public void Test04_SquareRoot()
{
TestContext.WriteLine($"ENTER: {nameof(Test04_SquareRoot)}");
Assert.IsTrue(step03_passed, "IsTrue(step03_passed)");
Assert.IsNotNull(gnfs, "IsNotNull(gnfs)");
int maxSetSize = gnfs.CurrentRelationsProgress.FreeRelations.Max(lst => lst.Count);
List <Relation> choosenRelationSet = gnfs.CurrentRelationsProgress.FreeRelations.Where(lst => lst.Count == maxSetSize).First();
SquareFinder squareRootFinder = new SquareFinder(gnfs, choosenRelationSet);
squareRootFinder.CalculateRationalSide();
Tuple <BigInteger, BigInteger> foundFactors = squareRootFinder.CalculateAlgebraicSide(cancelToken);
/* Non-trivial factors also recoverable by doing the following:
*
* BigInteger min = BigInteger.Min(squareRootFinder.RationalSquareRootResidue, squareRootFinder.AlgebraicSquareRootResidue);
* BigInteger max = BigInteger.Max(squareRootFinder.RationalSquareRootResidue, squareRootFinder.AlgebraicSquareRootResidue);
* BigInteger R = max - min;
* BigInteger S = max + min;
* BigInteger P = GCD.FindGCD(gnfs.N, S);
* BigInteger Q = GCD.FindGCD(gnfs.N, R);
*
*/
BigInteger P = foundFactors.Item1;
BigInteger Q = foundFactors.Item2;
Assert.AreNotEqual(1, P, "AreNotEqual(1, P)");
Assert.AreNotEqual(1, Q, "AreNotEqual(1, Q)");
Assert.AreEqual(new BigInteger(1811), P, "AreEqual(1811, P)");
Assert.AreEqual(new BigInteger(1777), Q, "AreEqual(1777, Q)");
step04_passed = true;
TestContext.WriteLine($"{nameof(Test04_SquareRoot)} passed?: {step04_passed}");
TestContext.WriteLine($"LEAVE: {nameof(Test04_SquareRoot)}");
}
public static GNFS FindSquares(CancellationToken cancelToken, GNFS gnfs)
{
if (cancelToken.IsCancellationRequested)
{
return(gnfs);
}
Logging.LogMessage();
Logging.LogMessage($"# of solution sets: {gnfs.CurrentRelationsProgress.FreeRelations.Count}");
Logging.LogMessage();
BigInteger polyBase = gnfs.PolynomialBase;
List <List <Relation> > freeRelations = gnfs.CurrentRelationsProgress.FreeRelations;
// Below randomly selects a solution set to try and find a square root of the polynomial in.
// Each time this step is stopped and restarted, it will try a different solution set.
// Previous used sets are tracked with the List<int> triedFreeRelationIndices
if (triedFreeRelationIndices.Count == freeRelations.Count) // If we have exhausted our solution sets, alert the user. Number wont factor for some reason.
{
Logging.LogMessage("ERROR: ALL RELATION SETS HAVE BEEN TRIED...?");
Logging.LogMessage($"If the number of solution sets ({freeRelations.Count}) is low, you may need to sieve some more and then re-run the matrix solving step.");
Logging.LogMessage("If there are many solution sets, and you have tried them all without finding non-trivial factors, then something is wrong...");
Logging.LogMessage();
return(gnfs);
}
int freeRelationIndex = 0;
do
{
freeRelationIndex = StaticRandom.Next(0, freeRelations.Count);
}while (triedFreeRelationIndices.Contains(freeRelationIndex));
triedFreeRelationIndices.Add(freeRelationIndex); // Add current selection to our list
List <Relation> selectedRelationSet = freeRelations[freeRelationIndex]; // Get the solution set
SquareFinder squareRootFinder = new SquareFinder(gnfs, selectedRelationSet); // If you want to solve for a new solution set, create a new instance
Logging.LogMessage($"Selected solution set # {freeRelationIndex + 1}");
Logging.LogMessage();
Logging.LogMessage($"Selected set (a,b) pairs (count: {selectedRelationSet.Count}): {string.Join(" ", selectedRelationSet.Select(rel => $"({rel.A},{rel.B})"))}");
Logging.LogMessage();
Logging.LogMessage();
Logging.LogMessage();
Logging.LogMessage($"ƒ'(m) = {squareRootFinder.PolynomialDerivative}");
Logging.LogMessage($"ƒ'(m)^2 = {squareRootFinder.PolynomialDerivativeSquared}");
Logging.LogMessage();
Logging.LogMessage("Calculating Rational Square Root.");
Logging.LogMessage("Please wait...");
squareRootFinder.CalculateRationalSide();
Logging.LogMessage("Completed.");
Logging.LogMessage();
Logging.LogMessage($"γ² = {squareRootFinder.RationalProduct} IsSquare? {squareRootFinder.RationalProduct.IsSquare()}");
Logging.LogMessage($"(γ · ƒ'(m))^2 = {squareRootFinder.RationalSquare} IsSquare? {squareRootFinder.RationalSquare.IsSquare()}");
Logging.LogMessage();
Logging.LogMessage();
Logging.LogMessage("Calculating Algebraic Square Root.");
Logging.LogMessage("Please wait...");
Tuple <BigInteger, BigInteger> foundFactors = squareRootFinder.CalculateAlgebraicSide(cancelToken);
BigInteger P = foundFactors.Item1;
BigInteger Q = foundFactors.Item2;
if (cancelToken.IsCancellationRequested && P == 1 && Q == 1)
{
Logging.LogMessage("Square root search aborted!");
return(gnfs);
}
Logging.LogMessage("Completed.");
Logging.LogMessage();
Logging.LogMessage();
if (P != 1 || Q != 1)
{
Logging.LogMessage("NON-TRIVIAL FACTORS FOUND!");
Logging.LogMessage();
}
IPolynomial S = squareRootFinder.S;
IPolynomial SRingSquare = squareRootFinder.SRingSquare;
BigInteger prodS = SRingSquare.Evaluate(polyBase);
IPolynomial reducedS = Polynomial.Modulus(S, gnfs.N);
BigInteger totalProdS = squareRootFinder.TotalS.Evaluate(polyBase) * squareRootFinder.PolynomialDerivative;
BigInteger totalProdModN = totalProdS % gnfs.N;
BigInteger prodSmodN = prodS % gnfs.N;
List <BigInteger> algebraicNumberFieldSquareRoots = squareRootFinder.AlgebraicResults;
BigInteger rationalSquareRoot = squareRootFinder.RationalSquareRootResidue;
BigInteger algebraicSquareRoot = squareRootFinder.AlgebraicSquareRootResidue;
Logging.LogMessage($"∏ Sᵢ =");
Logging.LogMessage($"{squareRootFinder.TotalS}");
Logging.LogMessage();
Logging.LogMessage($"∏ Sᵢ (mod ƒ) =");
Logging.LogMessage($"{reducedS}");
Logging.LogMessage();
Logging.LogMessage("Polynomial ring:");
Logging.LogMessage($"({string.Join(") * (", squareRootFinder.PolynomialRing.Select(ply => ply.ToString()))})");
Logging.LogMessage();
Logging.LogMessage("Primes:");
Logging.LogMessage($"{string.Join(" * ", squareRootFinder.AlgebraicPrimes)}"); // .RelationsSet.Select(rel => rel.B).Distinct().OrderBy(relB => relB))
Logging.LogMessage();
Logging.LogMessage();
Logging.LogMessage($"X² / ƒ(m) = {squareRootFinder.AlgebraicProductModF} IsSquare? {squareRootFinder.AlgebraicProductModF.IsSquare()}");
Logging.LogMessage();
Logging.LogMessage($"");
Logging.LogMessage($"AlgebraicPrimes: {squareRootFinder.AlgebraicPrimes.FormatString(false)}");
Logging.LogMessage($"AlgebraicResults: {squareRootFinder.AlgebraicResults.FormatString(false)}");
Logging.LogMessage($"");
Logging.LogMessage($"*****************************");
Logging.LogMessage($"");
Logging.LogMessage($"AlgebraicSquareRootResidue: {squareRootFinder.AlgebraicSquareRootResidue}");
Logging.LogMessage($"");
Logging.LogMessage($"AlgebraicNumberFieldSquareRoots: {algebraicNumberFieldSquareRoots.FormatString(false)}");
Logging.LogMessage($"");
Logging.LogMessage($" RationalSquareRoot : {rationalSquareRoot}");
Logging.LogMessage($" AlgebraicSquareRoot: {algebraicSquareRoot} ");
Logging.LogMessage($"");
Logging.LogMessage($"*****************************");
Logging.LogMessage($"S (x) = {prodSmodN} IsSquare? {prodSmodN.IsSquare()}");
Logging.LogMessage();
Logging.LogMessage("Roots of S(x):");
Logging.LogMessage($"{{{string.Join(", ", squareRootFinder.RootsOfS.Select(tup => (tup.Item2 > 1) ? $"{tup.Item1}/{tup.Item2}" : $"{tup.Item1}"))}}}");