7 #include "cln/numtheory.h"
13 #include "cln/abort.h"
17 #define floor cln_floor
21 // Algorithm 1 (for very small p only):
22 // Try different values.
23 // Assume p is prime and a nonzero square in Z/pZ.
24 static uint32 search_sqrt (uint32 p, uint32 a)
29 // 0 < x <= p/2, x2 = x^2 mod p.
32 x2 += x; x++; x2 += x;
38 // Algorithm 2 (for p > 2 only):
40 // [Beth et al.: Computer Algebra, 1988, Kapitel 5.3.3.]
41 // [Cohen, A Course in Computational Algebraic Number Theory,
42 // Section 3.4.4., Algorithm 3.4.6.]
43 // Input: R = Z/pZ with p>2, and a (nonzero square in R).
44 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a);
45 // Compute in the polynomial ring R[X]/(X^2-a).
47 // A polynomial c0+c1*X mod (X^2-a)
51 pol2 (const cl_MI& _c0, const cl_MI& _c1) : c0 (_c0), c1 (_c1) {}
54 const cl_modint_ring& R;
58 return pol2(R->zero(),R->zero());
62 return pol2(R->one(),R->zero());
64 const pol2 plus (const pol2& u, const pol2& v)
66 return pol2(u.c0+v.c0, u.c1+v.c1);
68 const pol2 minus (const pol2& u, const pol2& v)
70 return pol2(u.c0-v.c0, u.c1-v.c1);
72 const pol2 mul (const pol2& u, const pol2& v)
74 return pol2(u.c0*v.c0+u.c1*v.c1*a, u.c0*v.c1+u.c1*v.c0);
76 const pol2 square (const pol2& u)
78 return pol2(cln::square(u.c0) + cln::square(u.c1)*a, (u.c0*u.c1)<<1);
80 const pol2 expt_pos (const pol2& x, const cl_I& y)
82 // Right-Left Binary, [Cohen, Algorithm 1.2.1.]
85 while (!oddp(b)) { a = square(a); b = b = b >> 1; } // a^b = x^y
98 return pol2(R->random(),R->random());
100 // Computes the degree of gcd(u(X),X^2-a) and, if it is 1,
101 // also the zero if this polynomial of degree 1.
103 cl_composite_condition* condition;
107 gcd_result (cl_composite_condition* c) : condition (c) {}
108 gcd_result (int deg) : condition (NULL), gcd_degree (deg) {}
109 gcd_result (int deg, const cl_MI& sol) : condition (NULL), gcd_degree (deg), solution (sol) {}
111 const gcd_result gcd (const pol2& u)
114 // constant polynomial u(X)
116 return gcd_result(2);
118 return gcd_result(0);
119 // u(X) = c0 + c1*X has zero -c0/c1.
120 var cl_MI_x c1inv = R->recip(u.c1);
122 return c1inv.condition;
123 var cl_MI z = -u.c0*c1inv;
124 if (cln::square(z) == a)
125 return gcd_result(1,z);
127 return gcd_result(0);
130 pol2ring (const cl_modint_ring& _R, const cl_MI& _a) : R (_R), a (_a) {}
132 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a)
134 var pol2ring PR = pol2ring(R,a);
135 var cl_I& p = R->modulus;
136 // Assuming p is a prime, then R[X]/(X^2-a) is the direct product of
137 // two rings R[X]/(X-sqrt(a)), each being isomorphic to R. Thus taking
138 // a (p-1)/2-th power in this ring will return one of (0,+1,-1) in
139 // each ring, with independent probabilities (1/p, (p-1)/2p, (p-1)/2p).
140 // For any polynomial u(X), setting v(X) := u(X)^((p-1)/2) yields
141 // gcd(u(X),X^2-a) * gcd(v(X)-1,X^2-a) * gcd(v(X)+1,X^2-a) = X^2-a.
142 // If p is not prime, all of these gcd's are likely to be 1.
143 var cl_I e = (p-1) >> 1;
145 // Choose a random polynomial u(X) in the ring.
146 var pol2 u = PR.random();
147 // Compute v(X) = u(X)^((p-1)/2).
148 var pol2 v = PR.expt_pos(u,e);
149 // Compute the three gcds.
150 var pol2ring::gcd_result g1 = PR.gcd(PR.minus(v,PR.one()));
153 if (g1.gcd_degree == 1)
154 return sqrt_mod_p_t(2,g1.solution,-g1.solution);
155 if (g1.gcd_degree == 2)
157 var pol2ring::gcd_result g2 = PR.gcd(PR.plus(v,PR.one()));
160 if (g2.gcd_degree == 1)
161 return sqrt_mod_p_t(2,g2.solution,-g2.solution);
162 if (g2.gcd_degree == 2)
164 var pol2ring::gcd_result g3 = PR.gcd(u);
167 if (g3.gcd_degree == 1)
168 return sqrt_mod_p_t(2,g3.solution,-g3.solution);
169 if (g1.gcd_degree + g2.gcd_degree + g3.gcd_degree < 2)
170 // If the sum of the degrees of the gcd is != 2,
171 // p cannot be prime.
172 return new cl_composite_condition(p);
176 // Algorithm 3 (for p > 2 only):
178 // [Cohen, A Course in Computational Algebraic Number Theory,
179 // Section 1.5.1., Algorithm 1.5.1.]
180 static const sqrt_mod_p_t tonelli_shanks_sqrt (const cl_modint_ring& R, const cl_MI& a)
183 // Write p-1 = 2^e*m, m odd. G = (Z/pZ)^* (cyclic of order p-1) has
184 // subgroups G_0 < G_1 < ... < G_e, G_j of order 2^j. (G_e is called
185 // the "2-Sylow subgroup" of G.) More precisely
186 // G_j = { x in (Z/pZ)^* : x^(2^j) = 1 },
187 // G/G_j = { x^(2^j) : x in (Z/pZ)^* }.
188 // We compute the square root of a first in G/G_e, then lift it to
189 // G/G_(e-1), etc., up to G/G_0.
190 // Start with b = a^((m+1)/2), then (a^-1*b^2)^(2^e) = 1, i.e.
192 // Lifting from G/G_j to G/G_(j-1) is easy: Assume a = b^2 in G/G_j.
193 // If a = b^2 in G/G_(j-1), then nothing needs to be done. Else
194 // a^-1*b^2 is in G_j \ G_(j-1). If j=e, a^-1*b^2 is a non-square
195 // mod p, hence a is a non-square as well, contradiction. If j<e,
196 // take h in G_(j+1) \ G_j, so that h^2 in G_j \ G_(j-1), and
197 // a^-1*b^2*h^2 is in G_(j-1). So multiply b with h.
198 var cl_I& p = R->modulus;
199 var uintL e = ord2(p-1);
200 var cl_I m = (p-1) >> e;
201 // p-1 = 2^e*m, m odd.
202 // We will have the invariant c = a^-1*b^2 in G/G_j.
204 // Initialize b = a^((m+1)/2), c = a^m, but avoid to divide by a.
205 var cl_MI c = R->expt_pos(a,(m-1)>>1);
206 var cl_MI b = R->mul(a,c);
208 // Find h in G_e \ G_(e-1): h = h'^m, where h' is any non-square.
213 // Since this computation is a bit costly, we cache its result
214 // on the ring's property list.
215 static const cl_symbol key = (cl_symbol)(cl_string)"generator of 2-Sylow subgroup of (Z/pZ)^*";
216 struct cl_sylow2gen_property : public cl_property {
217 SUBCLASS_cl_property();
221 cl_sylow2gen_property (const cl_symbol& k, const cl_MI& h) : cl_property (k), h_rep (h.rep) {}
223 var cl_sylow2gen_property* prop = (cl_sylow2gen_property*) R->get_property(key);
225 h = cl_MI(R,prop->h_rep);
227 do { h = R->random(); }
228 until (jacobi(R->retract(h),p) == -1);
229 h = R->expt_pos(h,m);
230 R->add_property(new cl_sylow2gen_property(key,h));
234 // Now c = a^-1*b^2 in G_j, h in G_j \ G_(j-1).
235 // Determine the smallest i such that c in G_i.
237 var cl_MI ci = c; // c_i = c^(2^i)
238 for ( ; i < j; i++, ci = R->square(ci))
242 // Some problem: if j=e, a non-square, if j<e, the
243 // previous iteration didn't do its job correctly.
244 // Indicates that p is not prime.
245 return new cl_composite_condition(p);
247 for (var uintL count = j-i-1; count > 0; count--)
249 // Now h in G_(i+1) \ G_i.
253 // Now c = a^-1*b^2 in G_(i-1), h in G_i \ G_(i-1).
256 if (R->square(b) != a)
258 return new cl_composite_condition(p);
259 return sqrt_mod_p_t(2,b,-b);
262 // Break-Even-Points (on a i486 with 33 MHz):
263 // Algorithm 1 fastest for p < 1500..2000
264 // Algorithm 3 generally fastest for p > 2000.
265 // But the running time of algorithm 3 is proportional to e^2.
266 // For large e, algorithm 2 becomes faster.
267 // l=50 bits: for e >= 40
268 // l=100 bits: for e >= 55
269 // l=200 bits: for e >= 80
270 // l=400 bits: for e >= 130
271 // in general something like e > l/(log(l)/(2*log(2))-1).
273 const sqrt_mod_p_t sqrt_mod_p (const cl_modint_ring& R, const cl_MI& a)
275 if (!(a.ring() == R)) cl_abort();
276 var cl_I& p = R->modulus;
277 var cl_I aa = R->retract(a);
278 switch (jacobi(aa,p)) {
279 case -1: // no solution
280 return sqrt_mod_p_t(0);
281 case 0: // gcd(aa,p) > 1
284 return sqrt_mod_p_t(1,a);
287 return new cl_composite_condition(p,gcd(aa,p));
288 case 1: // two solutions
293 var cl_I x1 = search_sqrt(cl_I_to_UL(p),cl_I_to_UL(aa));
295 if (x1==x2) // can only happen when p = 2
296 return sqrt_mod_p_t(1,R->canonhom(x1));
298 return sqrt_mod_p_t(2,R->canonhom(x1),R->canonhom(x2));
300 var uintL l = integer_length(p);
301 var uintL e = ord2(p-1);
302 //if (e > 30 && e > l/(log((double)l)*0.72-1))
303 if (e > 30 && e > l/(::log((double)l)*0.92-2.41))
305 return cantor_zassenhaus_sqrt(R,a);
308 return tonelli_shanks_sqrt(R,a);