PolynomialGF2mSmallM[] {a,b} with b*this = a mod g and deg(a)<= deg(g)/2

modSquareMatrix back to summary

modSquareMatrix	back to summary
public PolynomialGF2mSmallM modSquareMatrix(PolynomialGF2mSmallM[] matrix) Square this polynomial using a squaring matrix. Parameters matrix:PolynomialGF2mSmallM[] the squaring matrix Returns:PolynomialGF2mSmallM `this^2` modulo the reduction polynomial implicitly given via the squaring matrix

public PolynomialGF2mSmallM modSquareMatrix(PolynomialGF2mSmallM[] matrix)

Square this polynomial using a squaring matrix.

Parameters

matrix:PolynomialGF2mSmallM[]: the squaring matrix

Returns:PolynomialGF2mSmallM

this^2 modulo the reduction polynomial implicitly given via the squaring matrix

modSquareRoot back to summary

modSquareRoot	back to summary
public PolynomialGF2mSmallM modSquareRoot(PolynomialGF2mSmallM a) Compute the square root of this polynomial modulo the given polynomial. Parameters a:PolynomialGF2mSmallM the reduction polynomial Returns:PolynomialGF2mSmallM `this^(1/2) mod a`

public PolynomialGF2mSmallM modSquareRoot(PolynomialGF2mSmallM a)

Compute the square root of this polynomial modulo the given polynomial.

Parameters

a:PolynomialGF2mSmallM: the reduction polynomial

Returns:PolynomialGF2mSmallM

this^(1/2) mod a

modSquareRootMatrix back to summary

modSquareRootMatrix	back to summary
public PolynomialGF2mSmallM modSquareRootMatrix(PolynomialGF2mSmallM[] matrix) Compute the square root of this polynomial using a square root matrix. Parameters matrix:PolynomialGF2mSmallM[] the matrix for computing square roots in `(GF(2^m))^t` the polynomial ring defining the square root matrix Returns:PolynomialGF2mSmallM `this^(1/2)` modulo the reduction polynomial implicitly given via the square root matrix

public PolynomialGF2mSmallM modSquareRootMatrix(PolynomialGF2mSmallM[] matrix)

Compute the square root of this polynomial using a square root matrix.

Parameters

matrix:PolynomialGF2mSmallM[]: the matrix for computing square roots in (GF(2^m))^t the polynomial ring defining the square root matrix

Returns:PolynomialGF2mSmallM

this^(1/2) modulo the reduction polynomial implicitly given via the square root matrix

multiply back to summary

multiply	back to summary
public PolynomialGF2mSmallM multiply(PolynomialGF2mSmallM factor) Compute the product of this polynomial and the given factor using a Karatzuba like scheme. Parameters factor:PolynomialGF2mSmallM the polynomial Returns:PolynomialGF2mSmallM `this * factor`

public PolynomialGF2mSmallM multiply(PolynomialGF2mSmallM factor)

Compute the product of this polynomial and the given factor using a Karatzuba like scheme.

Parameters

factor:PolynomialGF2mSmallM: the polynomial

Returns:PolynomialGF2mSmallM

this * factor

multiply back to summary

multiply	back to summary
private int[] multiply(int[] a, int[] b) Compute the product of two polynomials over the field `GF(2^m)` using a Karatzuba like multiplication. Parameters a:int[] the first polynomial b:int[] the second polynomial Returns:int[] a * b

private int[] multiply(int[] a, int[] b)

Compute the product of two polynomials over the field GF(2^m) using a Karatzuba like multiplication.

Parameters

a:int[]: the first polynomial
b:int[]: the second polynomial

Returns:int[]

a * b

multThisWithElement back to summary

multThisWithElement	back to summary
public void multThisWithElement(int element) Multiply this polynomial with an element from GF(2^m). Parameters element:int an element of the finite field GF(2^m) Exceptions ArithmeticException: if `element` is not an element of the finite field this polynomial is defined over.

public void multThisWithElement(int element)

Multiply this polynomial with an element from GF(2^m).

Parameters

element:int: an element of the finite field GF(2^m)

Exceptions

ArithmeticException:: if element is not an element of the finite field this polynomial is defined over.

multWithElement back to summary

multWithElement	back to summary
public PolynomialGF2mSmallM multWithElement(int element) Compute the product of this polynomial with an element from GF(2^m). Parameters element:int an element of the finite field GF(2^m) Returns:PolynomialGF2mSmallM `this * element` (newly created) Exceptions ArithmeticException: if `element` is not an element of the finite field this polynomial is defined over.

public PolynomialGF2mSmallM multWithElement(int element)

Compute the product of this polynomial with an element from GF(2^m).

Parameters

element:int: an element of the finite field GF(2^m)

Returns:PolynomialGF2mSmallM

this * element (newly created)

Exceptions

ArithmeticException:: if element is not an element of the finite field this polynomial is defined over.

multWithElement back to summary

private int[] multWithElement(int[] a, int element)

Compute the product of a polynomial a with an element from the finite field GF(2^m).

Parameters

a:int[]: the polynomial
element:int: an element of the finite field GF(2^m)

Returns:int[]

a * element

multWithMonomial back to summary

public PolynomialGF2mSmallM multWithMonomial(int k)

Compute the product of this polynomial with a monomial X^k.

Parameters

k:int: the degree of the monomial

Returns:PolynomialGF2mSmallM

this * X^k

multWithMonomial back to summary

private static int[] multWithMonomial(int[] a, int k)

Compute the product of a polynomial with a monomial X^k.

Parameters

a:int[]: the polynomial
k:int: the degree of the monomial

Returns:int[]

a * X^k

normalForm

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private static int[] normalForm(int[] a)

Strip leading zero coefficients from the given polynomial.

Parameters

a:int[]: the polynomial

Returns:int[]

the reduced polynomial

toString

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public String toString()

Overrides java.lang.Object.toString.

Returns a human readable form of the polynomial.

Returns:String: a human readable form of the polynomial.

public Class PolynomialGF2mSmallM

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