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common.cpp
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common.cpp
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/*******************************************************************************
* Header File: common.cpp
* Contains definitions of all common functions required
* for lattice-based cryptography
* Copyright (C) 2023; Sanjay Bhattacherjee, Jack Moyler.
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <https://www.gnu.org/licenses/>.
*******************************************************************************/
#include "common.h"
// #define __DEBUG__
int setPrecision(int m) {
if (m <= 40) {
RR::SetPrecision(1100);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (400)
} else if (m <= 45) {
RR::SetPrecision(1250);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (450)
} else if (m <= 50) {
RR::SetPrecision(1325);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (500)
} else if (m <= 55) {
RR::SetPrecision(1375);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (550)
} else if (m <= 60) {
RR::SetPrecision(1550);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (600)
} else if (m <= 65) {
RR::SetPrecision(1650);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (650)
} else if (m <= 70) {
RR::SetPrecision(1750);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (700)
} else if (m <= 75) {
RR::SetPrecision(1900);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (750)
} else if (m <= 80) {
RR::SetPrecision(2000);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (800)
} else if (m <= 85) {
RR::SetPrecision(2200);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (850)
} else if (m <= 90) {
RR::SetPrecision(2300);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (900)
} else if (m <= 100) {
RR::SetPrecision(3000);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1000)
} else if (m <= 105) {
RR::SetPrecision(3200);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1100)
} else if (m <= 110) {
RR::SetPrecision(3400);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1100)
} else if (m <= 115) {
RR::SetPrecision(3600);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1150)
} else if (m <= 120) {
RR::SetPrecision(3800);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1200)
} else if (m <= 125) {
RR::SetPrecision(4000);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1250)
} else if (m <= 130) {
RR::SetPrecision(4500);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1300)
} else if (m <= 140) {
RR::SetPrecision(5000);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1400)
} else if (m <= 150) {
RR::SetPrecision(5500);
#ifdef NTL_ZZ_NBITS
#undef NTL_ZZ_NBITS
#endif
#define NTL_ZZ_NBITS (1500)
} else {
printf ("\nDimension not yet tested to find precisions. \n");
exit (3);
}
return 0;
}
// Deallocate ppBasis
int deleteBasis (int ** ppBasis, int n) {
for (int i=0; i<n; i++) {
free (ppBasis[i]);
}
free (ppBasis);
return 0;
}
// Deallocate ppBasisDouble
template <typename T1>
int deleteBasisDouble (T1 ** ppBasisDouble, int n) {
for (int i=0; i<n; i++) {
free (ppBasisDouble[i]);
}
free (ppBasisDouble);
return 0;
}
// Compute SVP Challenge factor
RR SVP_Challenge_Factor(int n, ZZ volume) {
RR n_RR;
conv(n_RR, n);
RR power, volume_RR;
div(power, (RR) 1.0, n_RR);
conv(volume_RR, volume);
double gamma;
RR gamma_RR, nth_root_vol, nth_root_gamma;
gamma = tgamma(n/2 + 1);
conv(gamma_RR, gamma);
nth_root_gamma = pow(gamma_RR, power);
RR pi_RR, root_pi_RR;
ComputePi(pi_RR);
root_pi_RR = SqrRoot(pi_RR);
pow(nth_root_vol, volume_RR, power);
RR RRfactor;
RRfactor = (nth_root_gamma / root_pi_RR) * nth_root_vol;
return RRfactor;
}
// Comparator function to sort pairs according to second value
bool cmp_double(pair<string, double>& a, pair<string, double>& b){
return a.second < b.second;
}
// Comparator function to sort pairs according to second value
bool cmp_long_double(pair<string, long double>& a, pair<string, long double>& b){
return a.second < b.second;
}
// Function to sort the map according
// to value in a (key-value) pairs (fix)
void map_sort_long_double_fix(map<string, long double>& M){
// Declare vector of pairs
vector<pair<string, long double> > A;
// Copy key-value pair from Map to vector of pairs
for (auto& it : M) {
A.push_back(it);
}
// Sort using comparator function
sort(A.begin(), A.end(), cmp_long_double);
long double min_elem = A[0].second;
string min_label = A[0].first;
cout << std::fixed;
// Print the sorted value
for (auto& it : A) {
if (it.first == min_label) {
cout << std::setprecision(15) << it.first << "\t" << it.second << " = X " << endl;
} else {
long double mult = (it.second / min_elem);
cout << std::setprecision(15) << it.first << "\t" << it.second << " = " << mult << "X " << endl;
}
}
}
// Function to sort the map according
// to value in a (key-value) pairs (fix)
void map_sort_double_fix(map<string, double>& M){
// Declare vector of pairs
vector<pair<string, double> > A;
// Copy key-value pair from Map to vector of pairs
for (auto& it : M) {
A.push_back(it);
}
// Sort using comparator function
sort(A.begin(), A.end(), cmp_double);
double min_elem = A[0].second;
string min_label = A[0].first;
cout << std::fixed;
// Print the sorted value
for (auto& it : A) {
if (it.first == min_label) {
cout << std::setprecision(15) << it.first << "\t" << it.second << " = X " << endl;
} else {
double mult = (it.second/min_elem);
cout << std::setprecision(15) << it.first << "\t" << it.second << " = " << mult << "X " << endl;
}
}
}
// Compare elements for use in qsort algorithm
int compare (const void * a, const void * b) {
if ( *(long double *) a < *(long double *) b ) return -1;
if ( *(long double *) a == *(long double *) b ) return 0;
if ( *(long double *) a > *(long double *) b ) return 1;
return -2;
}
// Generate a random NTL basis (mat_ZZ) with n integer vectors, each of dimension m
mat_ZZ generateNTLChallengeBasis (int n) {
mat_ZZ NTLBasis;
ZZ prime;
long length, err;
length = 10*n;
GenPrime(prime, length, err = 80);
NTLBasis.SetDims(n,n);
NTLBasis[0][0] = prime;
for (int i=1; i<n; i++) {
ZZ r;
RandomBnd(r, prime);
NTLBasis[i][0] = r;
for (int j=1; j<n; j++){
if (j == i) {
NTLBasis[i][j] = 1;
} else {
NTLBasis[i][j] = 0;
}
}
}
return NTLBasis;
}
// Find the inner products of vectors pV1 with pV2
template <typename T1, typename T2>
long double inner_product_template (T1 * pV1, T2 * pV2, int m) {
long double result = 0;
for (int i=0; i<m; i++) {
result += (long double)pV1[i] * (long double)pV2[i];
}
return result;
}
// Compute norm of a vec_ZZ
RR NTL_vector_norm (vec_ZZ V1, int m) {
ZZ norm_squared;
InnerProduct(norm_squared, V1, V1);
RR RR_norm_squared, norm;
conv(RR_norm_squared, norm_squared);
SqrRoot(norm, RR_norm_squared);
return norm;
}
// Reduce vector pV2 with pV1
int reduce (int * pV1, int * pV2, int m, int closest_integer_to_mu) {
for (int i=0; i<m; i++) {
pV2[i] -= closest_integer_to_mu * pV1[i];
}
return 0;
}
template <typename T1>
int reduceDouble (T1 * pVDouble1, T1 * pVDouble2, int m, T1 mu) {
for (int i=0; i<m; i++) {
pVDouble2[i] -= mu * pVDouble1[i];
}
return 0;
}
// Reduce vec_ZZ V2 with vec_ZZ V1 (only NTL)
int NTL_reduce (vec_ZZ& V1, vec_ZZ& V2, int m, ZZ closest_ZZ_to_mu) {
for (int i=0; i<m; i++) {
ZZ V1_mu;
mul(V1_mu, closest_ZZ_to_mu, V1[i]);
sub(V2[i], V2[i], V1_mu);
}
return 0;
}
// Copy int **ppBasis to mat_ZZ BasisNTL and return
mat_ZZ copyBasisToNTL (int ** ppBasis, int m, int n) {
mat_ZZ BasisNTL;
BasisNTL.SetDims(n,m);
for (int i=0; i<m; i++) {
for (int j=0; j<n; j++) {
BasisNTL[i][j] = ppBasis[i][j];
}
}
return BasisNTL;
}
// Copy mat_ZZ ppBasis to mat_RR BasisNTL and return
mat_RR copyNTLBasisToRR (mat_ZZ ppBasis, int m, int n) {
mat_RR BasisRR;
BasisRR.SetDims(n,m);
for (int i=0; i<m; i++) {
for (int j=0; j<n; j++) {
conv(BasisRR[i][j], ppBasis[i][j]);
}
}
return BasisRR;
}
// Copy pBasis into ppBasisDouble and return
long double ** copyBasisToDouble (int ** ppBasis, int m, int n) {
long double ** ppBasisDouble;
// Create the 2-dimensional array
ppBasisDouble = (long double **) calloc (n, sizeof(long double *));
for (int i=0; i<n; i++) {
ppBasisDouble[i] = (long double *) calloc (m, sizeof(long double));
// Copy the values
for (int j=0; j<m; j++) {
ppBasisDouble[i][j] = (long double) ppBasis[i][j];
}
}
return ppBasisDouble;
}
// Rounds a/d to nearest integer, breaking ties
// by rounding towards zero. Assumes d > 0.
// Stored in q
// Copied from NTL Library (github)
int BalDiv(ZZ& q, const ZZ& a, const ZZ& d) {
NTL_ZZRegister(r);
DivRem(q, r, a, d);
add(r, r, r);
long cmp = compare(r, d);
if (cmp > 0 || (cmp == 0 && q < 0))
add(q, q, 1);
return 0;
}
// Exact division: q = a/b where b|a
// Copied from NTL Library (github)
int ExactDiv(ZZ& qq, const ZZ& a, const ZZ& b) {
NTL_ZZRegister(q);
NTL_ZZRegister(r);
DivRem(q, r, a, b);
if (!IsZero(r)) {
cerr << "a = " << a << "\n";
cerr << "b = " << b << "\n";
LogicError("ExactDiv: nonzero remainder");
}
qq = q;
return 0;
}
// Gram-Schmidt Orthogonalisation
long double ** GSO (int ** pBasis, int m, int n) {
long double ** ppBasisGSO;
ppBasisGSO = copyBasisToDouble (pBasis, m, n);
for (int i=1; i<n; i++) {
for (int j=i-1; j>=0; j--) {
// Find the value of mu_ij
long double mu_ij =
inner_product_template (pBasis[i], ppBasisGSO[j], m) /
inner_product_template (ppBasisGSO[j], ppBasisGSO[j], m);
// Reduce vector ppBasisGSO[i] with ppBasisGSO[j]
// Compute b_i - mu_ij x b_j*
reduceDouble (ppBasisGSO[j], ppBasisGSO[i], m, mu_ij);
}
}
return ppBasisGSO;
}
// Gram-Schmidt Orthogonalisation (NTL datatypes)
mat_RR GSO_RR (mat_ZZ ppBasis, int m, int n) {
mat_RR ppBasisGSO;
mat_RR ppBasisRR;
ppBasisGSO.SetDims(m, n);
ppBasisRR.SetDims(m, n);
for (int i=0; i<m; i++) {
for (int j=0; j<n; j++) {
conv(ppBasisGSO[i][j], ppBasis[i][j]);
conv(ppBasisRR[i][j], ppBasis[i][j]);
}
}
for (int i=0; i<n; i++) {
for (int j=i-1; j>=0; j--) {
RR mu_ij, num, den;
InnerProduct(num, ppBasisRR[i], ppBasisGSO[j]);
InnerProduct(den, ppBasisGSO[j], ppBasisGSO[j]);
div(mu_ij, num, den);
vec_RR product;
mul(product, mu_ij, ppBasisGSO[j]);
sub(ppBasisGSO[i], ppBasisGSO[i], product);
}
}
return ppBasisGSO;
}
// Update the GSO information in the way required for DeepLLL
// Algorithm 4 in Yamaguchi Yasuda NuTMic paper (NuTMic 2017)
template <typename T1>
int deep_LLL_GSO_update (T1 ** ppM, T1 * pB, int k , int i, int n) {
// Initialise arrays pP, pS, pD as described in [YY18]
T1 * pP;
pP = (T1 *) calloc (n, sizeof(T1));
T1 * pS;
pS = (T1 *) calloc (n, sizeof(T1));
T1 * pD;
pD = (T1 *) calloc (n, sizeof(T1));
for (int j=0; j<n; j++) {
pP[j] = 0.0;
pS[j] = 0.0;
pD[j] = 0.0;
}
pP[k] = pB[k];
pD[k] = pB[k];
// Lines 2-4 of Algorithm 4 in YY18
for (int j=k-1; j>=i; j--) {
pP[j] = ppM[k][j] * pB[j];
pD[j] = pD[j+1] + (ppM[k][j]*pP[j]);
}
// Line 5 of Algorithm 4 in YY18 - Intialising values of S
for (int j=i; j<n; j++) {
pS[j] = (T1) 0.0;
}
// Lines 6 - 18 of Algorithm 4 in YY18
for (int j=k; j>i; j--) {
T1 T = 0.0;
T = ppM[k][j-1] / pD[j];
for (int l=n-1; l>k;l--) {
pS[l] = pS[l] + (ppM[l][j] * pP[j]);
ppM[l][j] = ppM[l][j-1] - (T * pS[l]);
}
for (int l=k; l>=j+2;l--) {
pS[l] = pS[l] + (ppM[l-1][j] * pP[j]);
ppM[l][j] = ppM[l-1][j-1] - (T * pS[l]);
}
if (j != k) {
pS[j+1] = pP[j];
ppM[j+1][j] = ppM[j][j-1] - (T * pS[j+1]);
}
}
// Lines 20 - 22 of Algorithm 4 in YY18
T1 T = 0.0;
T = (T1) 1.0 / pD[i];
for (int l=n-1; l>k; l--) {
ppM[l][i] = T * (pS[l] + ppM[l][i] * pP[i]);
}
for (int l=k; l>=i+2; l--) {
ppM[l][i] = T * (pS[l] + ppM[l-1][i] * pP[i]);
}
// Lines 23 - 29 of Algorithm 4 in YY18
ppM[i+1][i] = T * pP[i];
for (int j=0; j<i; j++) {
T1 epsilon = 0.0;
epsilon = ppM[k][j];
for (int l=k; l>i; l--) {
ppM[l][j] = ppM[l-1][j];
}
ppM[i][j] = epsilon;
}
// Lines 31 - 32 of Algorithm 4 in YY18
for (int j=k; j>i;j--) {
pB[j] = (pD[j] * pB[j-1]) / pD[j-1];
}
pB[i] = pD[i];
return 0;
}
// Update the GSO information in the way required for DeepLLL (RR datatypes)
// Algorithm 4 in Yamaguchi Yasuda NuTMic paper (NuTMic 2017)
int deep_LLL_GSO_update_RR (mat_RR &ppM, vec_RR &pB, int k , int i, int n) {
// Initialise arrays pP, pS, pD as described in [YY18]
vec_RR pP;
pP.SetLength(n);
vec_RR pS;
pS.SetLength(n);
vec_RR pD;
pD.SetLength(n);
for (int j=0; j<n; j++) {
pP[j] = 0;
pS[j] = 0;
pD[j] = 0;
}
pP[k] = pB[k];
pD[k] = pB[k];
// Lines 2-4 of Algorithm 4 in YY18
for (int j=k-1; j>=i; j--) {
pP[j] = ppM[k][j] * pB[j];
pD[j] = pD[j+1] + (ppM[k][j]*pP[j]);
}
// Line 5 of Algorithm 4 in YY18 - Intialising values of S
for (int j=i; j<n; j++) {
pS[j] = 0;
}
// Lines 6 - 18 of Algorithm 4 in YY18
for (int j=k; j>i; j--) {
RR T;
T = 0;
T = ppM[k][j-1] / pD[j];
for (int l=n-1; l>k;l--) {
pS[l] = pS[l] + (ppM[l][j] * pP[j]);
ppM[l][j] = ppM[l][j-1] - (T * pS[l]);
}
for (int l=k; l>=j+2;l--) {
pS[l] = pS[l] + (ppM[l-1][j] * pP[j]);
ppM[l][j] = ppM[l-1][j-1] - (T * pS[l]);
}
if (j != k) {
pS[j+1] = pP[j];
ppM[j+1][j] = ppM[j][j-1] - (T * pS[j+1]);
}
}
// Lines 20 - 22 of Algorithm 4 in YY18
RR T;
inv(T, pD[i]);
for (int l=n-1; l>k; l--) {
ppM[l][i] = T * (pS[l] + ppM[l][i] * pP[i]);
}
for (int l=k; l>=i+2; l--) {
ppM[l][i] = T * (pS[l] + ppM[l-1][i] * pP[i]);
}
// Lines 23 - 29 of Algorithm 4 in YY18
ppM[i+1][i] = T * pP[i];
for (int j=0; j<i; j++) {
RR epsilon;
epsilon = ppM[k][j];
for (int l=k; l>i; l--) {
ppM[l][j] = ppM[l-1][j];
}
ppM[i][j] = epsilon;
}
// Lines 31 - 32 of Algorithm 4 in YY18
for (int j=k; j>i;j--) {
pB[j] = (pD[j] * pB[j-1]) / pD[j-1];
}
pB[i] = pD[i];
return 0;
}
// Find the indices k,i where we perform a deep insertion for SS-GGLLL (standard C datatypes)
template <typename T1>
T1 find_insertion_indices_SSGGLLL (T1 ** ppM, int ** ppBasis, T1 * pB, int n, int m, int * pK, int * pJ) {
int k_max = 1;
int i_max = 0;
T1 S_max = 0.0;
T1 S_ik = 0.0;
// For k = 1,...,n and j = k-1,...,1,0, we find the indices k, i where we insert b_k to reduce the value of SS the most
for (int k=1; k<n; k++) {
// Initialise to the k-1 case
T1 delta_k_1 = 0.0;
delta_k_1 = (pB[k] + (ppM[k][k-1] * ppM[k][k-1] * pB[k-1])) / pB[k-1];
// Initialise D_j = D_{k-1}
T1 D_j = 0.0;
D_j = (ppM[k][k-1] * ppM[k][k-1] * pB[k-1]) + pB[k];
// Initialise S_ik = S_{i-1,k}
// After initialising S and D, we can update them in the 'for' loop below for general S_jk and D_j for 0 <= j < k-1
T1 S_ik = 0.0;
S_ik = ppM[k][k-1] * ppM[k][k-1] * (1 - delta_k_1);
// Check if k, k-1 case improves upon current best
if (S_ik > S_max) {
S_max = S_ik;
k_max = k;
i_max = k-1;
}
for (int l=k-2; l>=0; l--) {
D_j += ppM[k][l] * ppM[k][l] * pB[l];
S_ik += ppM[k][l] * ppM[k][l] * pB[l] * ((pB[l]/D_j) - 1);
if (S_ik > S_max) {
k_max = k;
i_max = l;
S_max = S_ik;
}
}
}
*pK = k_max;
*pJ = i_max;
return S_max;
}
RR RR_find_insertion_indices_SSGGLLL (mat_RR ppM, mat_ZZ ppBasis, vec_RR pB, int n, int m, int * pK, int * pJ) {
int k_max = 1;
int i_max = 0;
RR S_max, S_ik;
S_max = 0;
S_ik = 0;
// For k = 1,...,n and j = k-1,...,1,0, we find the indices k, i where we insert b_k to reduce the value of SS the most
for (int k=1; k<n; k++) {
// Initialise to the k-1 case
RR delta_k_1;
delta_k_1 = 0;
delta_k_1 = (pB[k] + (ppM[k][k-1] * ppM[k][k-1] * pB[k-1])) / pB[k-1];
// Initialise D_j = D_{k-1}
RR D_j;
D_j = 0;
D_j = (ppM[k][k-1] * ppM[k][k-1] * pB[k-1]) + pB[k];
// Initialise S_ik = S_{i-1,k}
// After initialising S and D, we can update them in the 'for' loop below for general S_jk and D_j for 0 <= j < k-1
RR S_ik;
S_ik = 0;
S_ik = ppM[k][k-1] * ppM[k][k-1] * (1 - delta_k_1);
// Check if k, k-1 case improves upon current best
if (S_ik > S_max) {
S_max = S_ik;
k_max = k;
i_max = k-1;
}
for (int l=k-2; l>=0; l--) {
D_j += ppM[k][l] * ppM[k][l] * pB[l];
S_ik += ppM[k][l] * ppM[k][l] * pB[l] * ((pB[l]/D_j) - 1);
if (S_ik > S_max) {
k_max = k;
i_max = l;
S_max = S_ik;
}
}
}
*pK = k_max;
*pJ = i_max;
return S_max;
}
// Find insertion indices for Pot-GGLLL (standard C datatypes)
long double find_insertion_indices_PotGGLLL (long double ** ppM, int ** ppBasis, long double * pB, int n, int m, int * pK, int * pJ) {
int k_min = 0;
int i_min = 0;
// Initialise P = 1.0, P_min = 1.0
// We reinitialise P to be 1.0 every time we increment k (in standard PotLLL, for each k, we start with P = 1)
long double P = 1.0;
long double P_min = 1.0;
// Initialise P_min to be the k=1, i=0 case
long double P_init = 1.0;
P_init = (pB[1] + ppM[1][0]*ppM[1][0]*pB[0]) / pB[0];
if (P_init < P_min) {
P_min = P_init;
k_min = 1;
i_min = 0;
}
long double sum = 0.0;
long double P_mult = 0.0;
for (int k=2;k<n;k++) {
P = 1;
for (int j=k-1; j>=0; j--) {
sum = 0.0;
for (int r=j; r<k; r++) {
sum += ppM[k][r]*ppM[k][r]*pB[r];
}
P_mult = (pB[k] + sum) / pB[j];
P = P * P_mult;
if (P<P_min) {
k_min = k;
i_min = j;
P_min = P;
}
}
}
*pK = k_min;
*pJ = i_min;
return P_min;
}
// Find insertion indices for Pot-GGLLL (using NTL datatypes)
RR RR_find_insertion_indices_PotGGLLL (mat_RR ppM, mat_RR ppDelta, mat_ZZ ppBasis, vec_RR pB, int n, int m, int * pK, int * pJ) {
int k_min = 0;
int i_min = 0;
// Initialise P = 1.0, P_min = 1.0
// We reinitialise P to be 1.0 every time we increment k (in standard PotLLL, for each k, we start with P = 1)
RR P, P_min;
P = 1;
P_min = 1;
// Initialise P_min to be the k=1, i=0 case
RR P_init;
P_init = (pB[1] + ppM[1][0]*ppM[1][0]*pB[0]) / pB[0];
if (P_init < P_min) {
P_min = P_init;
k_min = 1;
i_min = 0;
}
for (int k=2;k<n;k++) {
P = 1;
for (int j=k-1; j>=0; j--) {
RR sum;
sum = 0;
for (int r=j; r<k; r++) {
sum += ppM[k][r]*ppM[k][r]*pB[r];
}
RR P_mult;
P_mult = (pB[k] + sum) / pB[j];
P = P * P_mult;
if (P<P_min) {
k_min = k;
i_min = j;
P_min = P;
}
}
}
*pK = k_min;
*pJ = i_min;
return P_min;
}
// RR Root Hermite Factor
RR NTLrootHermiteFactor (mat_ZZ ppBasis, int m, int n) {
RR RR_n;
conv(RR_n, n);
vec_RR V1;
V1.SetLength(m);
mat_RR ppBasisGSO;
ppBasisGSO = GSO_RR (ppBasis, m, n);
for (int i=0; i<m; i++){
conv(V1[i], ppBasis[0][i]);
}
RR volume;
volume = 1;
for (int i=0; i<n; i++) {
RR gsoVectorNormSquared, gsoVectorNorm;
InnerProduct(gsoVectorNormSquared, ppBasisGSO[i],ppBasisGSO[i]);
SqrRoot(gsoVectorNorm, gsoVectorNormSquared);
mul(volume, volume, gsoVectorNorm);
}
RR power;
inv(power, RR_n);
RR nthRootVolume;
pow(nthRootVolume, volume, power);
RR b1NormSquared;
InnerProduct (b1NormSquared, V1, V1);
RR b1Norm;
SqrRoot(b1Norm, b1NormSquared);
RR hermiteFactor, RHF;
div(hermiteFactor, b1Norm, nthRootVolume);
pow(RHF, hermiteFactor, power);
return RHF;
}
RR NTLorthogonalityDefect (mat_ZZ ppBasis, int m, int n) {
mat_RR ppBasisGSO;
ppBasisGSO = GSO_RR(ppBasis, m, n);
RR volumeGSO;
volumeGSO = 1;
for (int i=0; i<n; i++) {
RR gsoVectorNormSquared;
InnerProduct(gsoVectorNormSquared, ppBasisGSO[i], ppBasisGSO[i]);
mul(volumeGSO, volumeGSO, gsoVectorNormSquared);
}
SqrRoot(volumeGSO, volumeGSO);
RR volumeBasis;
ZZ volumeBasisSquared;
volumeBasisSquared = 1;
for (int j=0; j<n; j++) {
ZZ vectorNormSquared;
InnerProduct(vectorNormSquared, ppBasis[j], ppBasis[j]);
mul(volumeBasisSquared, volumeBasisSquared, vectorNormSquared);
}
conv (volumeBasis, volumeBasisSquared);
SqrRoot(volumeBasis,volumeBasis);
RR orthDefect;
div(orthDefect, volumeBasis, volumeGSO);
return orthDefect;
}
RR NTLsquaredSum (mat_ZZ ppBasis, int m, int n) {
mat_RR ppBasisGSO;
ppBasisGSO = GSO_RR(ppBasis, m, n);
RR squaredSum;
squaredSum = 0;
for (int i=0; i<n; i++) {
RR gsoNormSquared;
InnerProduct(gsoNormSquared, ppBasisGSO[i], ppBasisGSO[i]);
add(squaredSum, squaredSum, gsoNormSquared);
}
return squaredSum;
}
RR NTLbasisPotential (mat_ZZ ppBasis, int m, int n) {
mat_RR ppBasisGSO;
ppBasisGSO = GSO_RR(ppBasis, m, n);
RR potential;
potential = 0;
for (int i=0; i<n; i++) {
// The potential term for b_i is ||b_i*||^(2(n-i)|| (since indexing from 0)
// This is the same as (||b_i*||^2)^(n-i)
// Log of the potential term is (n-i) * log(||b_i*||^2)
RR gsoVectorNormSquared;
InnerProduct(gsoVectorNormSquared, ppBasisGSO[i], ppBasisGSO[i]);
RR RR_power;
int power = n-i;
conv(RR_power, power);
RR logNormSquared;
log(logNormSquared, gsoVectorNormSquared);
RR potentialTerm;
mul(potentialTerm, RR_power, logNormSquared);
potential += potentialTerm;
}
return potential;
}
// Check if a Basis is LLL reduced
int NTL_LLL_check (mat_ZZ ppBasis, int m, int n, RR delta_threshold) {
int isLLLReduced = TRUE;
mat_RR ppBasisGSO;
mat_RR ppBasisRR;
ppBasisRR = copyNTLBasisToRR(ppBasis, m, n);
// Find the GSO of the Basis
ppBasisGSO = GSO_RR (ppBasis, m, n);
#ifdef __DEBUG__
cout << "\n\n\nThe GSO of the basis is:\n" << ppBasisGSO;
#endif
// This 1-dimensional array pB will store all the values of ||b_i*||^2
vec_RR pB;
pB.SetLength(n);
#ifdef __DEBUG__
printf ("\nThe values of ||b_i*||^2 are:\n");
#endif
for (int i=0; i<n; i++) {
InnerProduct(pB[i], ppBasisGSO[i], ppBasisGSO[i]);
#ifdef __DEBUG__
cout << pB[i] << endl;