1 | |
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2 | #include <NTL/ZZ_pX.h> |
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3 | |
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4 | #include <NTL/new.h> |
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5 | |
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6 | NTL_START_IMPL |
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7 | |
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8 | |
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9 | |
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10 | |
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11 | long divide(ZZ_pX& q, const ZZ_pX& a, const ZZ_pX& b) |
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12 | { |
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13 | if (IsZero(b)) { |
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14 | if (IsZero(a)) { |
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15 | clear(q); |
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16 | return 1; |
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17 | } |
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18 | else |
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19 | return 0; |
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20 | } |
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21 | |
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22 | ZZ_pX lq, r; |
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23 | DivRem(lq, r, a, b); |
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24 | if (!IsZero(r)) return 0; |
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25 | q = lq; |
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26 | return 1; |
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27 | } |
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28 | |
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29 | long divide(const ZZ_pX& a, const ZZ_pX& b) |
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30 | { |
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31 | if (IsZero(b)) return IsZero(a); |
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32 | ZZ_pX lq, r; |
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33 | DivRem(lq, r, a, b); |
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34 | if (!IsZero(r)) return 0; |
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35 | return 1; |
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36 | } |
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37 | |
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38 | |
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39 | |
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40 | |
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41 | void ZZ_pXMatrix::operator=(const ZZ_pXMatrix& M) |
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42 | { |
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43 | elts[0][0] = M.elts[0][0]; |
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44 | elts[0][1] = M.elts[0][1]; |
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45 | elts[1][0] = M.elts[1][0]; |
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46 | elts[1][1] = M.elts[1][1]; |
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47 | } |
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48 | |
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49 | |
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50 | void RightShift(ZZ_pX& x, const ZZ_pX& a, long n) |
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51 | { |
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52 | if (n < 0) { |
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53 | if (n < -NTL_MAX_LONG) Error("overflow in RightShift"); |
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54 | LeftShift(x, a, -n); |
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55 | return; |
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56 | } |
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57 | |
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58 | long da = deg(a); |
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59 | long i; |
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60 | |
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61 | if (da < n) { |
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62 | clear(x); |
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63 | return; |
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64 | } |
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65 | |
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66 | if (&x != &a) |
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67 | x.rep.SetLength(da-n+1); |
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68 | |
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69 | for (i = 0; i <= da-n; i++) |
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70 | x.rep[i] = a.rep[i+n]; |
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71 | |
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72 | if (&x == &a) |
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73 | x.rep.SetLength(da-n+1); |
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74 | |
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75 | x.normalize(); |
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76 | } |
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77 | |
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78 | void LeftShift(ZZ_pX& x, const ZZ_pX& a, long n) |
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79 | { |
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80 | if (n < 0) { |
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81 | if (n < -NTL_MAX_LONG) Error("overflow in LeftShift"); |
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82 | RightShift(x, a, -n); |
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83 | return; |
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84 | } |
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85 | |
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86 | if (n >= (1L << (NTL_BITS_PER_LONG-4))) |
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87 | Error("overflow in LeftShift"); |
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88 | |
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89 | if (IsZero(a)) { |
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90 | clear(x); |
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91 | return; |
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92 | } |
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93 | |
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94 | long m = a.rep.length(); |
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95 | |
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96 | x.rep.SetLength(m+n); |
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97 | |
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98 | long i; |
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99 | for (i = m-1; i >= 0; i--) |
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100 | x.rep[i+n] = a.rep[i]; |
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101 | |
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102 | for (i = 0; i < n; i++) |
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103 | clear(x.rep[i]); |
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104 | } |
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105 | |
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106 | |
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107 | void ShiftAdd(ZZ_pX& U, const ZZ_pX& V, long n) |
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108 | // assumes input does not alias output |
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109 | { |
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110 | if (IsZero(V)) |
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111 | return; |
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112 | |
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113 | long du = deg(U); |
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114 | long dv = deg(V); |
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115 | |
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116 | long d = max(du, n+dv); |
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117 | |
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118 | U.rep.SetLength(d+1); |
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119 | long i; |
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120 | |
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121 | for (i = du+1; i <= d; i++) |
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122 | clear(U.rep[i]); |
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123 | |
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124 | for (i = 0; i <= dv; i++) |
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125 | add(U.rep[i+n], U.rep[i+n], V.rep[i]); |
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126 | |
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127 | U.normalize(); |
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128 | } |
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129 | |
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130 | void ShiftSub(ZZ_pX& U, const ZZ_pX& V, long n) |
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131 | // assumes input does not alias output |
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132 | { |
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133 | if (IsZero(V)) |
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134 | return; |
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135 | |
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136 | long du = deg(U); |
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137 | long dv = deg(V); |
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138 | |
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139 | long d = max(du, n+dv); |
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140 | |
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141 | U.rep.SetLength(d+1); |
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142 | long i; |
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143 | |
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144 | for (i = du+1; i <= d; i++) |
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145 | clear(U.rep[i]); |
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146 | |
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147 | for (i = 0; i <= dv; i++) |
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148 | sub(U.rep[i+n], U.rep[i+n], V.rep[i]); |
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149 | |
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150 | U.normalize(); |
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151 | } |
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152 | |
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153 | |
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154 | void mul(ZZ_pX& U, ZZ_pX& V, const ZZ_pXMatrix& M) |
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155 | // (U, V)^T = M*(U, V)^T |
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156 | { |
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157 | long d = deg(U) - deg(M(1,1)); |
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158 | long k = NextPowerOfTwo(d - 1); |
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159 | |
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160 | // When the GCD algorithm is run on polynomials of degree n, n-1, |
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161 | // where n is a power of two, then d-1 is likely to be a power of two. |
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162 | // It would be more natural to set k = NextPowerOfTwo(d+1), but this |
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163 | // would be much less efficient in this case. |
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164 | |
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165 | // We optimize this case, as it does sometimes arise naturally |
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166 | // in some situations. |
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167 | |
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168 | long n = (1L << k); |
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169 | long xx; |
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170 | ZZ_p a0, a1, b0, b1, c0, d0, u0, u1, v0, v1, nu0, nu1, nv0; |
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171 | static ZZ t1, t2; |
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172 | |
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173 | if (n == d-1) |
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174 | xx = 1; |
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175 | else if (n == d) |
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176 | xx = 2; |
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177 | else |
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178 | xx = 3; |
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179 | |
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180 | switch (xx) { |
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181 | case 1: |
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182 | GetCoeff(a0, M(0,0), 0); |
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183 | GetCoeff(a1, M(0,0), 1); |
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184 | GetCoeff(b0, M(0,1), 0); |
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185 | GetCoeff(b1, M(0,1), 1); |
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186 | GetCoeff(c0, M(1,0), 0); |
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187 | GetCoeff(d0, M(1,1), 0); |
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188 | |
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189 | GetCoeff(u0, U, 0); |
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190 | GetCoeff(u1, U, 1); |
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191 | GetCoeff(v0, V, 0); |
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192 | GetCoeff(v1, V, 1); |
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193 | |
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194 | mul(t1, rep(a0), rep(u0)); |
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195 | mul(t2, rep(b0), rep(v0)); |
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196 | add(t1, t1, t2); |
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197 | conv(nu0, t1); |
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198 | |
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199 | mul(t1, rep(a1), rep(u0)); |
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200 | mul(t2, rep(a0), rep(u1)); |
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201 | add(t1, t1, t2); |
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202 | mul(t2, rep(b1), rep(v0)); |
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203 | add(t1, t1, t2); |
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204 | mul(t2, rep(b0), rep(v1)); |
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205 | add(t1, t1, t2); |
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206 | conv(nu1, t1); |
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207 | |
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208 | mul(t1, rep(c0), rep(u0)); |
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209 | mul(t2, rep(d0), rep(v0)); |
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210 | add (t1, t1, t2); |
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211 | conv(nv0, t1); |
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212 | |
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213 | break; |
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214 | |
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215 | case 2: |
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216 | GetCoeff(a0, M(0,0), 0); |
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217 | GetCoeff(b0, M(0,1), 0); |
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218 | |
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219 | GetCoeff(u0, U, 0); |
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220 | GetCoeff(v0, V, 0); |
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221 | |
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222 | mul(t1, rep(a0), rep(u0)); |
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223 | mul(t2, rep(b0), rep(v0)); |
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224 | add(t1, t1, t2); |
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225 | conv(nu0, t1); |
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226 | |
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227 | break; |
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228 | |
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229 | case 3: |
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230 | break; |
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231 | |
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232 | } |
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233 | |
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234 | FFTRep RU(INIT_SIZE, k), RV(INIT_SIZE, k), R1(INIT_SIZE, k), |
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235 | R2(INIT_SIZE, k); |
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236 | |
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237 | ToFFTRep(RU, U, k); |
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238 | ToFFTRep(RV, V, k); |
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239 | |
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240 | ToFFTRep(R1, M(0,0), k); |
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241 | mul(R1, R1, RU); |
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242 | ToFFTRep(R2, M(0,1), k); |
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243 | mul(R2, R2, RV); |
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244 | add(R1, R1, R2); |
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245 | FromFFTRep(U, R1, 0, d); |
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246 | |
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247 | ToFFTRep(R1, M(1,0), k); |
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248 | mul(R1, R1, RU); |
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249 | ToFFTRep(R2, M(1,1), k); |
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250 | mul(R2, R2, RV); |
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251 | add(R1, R1, R2); |
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252 | FromFFTRep(V, R1, 0, d-1); |
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253 | |
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254 | // now fix-up results |
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255 | |
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256 | switch (xx) { |
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257 | case 1: |
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258 | GetCoeff(u0, U, 0); |
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259 | sub(u0, u0, nu0); |
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260 | SetCoeff(U, d-1, u0); |
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261 | SetCoeff(U, 0, nu0); |
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262 | |
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263 | GetCoeff(u1, U, 1); |
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264 | sub(u1, u1, nu1); |
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265 | SetCoeff(U, d, u1); |
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266 | SetCoeff(U, 1, nu1); |
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267 | |
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268 | GetCoeff(v0, V, 0); |
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269 | sub(v0, v0, nv0); |
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270 | SetCoeff(V, d-1, v0); |
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271 | SetCoeff(V, 0, nv0); |
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272 | |
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273 | break; |
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274 | |
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275 | |
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276 | case 2: |
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277 | GetCoeff(u0, U, 0); |
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278 | sub(u0, u0, nu0); |
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279 | SetCoeff(U, d, u0); |
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280 | SetCoeff(U, 0, nu0); |
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281 | |
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282 | break; |
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283 | |
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284 | } |
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285 | } |
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286 | |
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287 | |
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288 | void mul(ZZ_pXMatrix& A, ZZ_pXMatrix& B, ZZ_pXMatrix& C) |
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289 | // A = B*C, B and C are destroyed |
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290 | { |
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291 | long db = deg(B(1,1)); |
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292 | long dc = deg(C(1,1)); |
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293 | long da = db + dc; |
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294 | |
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295 | long k = NextPowerOfTwo(da+1); |
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296 | |
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297 | FFTRep B00, B01, B10, B11, C0, C1, T1, T2; |
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298 | |
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299 | ToFFTRep(B00, B(0,0), k); B(0,0).kill(); |
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300 | ToFFTRep(B01, B(0,1), k); B(0,1).kill(); |
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301 | ToFFTRep(B10, B(1,0), k); B(1,0).kill(); |
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302 | ToFFTRep(B11, B(1,1), k); B(1,1).kill(); |
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303 | |
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304 | ToFFTRep(C0, C(0,0), k); C(0,0).kill(); |
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305 | ToFFTRep(C1, C(1,0), k); C(1,0).kill(); |
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306 | |
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307 | mul(T1, B00, C0); |
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308 | mul(T2, B01, C1); |
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309 | add(T1, T1, T2); |
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310 | FromFFTRep(A(0,0), T1, 0, da); |
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311 | |
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312 | mul(T1, B10, C0); |
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313 | mul(T2, B11, C1); |
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314 | add(T1, T1, T2); |
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315 | FromFFTRep(A(1,0), T1, 0, da); |
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316 | |
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317 | ToFFTRep(C0, C(0,1), k); C(0,1).kill(); |
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318 | ToFFTRep(C1, C(1,1), k); C(1,1).kill(); |
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319 | |
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320 | mul(T1, B00, C0); |
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321 | mul(T2, B01, C1); |
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322 | add(T1, T1, T2); |
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323 | FromFFTRep(A(0,1), T1, 0, da); |
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324 | |
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325 | mul(T1, B10, C0); |
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326 | mul(T2, B11, C1); |
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327 | add(T1, T1, T2); |
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328 | FromFFTRep(A(1,1), T1, 0, da); |
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329 | } |
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330 | |
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331 | void IterHalfGCD(ZZ_pXMatrix& M_out, ZZ_pX& U, ZZ_pX& V, long d_red) |
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332 | { |
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333 | M_out(0,0).SetMaxLength(d_red); |
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334 | M_out(0,1).SetMaxLength(d_red); |
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335 | M_out(1,0).SetMaxLength(d_red); |
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336 | M_out(1,1).SetMaxLength(d_red); |
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337 | |
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338 | set(M_out(0,0)); clear(M_out(0,1)); |
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339 | clear(M_out(1,0)); set(M_out(1,1)); |
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340 | |
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341 | long goal = deg(U) - d_red; |
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342 | |
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343 | if (deg(V) <= goal) |
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344 | return; |
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345 | |
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346 | ZZVec tmp(deg(U)+1, ZZ_pInfo->ExtendedModulusSize); |
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347 | ZZ_pX Q, t(INIT_SIZE, d_red); |
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348 | |
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349 | while (deg(V) > goal) { |
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350 | PlainDivRem(Q, U, U, V, tmp); |
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351 | swap(U, V); |
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352 | |
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353 | mul(t, Q, M_out(1,0)); |
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354 | sub(t, M_out(0,0), t); |
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355 | M_out(0,0) = M_out(1,0); |
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356 | M_out(1,0) = t; |
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357 | |
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358 | mul(t, Q, M_out(1,1)); |
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359 | sub(t, M_out(0,1), t); |
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360 | M_out(0,1) = M_out(1,1); |
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361 | M_out(1,1) = t; |
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362 | } |
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363 | } |
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364 | |
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365 | |
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366 | |
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367 | void HalfGCD(ZZ_pXMatrix& M_out, const ZZ_pX& U, const ZZ_pX& V, long d_red) |
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368 | { |
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369 | if (IsZero(V) || deg(V) <= deg(U) - d_red) { |
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370 | set(M_out(0,0)); clear(M_out(0,1)); |
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371 | clear(M_out(1,0)); set(M_out(1,1)); |
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372 | |
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373 | return; |
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374 | } |
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375 | |
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376 | |
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377 | long n = deg(U) - 2*d_red + 2; |
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378 | if (n < 0) n = 0; |
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379 | |
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380 | ZZ_pX U1, V1; |
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381 | |
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382 | RightShift(U1, U, n); |
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383 | RightShift(V1, V, n); |
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384 | |
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385 | if (d_red <= NTL_ZZ_pX_HalfGCD_CROSSOVER) { |
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386 | IterHalfGCD(M_out, U1, V1, d_red); |
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387 | return; |
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388 | } |
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389 | |
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390 | long d1 = (d_red + 1)/2; |
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391 | if (d1 < 1) d1 = 1; |
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392 | if (d1 >= d_red) d1 = d_red - 1; |
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393 | |
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394 | ZZ_pXMatrix M1; |
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395 | |
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396 | HalfGCD(M1, U1, V1, d1); |
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397 | mul(U1, V1, M1); |
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398 | |
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399 | long d2 = deg(V1) - deg(U) + n + d_red; |
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400 | |
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401 | if (IsZero(V1) || d2 <= 0) { |
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402 | M_out = M1; |
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403 | return; |
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404 | } |
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405 | |
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406 | |
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407 | ZZ_pX Q; |
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408 | ZZ_pXMatrix M2; |
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409 | |
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410 | DivRem(Q, U1, U1, V1); |
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411 | swap(U1, V1); |
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412 | |
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413 | HalfGCD(M2, U1, V1, d2); |
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414 | |
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415 | ZZ_pX t(INIT_SIZE, deg(M1(1,1))+deg(Q)+1); |
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416 | |
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417 | mul(t, Q, M1(1,0)); |
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418 | sub(t, M1(0,0), t); |
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419 | swap(M1(0,0), M1(1,0)); |
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420 | swap(M1(1,0), t); |
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421 | |
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422 | t.kill(); |
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423 | |
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424 | t.SetMaxLength(deg(M1(1,1))+deg(Q)+1); |
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425 | |
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426 | mul(t, Q, M1(1,1)); |
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427 | sub(t, M1(0,1), t); |
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428 | swap(M1(0,1), M1(1,1)); |
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429 | swap(M1(1,1), t); |
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430 | |
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431 | t.kill(); |
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432 | |
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433 | mul(M_out, M2, M1); |
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434 | } |
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435 | |
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436 | |
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437 | |
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438 | |
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439 | void XHalfGCD(ZZ_pXMatrix& M_out, ZZ_pX& U, ZZ_pX& V, long d_red) |
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440 | { |
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441 | if (IsZero(V) || deg(V) <= deg(U) - d_red) { |
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442 | set(M_out(0,0)); clear(M_out(0,1)); |
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443 | clear(M_out(1,0)); set(M_out(1,1)); |
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444 | |
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445 | return; |
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446 | } |
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447 | |
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448 | long du = deg(U); |
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449 | |
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450 | if (d_red <= NTL_ZZ_pX_HalfGCD_CROSSOVER) { |
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451 | IterHalfGCD(M_out, U, V, d_red); |
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452 | return; |
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453 | } |
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454 | |
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455 | long d1 = (d_red + 1)/2; |
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456 | if (d1 < 1) d1 = 1; |
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457 | if (d1 >= d_red) d1 = d_red - 1; |
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458 | |
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459 | ZZ_pXMatrix M1; |
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460 | |
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461 | HalfGCD(M1, U, V, d1); |
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462 | mul(U, V, M1); |
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463 | |
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464 | long d2 = deg(V) - du + d_red; |
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465 | |
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466 | if (IsZero(V) || d2 <= 0) { |
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467 | M_out = M1; |
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468 | return; |
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469 | } |
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470 | |
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471 | |
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472 | ZZ_pX Q; |
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473 | ZZ_pXMatrix M2; |
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474 | |
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475 | DivRem(Q, U, U, V); |
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476 | swap(U, V); |
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477 | |
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478 | XHalfGCD(M2, U, V, d2); |
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479 | |
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480 | ZZ_pX t(INIT_SIZE, deg(M1(1,1))+deg(Q)+1); |
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481 | |
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482 | mul(t, Q, M1(1,0)); |
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483 | sub(t, M1(0,0), t); |
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484 | swap(M1(0,0), M1(1,0)); |
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485 | swap(M1(1,0), t); |
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486 | |
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487 | t.kill(); |
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488 | |
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489 | t.SetMaxLength(deg(M1(1,1))+deg(Q)+1); |
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490 | |
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491 | mul(t, Q, M1(1,1)); |
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492 | sub(t, M1(0,1), t); |
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493 | swap(M1(0,1), M1(1,1)); |
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494 | swap(M1(1,1), t); |
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495 | |
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496 | t.kill(); |
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497 | |
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498 | mul(M_out, M2, M1); |
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499 | } |
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500 | |
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501 | void HalfGCD(ZZ_pX& U, ZZ_pX& V) |
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502 | { |
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503 | long d_red = (deg(U)+1)/2; |
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504 | |
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505 | if (IsZero(V) || deg(V) <= deg(U) - d_red) { |
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506 | return; |
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507 | } |
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508 | |
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509 | long du = deg(U); |
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510 | |
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511 | |
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512 | long d1 = (d_red + 1)/2; |
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513 | if (d1 < 1) d1 = 1; |
---|
514 | if (d1 >= d_red) d1 = d_red - 1; |
---|
515 | |
---|
516 | ZZ_pXMatrix M1; |
---|
517 | |
---|
518 | HalfGCD(M1, U, V, d1); |
---|
519 | mul(U, V, M1); |
---|
520 | |
---|
521 | long d2 = deg(V) - du + d_red; |
---|
522 | |
---|
523 | if (IsZero(V) || d2 <= 0) { |
---|
524 | return; |
---|
525 | } |
---|
526 | |
---|
527 | M1(0,0).kill(); |
---|
528 | M1(0,1).kill(); |
---|
529 | M1(1,0).kill(); |
---|
530 | M1(1,1).kill(); |
---|
531 | |
---|
532 | |
---|
533 | ZZ_pX Q; |
---|
534 | |
---|
535 | DivRem(Q, U, U, V); |
---|
536 | swap(U, V); |
---|
537 | |
---|
538 | HalfGCD(M1, U, V, d2); |
---|
539 | |
---|
540 | mul(U, V, M1); |
---|
541 | } |
---|
542 | |
---|
543 | |
---|
544 | void GCD(ZZ_pX& d, const ZZ_pX& u, const ZZ_pX& v) |
---|
545 | { |
---|
546 | ZZ_pX u1, v1; |
---|
547 | |
---|
548 | u1 = u; |
---|
549 | v1 = v; |
---|
550 | |
---|
551 | if (deg(u1) == deg(v1)) { |
---|
552 | if (IsZero(u1)) { |
---|
553 | clear(d); |
---|
554 | return; |
---|
555 | } |
---|
556 | |
---|
557 | rem(v1, v1, u1); |
---|
558 | } |
---|
559 | else if (deg(u1) < deg(v1)) { |
---|
560 | swap(u1, v1); |
---|
561 | } |
---|
562 | |
---|
563 | // deg(u1) > deg(v1) |
---|
564 | |
---|
565 | while (deg(u1) > NTL_ZZ_pX_GCD_CROSSOVER && !IsZero(v1)) { |
---|
566 | HalfGCD(u1, v1); |
---|
567 | |
---|
568 | if (!IsZero(v1)) { |
---|
569 | rem(u1, u1, v1); |
---|
570 | swap(u1, v1); |
---|
571 | } |
---|
572 | } |
---|
573 | |
---|
574 | PlainGCD(d, u1, v1); |
---|
575 | } |
---|
576 | |
---|
577 | |
---|
578 | |
---|
579 | void XGCD(ZZ_pX& d, ZZ_pX& s, ZZ_pX& t, const ZZ_pX& a, const ZZ_pX& b) |
---|
580 | { |
---|
581 | ZZ_p w; |
---|
582 | |
---|
583 | if (IsZero(a) && IsZero(b)) { |
---|
584 | clear(d); |
---|
585 | set(s); |
---|
586 | clear(t); |
---|
587 | return; |
---|
588 | } |
---|
589 | |
---|
590 | ZZ_pX U, V, Q; |
---|
591 | |
---|
592 | U = a; |
---|
593 | V = b; |
---|
594 | |
---|
595 | long flag = 0; |
---|
596 | |
---|
597 | if (deg(U) == deg(V)) { |
---|
598 | DivRem(Q, U, U, V); |
---|
599 | swap(U, V); |
---|
600 | flag = 1; |
---|
601 | } |
---|
602 | else if (deg(U) < deg(V)) { |
---|
603 | swap(U, V); |
---|
604 | flag = 2; |
---|
605 | } |
---|
606 | |
---|
607 | ZZ_pXMatrix M; |
---|
608 | |
---|
609 | XHalfGCD(M, U, V, deg(U)+1); |
---|
610 | |
---|
611 | d = U; |
---|
612 | |
---|
613 | if (flag == 0) { |
---|
614 | s = M(0,0); |
---|
615 | t = M(0,1); |
---|
616 | } |
---|
617 | else if (flag == 1) { |
---|
618 | s = M(0,1); |
---|
619 | mul(t, Q, M(0,1)); |
---|
620 | sub(t, M(0,0), t); |
---|
621 | } |
---|
622 | else { /* flag == 2 */ |
---|
623 | s = M(0,1); |
---|
624 | t = M(0,0); |
---|
625 | } |
---|
626 | |
---|
627 | // normalize |
---|
628 | |
---|
629 | inv(w, LeadCoeff(d)); |
---|
630 | mul(d, d, w); |
---|
631 | mul(s, s, w); |
---|
632 | mul(t, t, w); |
---|
633 | } |
---|
634 | |
---|
635 | |
---|
636 | |
---|
637 | |
---|
638 | |
---|
639 | |
---|
640 | |
---|
641 | void IterBuild(ZZ_p* a, long n) |
---|
642 | { |
---|
643 | long i, k; |
---|
644 | ZZ_p b, t; |
---|
645 | |
---|
646 | if (n <= 0) return; |
---|
647 | |
---|
648 | negate(a[0], a[0]); |
---|
649 | |
---|
650 | for (k = 1; k <= n-1; k++) { |
---|
651 | negate(b, a[k]); |
---|
652 | add(a[k], b, a[k-1]); |
---|
653 | for (i = k-1; i >= 1; i--) { |
---|
654 | mul(t, a[i], b); |
---|
655 | add(a[i], t, a[i-1]); |
---|
656 | } |
---|
657 | mul(a[0], a[0], b); |
---|
658 | } |
---|
659 | } |
---|
660 | |
---|
661 | void mul(ZZ_p* x, const ZZ_p* a, const ZZ_p* b, long n) |
---|
662 | { |
---|
663 | static ZZ t, accum; |
---|
664 | |
---|
665 | long i, j, jmin, jmax; |
---|
666 | |
---|
667 | long d = 2*n-1; |
---|
668 | |
---|
669 | for (i = 0; i <= d; i++) { |
---|
670 | jmin = max(0, i-(n-1)); |
---|
671 | jmax = min(n-1, i); |
---|
672 | clear(accum); |
---|
673 | for (j = jmin; j <= jmax; j++) { |
---|
674 | mul(t, rep(a[j]), rep(b[i-j])); |
---|
675 | add(accum, accum, t); |
---|
676 | } |
---|
677 | if (i >= n) { |
---|
678 | add(accum, accum, rep(a[i-n])); |
---|
679 | add(accum, accum, rep(b[i-n])); |
---|
680 | } |
---|
681 | |
---|
682 | conv(x[i], accum); |
---|
683 | } |
---|
684 | } |
---|
685 | |
---|
686 | |
---|
687 | void BuildFromRoots(ZZ_pX& x, const vec_ZZ_p& a) |
---|
688 | { |
---|
689 | long n = a.length(); |
---|
690 | |
---|
691 | if (n == 0) { |
---|
692 | set(x); |
---|
693 | return; |
---|
694 | } |
---|
695 | |
---|
696 | long k0 = NextPowerOfTwo(NTL_ZZ_pX_FFT_CROSSOVER); |
---|
697 | long crossover = 1L << k0; |
---|
698 | |
---|
699 | if (n <= crossover) { |
---|
700 | x.rep.SetMaxLength(n+1); |
---|
701 | x.rep = a; |
---|
702 | IterBuild(&x.rep[0], n); |
---|
703 | x.rep.SetLength(n+1); |
---|
704 | SetCoeff(x, n); |
---|
705 | return; |
---|
706 | } |
---|
707 | |
---|
708 | long k = NextPowerOfTwo(n); |
---|
709 | |
---|
710 | long m = 1L << k; |
---|
711 | long i, j; |
---|
712 | long l, width; |
---|
713 | |
---|
714 | ZZ_pX b(INIT_SIZE, m+1); |
---|
715 | |
---|
716 | b.rep = a; |
---|
717 | b.rep.SetLength(m+1); |
---|
718 | for (i = n; i < m; i++) |
---|
719 | clear(b.rep[i]); |
---|
720 | |
---|
721 | set(b.rep[m]); |
---|
722 | |
---|
723 | FFTRep R1(INIT_SIZE, k), R2(INIT_SIZE, k); |
---|
724 | |
---|
725 | |
---|
726 | ZZ_p t1, one; |
---|
727 | set(one); |
---|
728 | |
---|
729 | vec_ZZ_p G(INIT_SIZE, crossover), H(INIT_SIZE, crossover); |
---|
730 | ZZ_p *g = G.elts(); |
---|
731 | ZZ_p *h = H.elts(); |
---|
732 | ZZ_p *tmp; |
---|
733 | |
---|
734 | for (i = 0; i < m; i+= crossover) { |
---|
735 | for (j = 0; j < crossover; j++) |
---|
736 | negate(g[j], b.rep[i+j]); |
---|
737 | |
---|
738 | if (k0 > 0) { |
---|
739 | for (j = 0; j < crossover; j+=2) { |
---|
740 | mul(t1, g[j], g[j+1]); |
---|
741 | add(g[j+1], g[j], g[j+1]); |
---|
742 | g[j] = t1; |
---|
743 | } |
---|
744 | } |
---|
745 | |
---|
746 | for (l = 1; l < k0; l++) { |
---|
747 | width = 1L << l; |
---|
748 | |
---|
749 | for (j = 0; j < crossover; j += 2*width) |
---|
750 | mul(&h[j], &g[j], &g[j+width], width); |
---|
751 | |
---|
752 | tmp = g; g = h; h = tmp; |
---|
753 | } |
---|
754 | |
---|
755 | for (j = 0; j < crossover; j++) |
---|
756 | b.rep[i+j] = g[j]; |
---|
757 | } |
---|
758 | |
---|
759 | for (l = k0; l < k; l++) { |
---|
760 | width = 1L << l; |
---|
761 | for (i = 0; i < m; i += 2*width) { |
---|
762 | t1 = b.rep[i+width]; |
---|
763 | set(b.rep[i+width]); |
---|
764 | ToFFTRep(R1, b, l+1, i, i+width); |
---|
765 | b.rep[i+width] = t1; |
---|
766 | t1 = b.rep[i+2*width]; |
---|
767 | set(b.rep[i+2*width]); |
---|
768 | ToFFTRep(R2, b, l+1, i+width, i+2*width); |
---|
769 | b.rep[i+2*width] = t1; |
---|
770 | mul(R1, R1, R2); |
---|
771 | FromFFTRep(&b.rep[i], R1, 0, 2*width-1); |
---|
772 | sub(b.rep[i], b.rep[i], one); |
---|
773 | } |
---|
774 | } |
---|
775 | |
---|
776 | x.rep.SetLength(n+1); |
---|
777 | long delta = m-n; |
---|
778 | for (i = 0; i <= n; i++) |
---|
779 | x.rep[i] = b.rep[i+delta]; |
---|
780 | |
---|
781 | // no need to normalize |
---|
782 | } |
---|
783 | |
---|
784 | |
---|
785 | |
---|
786 | void eval(ZZ_p& b, const ZZ_pX& f, const ZZ_p& a) |
---|
787 | // does a Horner evaluation |
---|
788 | { |
---|
789 | ZZ_p acc; |
---|
790 | long i; |
---|
791 | |
---|
792 | clear(acc); |
---|
793 | for (i = deg(f); i >= 0; i--) { |
---|
794 | mul(acc, acc, a); |
---|
795 | add(acc, acc, f.rep[i]); |
---|
796 | } |
---|
797 | |
---|
798 | b = acc; |
---|
799 | } |
---|
800 | |
---|
801 | |
---|
802 | |
---|
803 | void eval(vec_ZZ_p& b, const ZZ_pX& f, const vec_ZZ_p& a) |
---|
804 | // naive algorithm: repeats Horner |
---|
805 | { |
---|
806 | if (&b == &f.rep) { |
---|
807 | vec_ZZ_p bb; |
---|
808 | eval(bb, f, a); |
---|
809 | b = bb; |
---|
810 | return; |
---|
811 | } |
---|
812 | |
---|
813 | long m = a.length(); |
---|
814 | b.SetLength(m); |
---|
815 | long i; |
---|
816 | for (i = 0; i < m; i++) |
---|
817 | eval(b[i], f, a[i]); |
---|
818 | } |
---|
819 | |
---|
820 | |
---|
821 | |
---|
822 | |
---|
823 | void interpolate(ZZ_pX& f, const vec_ZZ_p& a, const vec_ZZ_p& b) |
---|
824 | { |
---|
825 | long m = a.length(); |
---|
826 | if (b.length() != m) Error("interpolate: vector length mismatch"); |
---|
827 | |
---|
828 | if (m == 0) { |
---|
829 | clear(f); |
---|
830 | return; |
---|
831 | } |
---|
832 | |
---|
833 | vec_ZZ_p prod; |
---|
834 | prod = a; |
---|
835 | |
---|
836 | ZZ_p t1, t2; |
---|
837 | |
---|
838 | long k, i; |
---|
839 | |
---|
840 | vec_ZZ_p res; |
---|
841 | res.SetLength(m); |
---|
842 | |
---|
843 | for (k = 0; k < m; k++) { |
---|
844 | |
---|
845 | const ZZ_p& aa = a[k]; |
---|
846 | |
---|
847 | set(t1); |
---|
848 | for (i = k-1; i >= 0; i--) { |
---|
849 | mul(t1, t1, aa); |
---|
850 | add(t1, t1, prod[i]); |
---|
851 | } |
---|
852 | |
---|
853 | clear(t2); |
---|
854 | for (i = k-1; i >= 0; i--) { |
---|
855 | mul(t2, t2, aa); |
---|
856 | add(t2, t2, res[i]); |
---|
857 | } |
---|
858 | |
---|
859 | |
---|
860 | inv(t1, t1); |
---|
861 | sub(t2, b[k], t2); |
---|
862 | mul(t1, t1, t2); |
---|
863 | |
---|
864 | for (i = 0; i < k; i++) { |
---|
865 | mul(t2, prod[i], t1); |
---|
866 | add(res[i], res[i], t2); |
---|
867 | } |
---|
868 | |
---|
869 | res[k] = t1; |
---|
870 | |
---|
871 | if (k < m-1) { |
---|
872 | if (k == 0) |
---|
873 | negate(prod[0], prod[0]); |
---|
874 | else { |
---|
875 | negate(t1, a[k]); |
---|
876 | add(prod[k], t1, prod[k-1]); |
---|
877 | for (i = k-1; i >= 1; i--) { |
---|
878 | mul(t2, prod[i], t1); |
---|
879 | add(prod[i], t2, prod[i-1]); |
---|
880 | } |
---|
881 | mul(prod[0], prod[0], t1); |
---|
882 | } |
---|
883 | } |
---|
884 | } |
---|
885 | |
---|
886 | while (m > 0 && IsZero(res[m-1])) m--; |
---|
887 | res.SetLength(m); |
---|
888 | f.rep = res; |
---|
889 | } |
---|
890 | |
---|
891 | NTL_vector_impl(ZZ_pX,vec_ZZ_pX) |
---|
892 | |
---|
893 | NTL_eq_vector_impl(ZZ_pX,vec_ZZ_pX) |
---|
894 | |
---|
895 | NTL_io_vector_impl(ZZ_pX,vec_ZZ_pX) |
---|
896 | |
---|
897 | |
---|
898 | |
---|
899 | |
---|
900 | void InnerProduct(ZZ_pX& x, const vec_ZZ_p& v, long low, long high, |
---|
901 | const vec_ZZ_pX& H, long n, ZZVec& t) |
---|
902 | { |
---|
903 | static ZZ s; |
---|
904 | long i, j; |
---|
905 | |
---|
906 | for (j = 0; j < n; j++) |
---|
907 | clear(t[j]); |
---|
908 | |
---|
909 | high = min(high, v.length()-1); |
---|
910 | for (i = low; i <= high; i++) { |
---|
911 | const vec_ZZ_p& h = H[i-low].rep; |
---|
912 | long m = h.length(); |
---|
913 | const ZZ& w = rep(v[i]); |
---|
914 | |
---|
915 | for (j = 0; j < m; j++) { |
---|
916 | mul(s, w, rep(h[j])); |
---|
917 | add(t[j], t[j], s); |
---|
918 | } |
---|
919 | } |
---|
920 | |
---|
921 | x.rep.SetLength(n); |
---|
922 | for (j = 0; j < n; j++) |
---|
923 | conv(x.rep[j], t[j]); |
---|
924 | x.normalize(); |
---|
925 | } |
---|
926 | |
---|
927 | |
---|
928 | void CompMod(ZZ_pX& x, const ZZ_pX& g, const ZZ_pXArgument& A, |
---|
929 | const ZZ_pXModulus& F) |
---|
930 | { |
---|
931 | if (deg(g) <= 0) { |
---|
932 | x = g; |
---|
933 | return; |
---|
934 | } |
---|
935 | |
---|
936 | |
---|
937 | ZZ_pX s, t; |
---|
938 | ZZVec scratch(F.n, ZZ_pInfo->ExtendedModulusSize); |
---|
939 | |
---|
940 | long m = A.H.length() - 1; |
---|
941 | long l = ((g.rep.length()+m-1)/m) - 1; |
---|
942 | |
---|
943 | ZZ_pXMultiplier M; |
---|
944 | build(M, A.H[m], F); |
---|
945 | |
---|
946 | InnerProduct(t, g.rep, l*m, l*m + m - 1, A.H, F.n, scratch); |
---|
947 | for (long i = l-1; i >= 0; i--) { |
---|
948 | InnerProduct(s, g.rep, i*m, i*m + m - 1, A.H, F.n, scratch); |
---|
949 | MulMod(t, t, M, F); |
---|
950 | add(t, t, s); |
---|
951 | } |
---|
952 | |
---|
953 | x = t; |
---|
954 | } |
---|
955 | |
---|
956 | |
---|
957 | void build(ZZ_pXArgument& A, const ZZ_pX& h, const ZZ_pXModulus& F, long m) |
---|
958 | { |
---|
959 | if (m <= 0 || deg(h) >= F.n) Error("build: bad args"); |
---|
960 | |
---|
961 | if (m > F.n) m = F.n; |
---|
962 | |
---|
963 | long i; |
---|
964 | |
---|
965 | if (ZZ_pXArgBound > 0) { |
---|
966 | double sz = ZZ_p::storage(); |
---|
967 | sz = sz*F.n; |
---|
968 | sz = sz + NTL_VECTOR_HEADER_SIZE + sizeof(vec_ZZ_p); |
---|
969 | sz = sz/1024; |
---|
970 | m = min(m, long(ZZ_pXArgBound/sz)); |
---|
971 | m = max(m, 1); |
---|
972 | } |
---|
973 | |
---|
974 | ZZ_pXMultiplier M; |
---|
975 | |
---|
976 | build(M, h, F); |
---|
977 | |
---|
978 | A.H.SetLength(m+1); |
---|
979 | |
---|
980 | set(A.H[0]); |
---|
981 | A.H[1] = h; |
---|
982 | for (i = 2; i <= m; i++) |
---|
983 | MulMod(A.H[i], A.H[i-1], M, F); |
---|
984 | } |
---|
985 | |
---|
986 | |
---|
987 | |
---|
988 | |
---|
989 | long ZZ_pXArgBound = 0; |
---|
990 | |
---|
991 | |
---|
992 | void CompMod(ZZ_pX& x, const ZZ_pX& g, const ZZ_pX& h, const ZZ_pXModulus& F) |
---|
993 | // x = g(h) mod f |
---|
994 | { |
---|
995 | long m = SqrRoot(g.rep.length()); |
---|
996 | |
---|
997 | if (m == 0) { |
---|
998 | clear(x); |
---|
999 | return; |
---|
1000 | } |
---|
1001 | |
---|
1002 | ZZ_pXArgument A; |
---|
1003 | |
---|
1004 | build(A, h, F, m); |
---|
1005 | |
---|
1006 | CompMod(x, g, A, F); |
---|
1007 | } |
---|
1008 | |
---|
1009 | |
---|
1010 | |
---|
1011 | |
---|
1012 | void Comp2Mod(ZZ_pX& x1, ZZ_pX& x2, const ZZ_pX& g1, const ZZ_pX& g2, |
---|
1013 | const ZZ_pX& h, const ZZ_pXModulus& F) |
---|
1014 | |
---|
1015 | { |
---|
1016 | long m = SqrRoot(g1.rep.length() + g2.rep.length()); |
---|
1017 | |
---|
1018 | if (m == 0) { |
---|
1019 | clear(x1); |
---|
1020 | clear(x2); |
---|
1021 | return; |
---|
1022 | } |
---|
1023 | |
---|
1024 | ZZ_pXArgument A; |
---|
1025 | |
---|
1026 | build(A, h, F, m); |
---|
1027 | |
---|
1028 | ZZ_pX xx1, xx2; |
---|
1029 | |
---|
1030 | CompMod(xx1, g1, A, F); |
---|
1031 | CompMod(xx2, g2, A, F); |
---|
1032 | |
---|
1033 | x1 = xx1; |
---|
1034 | x2 = xx2; |
---|
1035 | } |
---|
1036 | |
---|
1037 | void Comp3Mod(ZZ_pX& x1, ZZ_pX& x2, ZZ_pX& x3, |
---|
1038 | const ZZ_pX& g1, const ZZ_pX& g2, const ZZ_pX& g3, |
---|
1039 | const ZZ_pX& h, const ZZ_pXModulus& F) |
---|
1040 | |
---|
1041 | { |
---|
1042 | long m = SqrRoot(g1.rep.length() + g2.rep.length() + g3.rep.length()); |
---|
1043 | |
---|
1044 | if (m == 0) { |
---|
1045 | clear(x1); |
---|
1046 | clear(x2); |
---|
1047 | clear(x3); |
---|
1048 | return; |
---|
1049 | } |
---|
1050 | |
---|
1051 | ZZ_pXArgument A; |
---|
1052 | |
---|
1053 | build(A, h, F, m); |
---|
1054 | |
---|
1055 | ZZ_pX xx1, xx2, xx3; |
---|
1056 | |
---|
1057 | CompMod(xx1, g1, A, F); |
---|
1058 | CompMod(xx2, g2, A, F); |
---|
1059 | CompMod(xx3, g3, A, F); |
---|
1060 | |
---|
1061 | x1 = xx1; |
---|
1062 | x2 = xx2; |
---|
1063 | x3 = xx3; |
---|
1064 | } |
---|
1065 | |
---|
1066 | |
---|
1067 | static void StripZeroes(vec_ZZ_p& x) |
---|
1068 | { |
---|
1069 | long n = x.length(); |
---|
1070 | while (n > 0 && IsZero(x[n-1])) |
---|
1071 | n--; |
---|
1072 | x.SetLength(n); |
---|
1073 | } |
---|
1074 | |
---|
1075 | |
---|
1076 | void PlainUpdateMap(vec_ZZ_p& xx, const vec_ZZ_p& a, |
---|
1077 | const ZZ_pX& b, const ZZ_pX& f) |
---|
1078 | { |
---|
1079 | long n = deg(f); |
---|
1080 | long i, m; |
---|
1081 | |
---|
1082 | if (IsZero(b)) { |
---|
1083 | xx.SetLength(0); |
---|
1084 | return; |
---|
1085 | } |
---|
1086 | |
---|
1087 | m = n-1 - deg(b); |
---|
1088 | |
---|
1089 | vec_ZZ_p x(INIT_SIZE, n); |
---|
1090 | |
---|
1091 | for (i = 0; i <= m; i++) |
---|
1092 | InnerProduct(x[i], a, b.rep, i); |
---|
1093 | |
---|
1094 | if (deg(b) != 0) { |
---|
1095 | ZZ_pX c(INIT_SIZE, n); |
---|
1096 | LeftShift(c, b, m); |
---|
1097 | |
---|
1098 | for (i = m+1; i < n; i++) { |
---|
1099 | MulByXMod(c, c, f); |
---|
1100 | InnerProduct(x[i], a, c.rep); |
---|
1101 | } |
---|
1102 | } |
---|
1103 | |
---|
1104 | xx = x; |
---|
1105 | } |
---|
1106 | |
---|
1107 | |
---|
1108 | void UpdateMap(vec_ZZ_p& x, const vec_ZZ_p& aa, |
---|
1109 | const ZZ_pXMultiplier& B, const ZZ_pXModulus& F) |
---|
1110 | { |
---|
1111 | long n = F.n; |
---|
1112 | long i; |
---|
1113 | |
---|
1114 | |
---|
1115 | vec_ZZ_p a; |
---|
1116 | a = aa; |
---|
1117 | StripZeroes(a); |
---|
1118 | |
---|
1119 | if (a.length() > n) Error("UpdateMap: bad args"); |
---|
1120 | |
---|
1121 | if (!B.UseFFT) { |
---|
1122 | PlainUpdateMap(x, a, B.b, F.f); |
---|
1123 | StripZeroes(x); |
---|
1124 | return; |
---|
1125 | } |
---|
1126 | |
---|
1127 | FFTRep R1(INIT_SIZE, F.k), R2(INIT_SIZE, F.l); |
---|
1128 | vec_ZZ_p V1(INIT_SIZE, n); |
---|
1129 | |
---|
1130 | |
---|
1131 | RevToFFTRep(R1, a, F.k, 0, a.length()-1, 0); |
---|
1132 | mul(R2, R1, F.FRep); |
---|
1133 | RevFromFFTRep(V1, R2, 0, n-2); |
---|
1134 | for (i = 0; i <= n-2; i++) negate(V1[i], V1[i]); |
---|
1135 | RevToFFTRep(R2, V1, F.l, 0, n-2, n-1); |
---|
1136 | mul(R2, R2, B.B1); |
---|
1137 | mul(R1, R1, B.B2); |
---|
1138 | |
---|
1139 | AddExpand(R2, R1); |
---|
1140 | RevFromFFTRep(x, R2, 0, n-1); |
---|
1141 | StripZeroes(x); |
---|
1142 | } |
---|
1143 | |
---|
1144 | |
---|
1145 | |
---|
1146 | void ProjectPowers(vec_ZZ_p& x, const vec_ZZ_p& a, long k, |
---|
1147 | const ZZ_pXArgument& H, const ZZ_pXModulus& F) |
---|
1148 | |
---|
1149 | { |
---|
1150 | long n = F.n; |
---|
1151 | |
---|
1152 | if (a.length() > n || k < 0 || k >= (1L << (NTL_BITS_PER_LONG-4))) |
---|
1153 | Error("ProjectPowers: bad args"); |
---|
1154 | |
---|
1155 | long m = H.H.length()-1; |
---|
1156 | long l = (k+m-1)/m - 1; |
---|
1157 | |
---|
1158 | ZZ_pXMultiplier M; |
---|
1159 | build(M, H.H[m], F); |
---|
1160 | |
---|
1161 | vec_ZZ_p s(INIT_SIZE, n); |
---|
1162 | s = a; |
---|
1163 | StripZeroes(s); |
---|
1164 | |
---|
1165 | x.SetLength(k); |
---|
1166 | |
---|
1167 | for (long i = 0; i <= l; i++) { |
---|
1168 | long m1 = min(m, k-i*m); |
---|
1169 | ZZ_p* w = &x[i*m]; |
---|
1170 | for (long j = 0; j < m1; j++) |
---|
1171 | InnerProduct(w[j], H.H[j].rep, s); |
---|
1172 | if (i < l) |
---|
1173 | UpdateMap(s, s, M, F); |
---|
1174 | } |
---|
1175 | } |
---|
1176 | |
---|
1177 | |
---|
1178 | |
---|
1179 | void ProjectPowers(vec_ZZ_p& x, const vec_ZZ_p& a, long k, |
---|
1180 | const ZZ_pX& h, const ZZ_pXModulus& F) |
---|
1181 | |
---|
1182 | { |
---|
1183 | if (a.length() > F.n || k < 0) Error("ProjectPowers: bad args"); |
---|
1184 | |
---|
1185 | if (k == 0) { |
---|
1186 | x.SetLength(0); |
---|
1187 | return; |
---|
1188 | } |
---|
1189 | |
---|
1190 | long m = SqrRoot(k); |
---|
1191 | |
---|
1192 | ZZ_pXArgument H; |
---|
1193 | |
---|
1194 | build(H, h, F, m); |
---|
1195 | ProjectPowers(x, a, k, H, F); |
---|
1196 | } |
---|
1197 | |
---|
1198 | |
---|
1199 | void BerlekampMassey(ZZ_pX& h, const vec_ZZ_p& a, long m) |
---|
1200 | { |
---|
1201 | ZZ_pX Lambda, Sigma, Temp; |
---|
1202 | long L; |
---|
1203 | ZZ_p Delta, Delta1, t1; |
---|
1204 | long shamt; |
---|
1205 | |
---|
1206 | // cerr << "*** " << m << "\n"; |
---|
1207 | |
---|
1208 | Lambda.SetMaxLength(m+1); |
---|
1209 | Sigma.SetMaxLength(m+1); |
---|
1210 | Temp.SetMaxLength(m+1); |
---|
1211 | |
---|
1212 | L = 0; |
---|
1213 | set(Lambda); |
---|
1214 | clear(Sigma); |
---|
1215 | set(Delta); |
---|
1216 | shamt = 0; |
---|
1217 | |
---|
1218 | long i, r, dl; |
---|
1219 | |
---|
1220 | for (r = 1; r <= 2*m; r++) { |
---|
1221 | // cerr << r << "--"; |
---|
1222 | clear(Delta1); |
---|
1223 | dl = deg(Lambda); |
---|
1224 | for (i = 0; i <= dl; i++) { |
---|
1225 | mul(t1, Lambda.rep[i], a[r-i-1]); |
---|
1226 | add(Delta1, Delta1, t1); |
---|
1227 | } |
---|
1228 | |
---|
1229 | if (IsZero(Delta1)) { |
---|
1230 | shamt++; |
---|
1231 | // cerr << "case 1: " << deg(Lambda) << " " << deg(Sigma) << " " << shamt << "\n"; |
---|
1232 | } |
---|
1233 | else if (2*L < r) { |
---|
1234 | div(t1, Delta1, Delta); |
---|
1235 | mul(Temp, Sigma, t1); |
---|
1236 | Sigma = Lambda; |
---|
1237 | ShiftSub(Lambda, Temp, shamt+1); |
---|
1238 | shamt = 0; |
---|
1239 | L = r-L; |
---|
1240 | Delta = Delta1; |
---|
1241 | // cerr << "case 2: " << deg(Lambda) << " " << deg(Sigma) << " " << shamt << "\n"; |
---|
1242 | } |
---|
1243 | else { |
---|
1244 | shamt++; |
---|
1245 | div(t1, Delta1, Delta); |
---|
1246 | mul(Temp, Sigma, t1); |
---|
1247 | ShiftSub(Lambda, Temp, shamt); |
---|
1248 | // cerr << "case 3: " << deg(Lambda) << " " << deg(Sigma) << " " << shamt << "\n"; |
---|
1249 | } |
---|
1250 | } |
---|
1251 | |
---|
1252 | // cerr << "finished: " << L << " " << deg(Lambda) << "\n"; |
---|
1253 | |
---|
1254 | dl = deg(Lambda); |
---|
1255 | h.rep.SetLength(L + 1); |
---|
1256 | |
---|
1257 | for (i = 0; i < L - dl; i++) |
---|
1258 | clear(h.rep[i]); |
---|
1259 | |
---|
1260 | for (i = L - dl; i <= L; i++) |
---|
1261 | h.rep[i] = Lambda.rep[L - i]; |
---|
1262 | } |
---|
1263 | |
---|
1264 | |
---|
1265 | |
---|
1266 | |
---|
1267 | void GCDMinPolySeq(ZZ_pX& h, const vec_ZZ_p& x, long m) |
---|
1268 | { |
---|
1269 | long i; |
---|
1270 | ZZ_pX a, b; |
---|
1271 | ZZ_pXMatrix M; |
---|
1272 | ZZ_p t; |
---|
1273 | |
---|
1274 | a.rep.SetLength(2*m); |
---|
1275 | for (i = 0; i < 2*m; i++) a.rep[i] = x[2*m-1-i]; |
---|
1276 | a.normalize(); |
---|
1277 | |
---|
1278 | SetCoeff(b, 2*m); |
---|
1279 | |
---|
1280 | HalfGCD(M, b, a, m+1); |
---|
1281 | |
---|
1282 | /* make monic */ |
---|
1283 | |
---|
1284 | inv(t, LeadCoeff(M(1,1))); |
---|
1285 | mul(h, M(1,1), t); |
---|
1286 | } |
---|
1287 | |
---|
1288 | |
---|
1289 | void MinPolySeq(ZZ_pX& h, const vec_ZZ_p& a, long m) |
---|
1290 | { |
---|
1291 | if (m < 0 || m >= (1L << (NTL_BITS_PER_LONG-4))) Error("MinPoly: bad args"); |
---|
1292 | if (a.length() < 2*m) Error("MinPoly: sequence too short"); |
---|
1293 | |
---|
1294 | if (m > NTL_ZZ_pX_BERMASS_CROSSOVER) |
---|
1295 | GCDMinPolySeq(h, a, m); |
---|
1296 | else |
---|
1297 | BerlekampMassey(h, a, m); |
---|
1298 | } |
---|
1299 | |
---|
1300 | |
---|
1301 | void DoMinPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F, long m, |
---|
1302 | const vec_ZZ_p& R) |
---|
1303 | { |
---|
1304 | vec_ZZ_p x; |
---|
1305 | |
---|
1306 | ProjectPowers(x, R, 2*m, g, F); |
---|
1307 | MinPolySeq(h, x, m); |
---|
1308 | } |
---|
1309 | |
---|
1310 | |
---|
1311 | void ProbMinPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F, long m) |
---|
1312 | { |
---|
1313 | long n = F.n; |
---|
1314 | if (m < 1 || m > n) Error("ProbMinPoly: bad args"); |
---|
1315 | |
---|
1316 | long i; |
---|
1317 | vec_ZZ_p R(INIT_SIZE, n); |
---|
1318 | |
---|
1319 | for (i = 0; i < n; i++) random(R[i]); |
---|
1320 | DoMinPolyMod(h, g, F, m, R); |
---|
1321 | } |
---|
1322 | |
---|
1323 | void MinPolyMod(ZZ_pX& hh, const ZZ_pX& g, const ZZ_pXModulus& F, long m) |
---|
1324 | { |
---|
1325 | ZZ_pX h, h1; |
---|
1326 | long n = F.n; |
---|
1327 | if (m < 1 || m > n) Error("MinPoly: bad args"); |
---|
1328 | |
---|
1329 | /* probabilistically compute min-poly */ |
---|
1330 | |
---|
1331 | ProbMinPolyMod(h, g, F, m); |
---|
1332 | if (deg(h) == m) { hh = h; return; } |
---|
1333 | CompMod(h1, h, g, F); |
---|
1334 | if (IsZero(h1)) { hh = h; return; } |
---|
1335 | |
---|
1336 | /* not completely successful...must iterate */ |
---|
1337 | |
---|
1338 | long i; |
---|
1339 | |
---|
1340 | ZZ_pX h2, h3; |
---|
1341 | ZZ_pXMultiplier H1; |
---|
1342 | vec_ZZ_p R(INIT_SIZE, n); |
---|
1343 | |
---|
1344 | for (;;) { |
---|
1345 | R.SetLength(n); |
---|
1346 | for (i = 0; i < n; i++) random(R[i]); |
---|
1347 | build(H1, h1, F); |
---|
1348 | UpdateMap(R, R, H1, F); |
---|
1349 | DoMinPolyMod(h2, g, F, m-deg(h), R); |
---|
1350 | |
---|
1351 | mul(h, h, h2); |
---|
1352 | if (deg(h) == m) { hh = h; return; } |
---|
1353 | CompMod(h3, h2, g, F); |
---|
1354 | MulMod(h1, h3, H1, F); |
---|
1355 | if (IsZero(h1)) { hh = h; return; } |
---|
1356 | } |
---|
1357 | } |
---|
1358 | |
---|
1359 | |
---|
1360 | |
---|
1361 | void IrredPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F, long m) |
---|
1362 | { |
---|
1363 | vec_ZZ_p R(INIT_SIZE, 1); |
---|
1364 | if (m < 1 || m > F.n) Error("IrredPoly: bad args"); |
---|
1365 | |
---|
1366 | set(R[0]); |
---|
1367 | DoMinPolyMod(h, g, F, m, R); |
---|
1368 | } |
---|
1369 | |
---|
1370 | |
---|
1371 | |
---|
1372 | void diff(ZZ_pX& x, const ZZ_pX& a) |
---|
1373 | { |
---|
1374 | long n = deg(a); |
---|
1375 | long i; |
---|
1376 | |
---|
1377 | if (n <= 0) { |
---|
1378 | clear(x); |
---|
1379 | return; |
---|
1380 | } |
---|
1381 | |
---|
1382 | if (&x != &a) |
---|
1383 | x.rep.SetLength(n); |
---|
1384 | |
---|
1385 | for (i = 0; i <= n-1; i++) { |
---|
1386 | mul(x.rep[i], a.rep[i+1], i+1); |
---|
1387 | } |
---|
1388 | |
---|
1389 | if (&x == &a) |
---|
1390 | x.rep.SetLength(n); |
---|
1391 | |
---|
1392 | x.normalize(); |
---|
1393 | } |
---|
1394 | |
---|
1395 | |
---|
1396 | void MakeMonic(ZZ_pX& x) |
---|
1397 | { |
---|
1398 | if (IsZero(x)) |
---|
1399 | return; |
---|
1400 | |
---|
1401 | if (IsOne(LeadCoeff(x))) |
---|
1402 | return; |
---|
1403 | |
---|
1404 | ZZ_p t; |
---|
1405 | |
---|
1406 | inv(t, LeadCoeff(x)); |
---|
1407 | mul(x, x, t); |
---|
1408 | } |
---|
1409 | |
---|
1410 | |
---|
1411 | |
---|
1412 | void PlainMulTrunc(ZZ_pX& x, const ZZ_pX& a, const ZZ_pX& b, long n) |
---|
1413 | { |
---|
1414 | ZZ_pX y; |
---|
1415 | mul(y, a, b); |
---|
1416 | trunc(x, y, n); |
---|
1417 | } |
---|
1418 | |
---|
1419 | |
---|
1420 | void FFTMulTrunc(ZZ_pX& x, const ZZ_pX& a, const ZZ_pX& b, long n) |
---|
1421 | { |
---|
1422 | if (IsZero(a) || IsZero(b)) { |
---|
1423 | clear(x); |
---|
1424 | return; |
---|
1425 | } |
---|
1426 | |
---|
1427 | long d = deg(a) + deg(b); |
---|
1428 | if (n > d + 1) |
---|
1429 | n = d + 1; |
---|
1430 | |
---|
1431 | long k = NextPowerOfTwo(d + 1); |
---|
1432 | FFTRep R1(INIT_SIZE, k), R2(INIT_SIZE, k); |
---|
1433 | |
---|
1434 | ToFFTRep(R1, a, k); |
---|
1435 | ToFFTRep(R2, b, k); |
---|
1436 | mul(R1, R1, R2); |
---|
1437 | FromFFTRep(x, R1, 0, n-1); |
---|
1438 | } |
---|
1439 | |
---|
1440 | void MulTrunc(ZZ_pX& x, const ZZ_pX& a, const ZZ_pX& b, long n) |
---|
1441 | { |
---|
1442 | if (n < 0) Error("MulTrunc: bad args"); |
---|
1443 | |
---|
1444 | if (deg(a) <= NTL_ZZ_pX_FFT_CROSSOVER || deg(b) <= NTL_ZZ_pX_FFT_CROSSOVER) |
---|
1445 | PlainMulTrunc(x, a, b, n); |
---|
1446 | else |
---|
1447 | FFTMulTrunc(x, a, b, n); |
---|
1448 | } |
---|
1449 | |
---|
1450 | |
---|
1451 | void PlainSqrTrunc(ZZ_pX& x, const ZZ_pX& a, long n) |
---|
1452 | { |
---|
1453 | ZZ_pX y; |
---|
1454 | sqr(y, a); |
---|
1455 | trunc(x, y, n); |
---|
1456 | } |
---|
1457 | |
---|
1458 | |
---|
1459 | void FFTSqrTrunc(ZZ_pX& x, const ZZ_pX& a, long n) |
---|
1460 | { |
---|
1461 | if (IsZero(a)) { |
---|
1462 | clear(x); |
---|
1463 | return; |
---|
1464 | } |
---|
1465 | |
---|
1466 | long d = 2*deg(a); |
---|
1467 | if (n > d + 1) |
---|
1468 | n = d + 1; |
---|
1469 | |
---|
1470 | long k = NextPowerOfTwo(d + 1); |
---|
1471 | FFTRep R1(INIT_SIZE, k); |
---|
1472 | |
---|
1473 | ToFFTRep(R1, a, k); |
---|
1474 | mul(R1, R1, R1); |
---|
1475 | FromFFTRep(x, R1, 0, n-1); |
---|
1476 | } |
---|
1477 | |
---|
1478 | void SqrTrunc(ZZ_pX& x, const ZZ_pX& a, long n) |
---|
1479 | { |
---|
1480 | if (n < 0) Error("SqrTrunc: bad args"); |
---|
1481 | |
---|
1482 | if (deg(a) <= NTL_ZZ_pX_FFT_CROSSOVER) |
---|
1483 | PlainSqrTrunc(x, a, n); |
---|
1484 | else |
---|
1485 | FFTSqrTrunc(x, a, n); |
---|
1486 | } |
---|
1487 | |
---|
1488 | |
---|
1489 | void FastTraceVec(vec_ZZ_p& S, const ZZ_pX& f) |
---|
1490 | { |
---|
1491 | long n = deg(f); |
---|
1492 | |
---|
1493 | if (n <= 0) |
---|
1494 | Error("FastTraceVec: bad args"); |
---|
1495 | |
---|
1496 | if (n == 0) { |
---|
1497 | S.SetLength(0); |
---|
1498 | return; |
---|
1499 | } |
---|
1500 | |
---|
1501 | if (n == 1) { |
---|
1502 | S.SetLength(1); |
---|
1503 | set(S[0]); |
---|
1504 | return; |
---|
1505 | } |
---|
1506 | |
---|
1507 | long i; |
---|
1508 | ZZ_pX f1; |
---|
1509 | |
---|
1510 | f1.rep.SetLength(n-1); |
---|
1511 | for (i = 0; i <= n-2; i++) |
---|
1512 | f1.rep[i] = f.rep[n-i]; |
---|
1513 | f1.normalize(); |
---|
1514 | |
---|
1515 | ZZ_pX f2; |
---|
1516 | f2.rep.SetLength(n-1); |
---|
1517 | for (i = 0; i <= n-2; i++) |
---|
1518 | mul(f2.rep[i], f.rep[n-1-i], i+1); |
---|
1519 | f2.normalize(); |
---|
1520 | |
---|
1521 | ZZ_pX f3; |
---|
1522 | InvTrunc(f3, f1, n-1); |
---|
1523 | MulTrunc(f3, f3, f2, n-1); |
---|
1524 | |
---|
1525 | S.SetLength(n); |
---|
1526 | |
---|
1527 | S[0] = n; |
---|
1528 | for (i = 1; i < n; i++) |
---|
1529 | negate(S[i], coeff(f3, i-1)); |
---|
1530 | } |
---|
1531 | |
---|
1532 | |
---|
1533 | void PlainTraceVec(vec_ZZ_p& S, const ZZ_pX& ff) |
---|
1534 | { |
---|
1535 | if (deg(ff) <= 0) |
---|
1536 | Error("TraceVec: bad args"); |
---|
1537 | |
---|
1538 | ZZ_pX f; |
---|
1539 | f = ff; |
---|
1540 | |
---|
1541 | MakeMonic(f); |
---|
1542 | |
---|
1543 | long n = deg(f); |
---|
1544 | |
---|
1545 | S.SetLength(n); |
---|
1546 | |
---|
1547 | if (n == 0) |
---|
1548 | return; |
---|
1549 | |
---|
1550 | long k, i; |
---|
1551 | ZZ acc, t; |
---|
1552 | ZZ_p t1; |
---|
1553 | |
---|
1554 | S[0] = n; |
---|
1555 | |
---|
1556 | for (k = 1; k < n; k++) { |
---|
1557 | mul(acc, rep(f.rep[n-k]), k); |
---|
1558 | |
---|
1559 | for (i = 1; i < k; i++) { |
---|
1560 | mul(t, rep(f.rep[n-i]), rep(S[k-i])); |
---|
1561 | add(acc, acc, t); |
---|
1562 | } |
---|
1563 | |
---|
1564 | conv(t1, acc); |
---|
1565 | negate(S[k], t1); |
---|
1566 | } |
---|
1567 | } |
---|
1568 | |
---|
1569 | void TraceVec(vec_ZZ_p& S, const ZZ_pX& f) |
---|
1570 | { |
---|
1571 | if (deg(f) <= NTL_ZZ_pX_TRACE_CROSSOVER) |
---|
1572 | PlainTraceVec(S, f); |
---|
1573 | else |
---|
1574 | FastTraceVec(S, f); |
---|
1575 | } |
---|
1576 | |
---|
1577 | void ComputeTraceVec(const ZZ_pXModulus& F) |
---|
1578 | { |
---|
1579 | vec_ZZ_p& S = *((vec_ZZ_p *) &F.tracevec); |
---|
1580 | |
---|
1581 | if (S.length() > 0) |
---|
1582 | return; |
---|
1583 | |
---|
1584 | if (!F.UseFFT) { |
---|
1585 | PlainTraceVec(S, F.f); |
---|
1586 | return; |
---|
1587 | } |
---|
1588 | |
---|
1589 | long i; |
---|
1590 | long n = F.n; |
---|
1591 | |
---|
1592 | FFTRep R; |
---|
1593 | ZZ_pX P, g; |
---|
1594 | |
---|
1595 | g.rep.SetLength(n-1); |
---|
1596 | for (i = 1; i < n; i++) |
---|
1597 | mul(g.rep[n-i-1], F.f.rep[n-i], i); |
---|
1598 | g.normalize(); |
---|
1599 | |
---|
1600 | ToFFTRep(R, g, F.l); |
---|
1601 | mul(R, R, F.HRep); |
---|
1602 | FromFFTRep(P, R, n-2, 2*n-4); |
---|
1603 | |
---|
1604 | S.SetLength(n); |
---|
1605 | |
---|
1606 | S[0] = n; |
---|
1607 | for (i = 1; i < n; i++) |
---|
1608 | negate(S[i], coeff(P, n-1-i)); |
---|
1609 | } |
---|
1610 | |
---|
1611 | void TraceMod(ZZ_p& x, const ZZ_pX& a, const ZZ_pXModulus& F) |
---|
1612 | { |
---|
1613 | long n = F.n; |
---|
1614 | |
---|
1615 | if (deg(a) >= n) |
---|
1616 | Error("trace: bad args"); |
---|
1617 | |
---|
1618 | if (F.tracevec.length() == 0) |
---|
1619 | ComputeTraceVec(F); |
---|
1620 | |
---|
1621 | InnerProduct(x, a.rep, F.tracevec); |
---|
1622 | } |
---|
1623 | |
---|
1624 | void TraceMod(ZZ_p& x, const ZZ_pX& a, const ZZ_pX& f) |
---|
1625 | { |
---|
1626 | if (deg(a) >= deg(f) || deg(f) <= 0) |
---|
1627 | Error("trace: bad args"); |
---|
1628 | |
---|
1629 | project(x, TraceVec(f), a); |
---|
1630 | } |
---|
1631 | |
---|
1632 | void PlainResultant(ZZ_p& rres, const ZZ_pX& a, const ZZ_pX& b) |
---|
1633 | { |
---|
1634 | ZZ_p res; |
---|
1635 | |
---|
1636 | if (IsZero(a) || IsZero(b)) |
---|
1637 | clear(res); |
---|
1638 | else if (deg(a) == 0 && deg(b) == 0) |
---|
1639 | set(res); |
---|
1640 | else { |
---|
1641 | long d0, d1, d2; |
---|
1642 | ZZ_p lc; |
---|
1643 | set(res); |
---|
1644 | |
---|
1645 | long n = max(deg(a),deg(b)) + 1; |
---|
1646 | ZZ_pX u(INIT_SIZE, n), v(INIT_SIZE, n); |
---|
1647 | ZZVec tmp(n, ZZ_pInfo->ExtendedModulusSize); |
---|
1648 | |
---|
1649 | u = a; |
---|
1650 | v = b; |
---|
1651 | |
---|
1652 | for (;;) { |
---|
1653 | d0 = deg(u); |
---|
1654 | d1 = deg(v); |
---|
1655 | lc = LeadCoeff(v); |
---|
1656 | |
---|
1657 | PlainRem(u, u, v, tmp); |
---|
1658 | swap(u, v); |
---|
1659 | |
---|
1660 | d2 = deg(v); |
---|
1661 | if (d2 >= 0) { |
---|
1662 | power(lc, lc, d0-d2); |
---|
1663 | mul(res, res, lc); |
---|
1664 | if (d0 & d1 & 1) negate(res, res); |
---|
1665 | } |
---|
1666 | else { |
---|
1667 | if (d1 == 0) { |
---|
1668 | power(lc, lc, d0); |
---|
1669 | mul(res, res, lc); |
---|
1670 | } |
---|
1671 | else |
---|
1672 | clear(res); |
---|
1673 | |
---|
1674 | break; |
---|
1675 | } |
---|
1676 | } |
---|
1677 | |
---|
1678 | rres = res; |
---|
1679 | } |
---|
1680 | } |
---|
1681 | |
---|
1682 | |
---|
1683 | void ResIterHalfGCD(ZZ_pXMatrix& M_out, ZZ_pX& U, ZZ_pX& V, long d_red, |
---|
1684 | vec_ZZ_p& cvec, vec_long& dvec) |
---|
1685 | { |
---|
1686 | M_out(0,0).SetMaxLength(d_red); |
---|
1687 | M_out(0,1).SetMaxLength(d_red); |
---|
1688 | M_out(1,0).SetMaxLength(d_red); |
---|
1689 | M_out(1,1).SetMaxLength(d_red); |
---|
1690 | |
---|
1691 | set(M_out(0,0)); clear(M_out(0,1)); |
---|
1692 | clear(M_out(1,0)); set(M_out(1,1)); |
---|
1693 | |
---|
1694 | long goal = deg(U) - d_red; |
---|
1695 | |
---|
1696 | if (deg(V) <= goal) |
---|
1697 | return; |
---|
1698 | |
---|
1699 | ZZVec tmp(deg(U)+1, ZZ_pInfo->ExtendedModulusSize); |
---|
1700 | ZZ_pX Q, t(INIT_SIZE, d_red); |
---|
1701 | |
---|
1702 | |
---|
1703 | while (deg(V) > goal) { |
---|
1704 | append(cvec, LeadCoeff(V)); |
---|
1705 | append(dvec, dvec[dvec.length()-1]-deg(U)+deg(V)); |
---|
1706 | PlainDivRem(Q, U, U, V, tmp); |
---|
1707 | swap(U, V); |
---|
1708 | |
---|
1709 | mul(t, Q, M_out(1,0)); |
---|
1710 | sub(t, M_out(0,0), t); |
---|
1711 | M_out(0,0) = M_out(1,0); |
---|
1712 | M_out(1,0) = t; |
---|
1713 | |
---|
1714 | mul(t, Q, M_out(1,1)); |
---|
1715 | sub(t, M_out(0,1), t); |
---|
1716 | M_out(0,1) = M_out(1,1); |
---|
1717 | M_out(1,1) = t; |
---|
1718 | } |
---|
1719 | } |
---|
1720 | |
---|
1721 | |
---|
1722 | |
---|
1723 | void ResHalfGCD(ZZ_pXMatrix& M_out, const ZZ_pX& U, const ZZ_pX& V, long d_red, |
---|
1724 | vec_ZZ_p& cvec, vec_long& dvec) |
---|
1725 | { |
---|
1726 | if (IsZero(V) || deg(V) <= deg(U) - d_red) { |
---|
1727 | set(M_out(0,0)); clear(M_out(0,1)); |
---|
1728 | clear(M_out(1,0)); set(M_out(1,1)); |
---|
1729 | |
---|
1730 | return; |
---|
1731 | } |
---|
1732 | |
---|
1733 | |
---|
1734 | long n = deg(U) - 2*d_red + 2; |
---|
1735 | if (n < 0) n = 0; |
---|
1736 | |
---|
1737 | ZZ_pX U1, V1; |
---|
1738 | |
---|
1739 | RightShift(U1, U, n); |
---|
1740 | RightShift(V1, V, n); |
---|
1741 | |
---|
1742 | if (d_red <= NTL_ZZ_pX_HalfGCD_CROSSOVER) { |
---|
1743 | ResIterHalfGCD(M_out, U1, V1, d_red, cvec, dvec); |
---|
1744 | return; |
---|
1745 | } |
---|
1746 | |
---|
1747 | long d1 = (d_red + 1)/2; |
---|
1748 | if (d1 < 1) d1 = 1; |
---|
1749 | if (d1 >= d_red) d1 = d_red - 1; |
---|
1750 | |
---|
1751 | ZZ_pXMatrix M1; |
---|
1752 | |
---|
1753 | ResHalfGCD(M1, U1, V1, d1, cvec, dvec); |
---|
1754 | mul(U1, V1, M1); |
---|
1755 | |
---|
1756 | long d2 = deg(V1) - deg(U) + n + d_red; |
---|
1757 | |
---|
1758 | if (IsZero(V1) || d2 <= 0) { |
---|
1759 | M_out = M1; |
---|
1760 | return; |
---|
1761 | } |
---|
1762 | |
---|
1763 | |
---|
1764 | ZZ_pX Q; |
---|
1765 | ZZ_pXMatrix M2; |
---|
1766 | |
---|
1767 | append(cvec, LeadCoeff(V1)); |
---|
1768 | append(dvec, dvec[dvec.length()-1]-deg(U1)+deg(V1)); |
---|
1769 | DivRem(Q, U1, U1, V1); |
---|
1770 | swap(U1, V1); |
---|
1771 | |
---|
1772 | ResHalfGCD(M2, U1, V1, d2, cvec, dvec); |
---|
1773 | |
---|
1774 | ZZ_pX t(INIT_SIZE, deg(M1(1,1))+deg(Q)+1); |
---|
1775 | |
---|
1776 | mul(t, Q, M1(1,0)); |
---|
1777 | sub(t, M1(0,0), t); |
---|
1778 | swap(M1(0,0), M1(1,0)); |
---|
1779 | swap(M1(1,0), t); |
---|
1780 | |
---|
1781 | t.kill(); |
---|
1782 | |
---|
1783 | t.SetMaxLength(deg(M1(1,1))+deg(Q)+1); |
---|
1784 | |
---|
1785 | mul(t, Q, M1(1,1)); |
---|
1786 | sub(t, M1(0,1), t); |
---|
1787 | swap(M1(0,1), M1(1,1)); |
---|
1788 | swap(M1(1,1), t); |
---|
1789 | |
---|
1790 | t.kill(); |
---|
1791 | |
---|
1792 | mul(M_out, M2, M1); |
---|
1793 | } |
---|
1794 | |
---|
1795 | void ResHalfGCD(ZZ_pX& U, ZZ_pX& V, vec_ZZ_p& cvec, vec_long& dvec) |
---|
1796 | { |
---|
1797 | long d_red = (deg(U)+1)/2; |
---|
1798 | |
---|
1799 | if (IsZero(V) || deg(V) <= deg(U) - d_red) { |
---|
1800 | return; |
---|
1801 | } |
---|
1802 | |
---|
1803 | long du = deg(U); |
---|
1804 | |
---|
1805 | |
---|
1806 | long d1 = (d_red + 1)/2; |
---|
1807 | if (d1 < 1) d1 = 1; |
---|
1808 | if (d1 >= d_red) d1 = d_red - 1; |
---|
1809 | |
---|
1810 | ZZ_pXMatrix M1; |
---|
1811 | |
---|
1812 | ResHalfGCD(M1, U, V, d1, cvec, dvec); |
---|
1813 | mul(U, V, M1); |
---|
1814 | |
---|
1815 | long d2 = deg(V) - du + d_red; |
---|
1816 | |
---|
1817 | if (IsZero(V) || d2 <= 0) { |
---|
1818 | return; |
---|
1819 | } |
---|
1820 | |
---|
1821 | M1(0,0).kill(); |
---|
1822 | M1(0,1).kill(); |
---|
1823 | M1(1,0).kill(); |
---|
1824 | M1(1,1).kill(); |
---|
1825 | |
---|
1826 | |
---|
1827 | ZZ_pX Q; |
---|
1828 | |
---|
1829 | append(cvec, LeadCoeff(V)); |
---|
1830 | append(dvec, dvec[dvec.length()-1]-deg(U)+deg(V)); |
---|
1831 | DivRem(Q, U, U, V); |
---|
1832 | swap(U, V); |
---|
1833 | |
---|
1834 | ResHalfGCD(M1, U, V, d2, cvec, dvec); |
---|
1835 | |
---|
1836 | mul(U, V, M1); |
---|
1837 | } |
---|
1838 | |
---|
1839 | |
---|
1840 | void resultant(ZZ_p& rres, const ZZ_pX& u, const ZZ_pX& v) |
---|
1841 | { |
---|
1842 | if (deg(u) <= NTL_ZZ_pX_GCD_CROSSOVER || deg(v) <= NTL_ZZ_pX_GCD_CROSSOVER) { |
---|
1843 | PlainResultant(rres, u, v); |
---|
1844 | return; |
---|
1845 | } |
---|
1846 | |
---|
1847 | ZZ_pX u1, v1; |
---|
1848 | |
---|
1849 | u1 = u; |
---|
1850 | v1 = v; |
---|
1851 | |
---|
1852 | ZZ_p res, t; |
---|
1853 | set(res); |
---|
1854 | |
---|
1855 | if (deg(u1) == deg(v1)) { |
---|
1856 | rem(u1, u1, v1); |
---|
1857 | swap(u1, v1); |
---|
1858 | |
---|
1859 | if (IsZero(v1)) { |
---|
1860 | clear(rres); |
---|
1861 | return; |
---|
1862 | } |
---|
1863 | |
---|
1864 | power(t, LeadCoeff(u1), deg(u1) - deg(v1)); |
---|
1865 | mul(res, res, t); |
---|
1866 | if (deg(u1) & 1) |
---|
1867 | negate(res, res); |
---|
1868 | } |
---|
1869 | else if (deg(u1) < deg(v1)) { |
---|
1870 | swap(u1, v1); |
---|
1871 | if (deg(u1) & deg(v1) & 1) |
---|
1872 | negate(res, res); |
---|
1873 | } |
---|
1874 | |
---|
1875 | // deg(u1) > deg(v1) && v1 != 0 |
---|
1876 | |
---|
1877 | vec_ZZ_p cvec; |
---|
1878 | vec_long dvec; |
---|
1879 | |
---|
1880 | cvec.SetMaxLength(deg(v1)+2); |
---|
1881 | dvec.SetMaxLength(deg(v1)+2); |
---|
1882 | |
---|
1883 | append(cvec, LeadCoeff(u1)); |
---|
1884 | append(dvec, deg(u1)); |
---|
1885 | |
---|
1886 | |
---|
1887 | while (deg(u1) > NTL_ZZ_pX_GCD_CROSSOVER && !IsZero(v1)) { |
---|
1888 | ResHalfGCD(u1, v1, cvec, dvec); |
---|
1889 | |
---|
1890 | if (!IsZero(v1)) { |
---|
1891 | append(cvec, LeadCoeff(v1)); |
---|
1892 | append(dvec, deg(v1)); |
---|
1893 | rem(u1, u1, v1); |
---|
1894 | swap(u1, v1); |
---|
1895 | } |
---|
1896 | } |
---|
1897 | |
---|
1898 | if (IsZero(v1) && deg(u1) > 0) { |
---|
1899 | clear(rres); |
---|
1900 | return; |
---|
1901 | } |
---|
1902 | |
---|
1903 | long i, l; |
---|
1904 | l = dvec.length(); |
---|
1905 | |
---|
1906 | if (deg(u1) == 0) { |
---|
1907 | // we went all the way... |
---|
1908 | |
---|
1909 | for (i = 0; i <= l-3; i++) { |
---|
1910 | power(t, cvec[i+1], dvec[i]-dvec[i+2]); |
---|
1911 | mul(res, res, t); |
---|
1912 | if (dvec[i] & dvec[i+1] & 1) |
---|
1913 | negate(res, res); |
---|
1914 | } |
---|
1915 | |
---|
1916 | power(t, cvec[l-1], dvec[l-2]); |
---|
1917 | mul(res, res, t); |
---|
1918 | } |
---|
1919 | else { |
---|
1920 | for (i = 0; i <= l-3; i++) { |
---|
1921 | power(t, cvec[i+1], dvec[i]-dvec[i+2]); |
---|
1922 | mul(res, res, t); |
---|
1923 | if (dvec[i] & dvec[i+1] & 1) |
---|
1924 | negate(res, res); |
---|
1925 | } |
---|
1926 | |
---|
1927 | power(t, cvec[l-1], dvec[l-2]-deg(v1)); |
---|
1928 | mul(res, res, t); |
---|
1929 | if (dvec[l-2] & dvec[l-1] & 1) |
---|
1930 | negate(res, res); |
---|
1931 | |
---|
1932 | PlainResultant(t, u1, v1); |
---|
1933 | mul(res, res, t); |
---|
1934 | } |
---|
1935 | |
---|
1936 | rres = res; |
---|
1937 | } |
---|
1938 | |
---|
1939 | void NormMod(ZZ_p& x, const ZZ_pX& a, const ZZ_pX& f) |
---|
1940 | { |
---|
1941 | if (deg(f) <= 0 || deg(a) >= deg(f)) |
---|
1942 | Error("norm: bad args"); |
---|
1943 | |
---|
1944 | if (IsZero(a)) { |
---|
1945 | clear(x); |
---|
1946 | return; |
---|
1947 | } |
---|
1948 | |
---|
1949 | ZZ_p t; |
---|
1950 | resultant(t, f, a); |
---|
1951 | if (!IsOne(LeadCoeff(f))) { |
---|
1952 | ZZ_p t1; |
---|
1953 | power(t1, LeadCoeff(f), deg(a)); |
---|
1954 | inv(t1, t1); |
---|
1955 | mul(t, t, t1); |
---|
1956 | } |
---|
1957 | |
---|
1958 | x = t; |
---|
1959 | } |
---|
1960 | |
---|
1961 | NTL_END_IMPL |
---|