さとしん satoshin-des

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🧑‍💻 I'm a master's student.
🌱 I’m currently learning lattice theory!
📫 How to reach me: Twitter - @satoshin_NCD

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C	C++	C#	Cython	HSP

maxima	processing	Python	Risa/Asir	SageMath

wenyan

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DXライブラリ	Eigen	IPython	matplotlib

pandas	plotly	scipy	sympy

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clang	GCC

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Android	Ubuntu	Windows

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Visual Studio

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AviUtl	ibis	Inkscape	MuseScore	NEUTRINO

SoundEngine	UTAU	VOICEVOX	COEIROINK	Domino

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Achievement

Lattice Algorithms

LLL-reduction

C

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

/* inner product */
double dot_dbl_dbl(double *x, double *y, const int n){
    double s = 0.0;
    for(int i = 0; i < n; ++i) s += x[i] * y[i];
    return s;
}
double dot_int_dbl(int *x, double *y, const int n){
    double s = 0.0;
    for(int i = 0; i < n; ++i) s += y[i] * x[i];
    return s;
}


/* Gram-Schmidt's method */
void GSO(int **b, double *B, double **mu, const int n, const int m){
    int i, j, k;
    double t, s, **GSOb;
    GSOb = (double **)malloc(n * sizeof(double *));
    for(i = 0; i < n; ++i) GSOb[i] = (double *)malloc(m * sizeof(double));

    for(i = 0; i < n; ++i){
        mu[i][i] = 1.0;
        for(j = 0; j < m; ++j) GSOb[i][j] = b[i][j];
        for(j = 0; j < i; ++j){
            mu[i][j] = dot_int_dbl(b[i], GSOb[j], m) / dot_dbl_dbl(GSOb[j], GSOb[j], m);
            for(k = 0; k < m; ++k) GSOb[i][k] -= mu[i][j] * GSOb[j][k];
        }
        B[i] = dot_dbl_dbl(GSOb[i], GSOb[i], m);
    }
}


/* size-reduction */
void SizeReduce(int **b, double **mu, const int i, const int j, const int m){
    int k;
    if(mu[i][j] > 0.5 || mu[i][j] < -0.5){
        const int q = round(mu[i][j]);
        for(k = 0; k < m; ++k) b[i][k] -= q * b[j][k];
        for(k = 0; k <= j; ++k) mu[i][k] -= mu[j][k] * q;
    }
}

/* LLL-reduction */
void LLLReduce(int **b, const double d, const int n, const int m){
    int j, i, h;
    double **mu, *B, nu, BB, C, t;
    mu = (double **)malloc(n * sizeof(double *));
    B = (double *)malloc(n * sizeof(double));
    for(i = 0; i < n; ++i) mu[i] = (double *)malloc(n * sizeof(double));
    GSO(b, B, mu, n, m);

    int tmp;
    for(int k = 1; k < n;){
        h = k - 1;
        for(j = h; j > -1; --j) SizeReduce(b, mu, k, j, m);

        if(k > 0 && B[k] < (d - mu[k][h] * mu[k][h]) * B[h]){
            for(i = 0; i < m; ++i){tmp = b[h][i]; b[h][i] = b[k][i]; b[k][i] = tmp;}
            
            nu = mu[k][k - 1]; BB = B[k] + nu * nu * B[k - 1]; C = 1.0 / BB;
            mu[k][k - 1] = nu * B[k - 1] * C; B[k] *= B[k - 1] * C; B[k - 1] = BB;

            for(i = 0; i <= k - 2; ++i){
                t = mu[k - 1][i]; mu[k - 1][i] = mu[k][i]; mu[k][i] = t;
            }
            for(i = k + 1; i < n; ++i){
                t = mu[i][k]; mu[i][k] = mu[i][k - 1] - nu * t;
                mu[i][k - 1] = t + mu[k][k - 1] * mu[i][k];
            }
            
            k = h;
        }else ++k;
    }
}

C++

#include <iostream>
#include <vector>
#include <tuple>

/* inner product */
double dot(const std::vector<int> x, const std::vector<double> y){
	double z = 0.0;
	const int n = x.size();
	for(int i = 0; i < n; ++i) z += x.at(i) * y.at(i);
	return z;
}
double dot(const std::vector<double> x, const std::vector<double> y){
	double z = 0.0;
	const int n = x.size();
	for(int i = 0; i < n; ++i) z += x.at(i) * y.at(i);
	return z;
}
double dot(const std::vector<int> x, const std::vector<int> y){
	double z = 0.0;
	const int n = x.size();
	for(int i = 0; i < n; ++i) z += x.at(i) * y.at(i);
	return z;
}


/* Gram-Schmidt's method */
std::tuple<std::vector<double>, std::vector<std::vector<double>>> Gram_Schmidt_squared(const std::vector<std::vector<int>> b){
	const int n = b.size(), m = b.at(0).size(); int i, j, k;
    std::vector<double> B(n);
	std::vector<std::vector<double>> GSOb(n, std::vector<double>(m)), mu(n, std::vector<double>(n));
	for(i = 0; i < n; ++i){
		mu.at(i).at(i)= 1.0;
		for(j = 0; j < m; ++j) GSOb.at(i).at(j) = b.at(i).at(j);
		for(j = 0; j < i; ++j){
			mu.at(i).at(j) = dot(b.at(i), GSOb.at(j)) / dot(GSOb.at(j), GSOb.at(j));
			for(k = 0; k < m; ++k) GSOb.at(i).at(k) -= mu.at(i).at(j) * GSOb.at(j).at(k);
		}
        B.at(i) = dot(GSOb.at(i), GSOb.at(i));
	}
	return std::forward_as_tuple(B, mu);
}


/* size-reduction */
void SizeReduce(std::vector<std::vector<int>>& b, std::vector<std::vector<double>>& mu, const int i, const int j){
	int q;
	const int m = b.at(0).size();
	if(mu.at(i).at(j) > 0.5 || mu.at(i).at(j) < -0.5){
		q = round(mu.at(i).at(j));
		for(int k = 0; k < m; ++k) b.at(i).at(k) -= q * b.at(j).at(k);
		for(int k = 0; k <= j; ++k) mu.at(i).at(k) -= mu.at(j).at(k) * q;
	}
}


/* LLL-reduction */
void LLLReduce(std::vector<std::vector<int>>& b, const float d = 0.99){
	const int n = b.size(), m = b.at(0).size(); int j, i, h;
	double t, nu, BB, C;
	std::vector<std::vector<double>> mu;
	std::vector<double> B; std::tie(B, mu) = Gram_Schmidt_squared(b);
	int tmp;
	for(int k = 1; k < n;){
		h = k - 1;

		for(j = h; j > -1; --j) SizeReduce(b, mu, k, j);

		//Checks if the lattice basis matrix b satisfies Lovasz condition.
		if(k > 0 && B.at(k) < (d - mu.at(k).at(h) * mu.at(k).at(h)) * B.at(h)){
			for(i = 0; i < m; ++i){tmp = b.at(h).at(i); b.at(h).at(i) = b.at(k).at(i); b.at(k).at(i) = tmp;}

			nu = mu.at(k).at(h); BB = B.at(k) + nu * nu * B.at(h); C = 1.0 / BB;
            mu.at(k).at(h) = nu * B.at(h) * C; B[k] *= B.at(h) * C; B.at(h) = BB;

            for(i = 0; i <= k - 2; ++i){
                t = mu.at(h).at(i); mu.at(h).at(i) = mu.at(k).at(i); mu.at(k).at(i) = t;
            }
            for(i = k + 1; i < n; ++i){
                t = mu.at(i).at(k); mu.at(i).at(k) = mu.at(i).at(h) - nu * t;
                mu.at(i).at(h) = t + mu.at(k).at(h) * mu.at(i).at(k);
            }

			--k;
		}else ++k;
	}
}

C++ with Eigen library

#include <iostream>
#include <eigen3/Eigen/Dense>

/* Gram-Schmidt's method */
void GSO(const Eigen::MatrixXi b, Eigen::VectorXd& B, Eigen::MatrixXd& mu, const int n, const int m){
    int j;
    Eigen::MatrixXd GSOb(n, m);

    for(int i = 0; i < n; ++i){
        mu.coeffRef(i, i) = 1.0;
        GSOb.row(i) = b.row(i).cast<double>();
        for(j = 0; j < i; ++j){
            mu.coeffRef(i, j) = b.row(i).cast<double>().dot(GSOb.row(j)) / GSOb.row(j).dot(GSOb.row(j));
            GSOb.row(i) -= mu.coeff(i, j) * GSOb.row(j);
        }
        B.coeffRef(i) = GSOb.row(i).dot(GSOb.row(i));
    }
}

/* size-reduction */
void SizeReduce(Eigen::MatrixXi& b, Eigen::MatrixXd& mu, const int i, const int j, const int m){
    if(mu.coeff(i, j) > 0.5 || mu.coeff(i, j) < -0.5){
        const int q = round(mu.coeff(i, j));
        b.row(i) -= q * b.row(j);
        mu.row(i).head(j + 1) -= (double)q * mu.row(j).head(j + 1);
    }
}

/* LLL-reduction */
void LLLReduce(Eigen::MatrixXi& b, const long double d, const int n, const int m){
    double nu, BB, C, t;
    Eigen::VectorXd B(n), logB(n);
    Eigen::MatrixXd mu(n, n);
    GSO(b, B, mu, n, m);
    
    for(int k = 1, j, i, k1; k < n;){
        k1 = k - 1;
        for(j = k1; j > -1; --j) SizeReduce(b, mu, k, j, m);

        if(k > 0 && B.coeff(k) < (d - mu.coeff(k, k1) * mu.coeff(k, k1)) * B.coeff(k1)){
            b.row(k).swap(b.row(k1));
            
            nu = mu.coeff(k, k1); BB = B.coeff(k) + nu * nu * B.coeff(k1); C = 1.0 / BB;
            mu.coeffRef(k, k1) = nu * B.coeff(k1) * C;
            B.coeffRef(k) *= B.coeff(k1) * C; B.coeffRef(k1) = BB;

            mu.row(k1).head(k - 1).swap(mu.row(k).head(k - 1));
            for(i = k + 1; i < n; ++i){
                t = mu.coeff(i, k); mu.coeffRef(i, k) = mu.coeff(i, k1) - nu * t;
                mu.coeffRef(i, k1) = t + mu.coeff(k, k1) * mu.coeff(i, k);
            }
            
            k = k1;
        }else ++k;
    }
}

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Lattice Algorithms

LLL-reduction

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