org.cryptimeleon.math.structures.rings.polynomial

Class LagrangeUtils

java.lang.Object

org.cryptimeleon.math.structures.rings.polynomial.LagrangeUtils

public class LagrangeUtils
extends java.lang.Object

Contains methods for calculating lagrange polynomials as well as interpolation in the exponent of group elements.

For interpolating a polynomial given a set of known evaluations, see PolynomialRing.getPoly(Map).

Constructor Summary

Constructors

Constructor and Description
LagrangeUtils()

Method Summary

Modifier and Type	Method and Description
static java.math.BigInteger	computeCoefficient(java.math.BigInteger i, java.util.Set<java.math.BigInteger> S, java.math.BigInteger x, Zp field) Compute the Lagrange coefficient \(\ell_j(x)\) over the specified field.
static Zp.ZpElement	computeCoefficient(Zp.ZpElement i, java.util.Set<Zp.ZpElement> S, Zp.ZpElement x) Compute the Lagrange coefficient \(\ell_j(x)\).
static GroupElement	interpolateInTheExponent(java.util.Map<java.math.BigInteger,GroupElement> givenElems, java.math.BigInteger newCoord) Given a set of group elements with a common basis whose exponents implicitly define a polynomial, returns that common basis to the power of that polynomial evaluated at point newPoint.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

LagrangeUtils

public LagrangeUtils()

Method Detail

interpolateInTheExponent

public static GroupElement interpolateInTheExponent(java.util.Map<java.math.BigInteger,GroupElement> givenElems,
                                                    java.math.BigInteger newCoord)

Given a set of group elements with a common basis whose exponents implicitly define a polynomial, returns that common basis to the power of that polynomial evaluated at point newPoint.

Specifically, let \(g_{a_1}\) to \(g_{a_m}\) be the group elements contained in map knownPoints where \(a_i\) is the BigInteger mapped to \(g_{a_i}\). Given some generator \(g\) of the group, \(g_{a_1}\) to \(g_{a_m}\) can be written as \(g_{a_1} = g^{p(a_1)}, ..., g_{a_m} = g^{p(a_m)}\) for some implicitly defined polynomial \(p\) of degree \(m-1\).

Let \(x\) be newPoint and let \(\ell_{a_i}\) be the i-th Lagrange polynomial. Then this method returns \(g^{p(x)}\) using Lagrange interpolation by computing \(g^{p(x)} = \prod_{a_i}{g_{a_i}^{\ell_{a_i}(x)}}\).

Parameters:

givenElems - maps \(a_i\) to \(g^{p(a_i)}\) for some implicit polynomial \(p\)

newCoord - the x-coordinate to return \(g\) to the power \(p(x)\) for

Returns:

the result of interpolating in the exponent, \(g^{p(x)}\)

computeCoefficient

public static Zp.ZpElement computeCoefficient(Zp.ZpElement i,
                                              java.util.Set<Zp.ZpElement> S,
                                              Zp.ZpElement x)

Compute the Lagrange coefficient \(\ell_j(x)\).

Parameters:

i - j-th x coordinate

S - set of x coordinates

x - x coordinate to evaluate the lagrange basis polynomial at

Returns:

the lagrange basis polynomial evaluated at coordinate x

computeCoefficient

public static java.math.BigInteger computeCoefficient(java.math.BigInteger i,
                                                      java.util.Set<java.math.BigInteger> S,
                                                      java.math.BigInteger x,
                                                      Zp field)

Compute the Lagrange coefficient \(\ell_j(x)\) over the specified field.

Parameters:

i - j-th x coordinate

S - set of x coordinates

x - x coordinate to evaluate the lagrange basis polynomial at

field - the field to do the computation over

Returns:

the lagrange basis polynomial evaluated at coordinate x in the given field