7 #include "cl_numtheory.h"
17 #define floor cln_floor
19 // Algorithm 1 (for very small p only):
20 // Try different values.
21 // Assume p is prime and a nonzero square in Z/pZ.
22 static uint32 search_sqrt (uint32 p, uint32 a)
27 // 0 < x <= p/2, x2 = x^2 mod p.
30 x2 += x; x++; x2 += x;
36 // Algorithm 2 (for p > 2 only):
38 // [Beth et al.: Computer Algebra, 1988, Kapitel 5.3.3.]
39 // [Cohen, A Course in Computational Algebraic Number Theory,
40 // Section 3.4.4., Algorithm 3.4.6.]
41 // Input: R = Z/pZ with p>2, and a (nonzero square in R).
42 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a);
43 // Compute in the polynomial ring R[X]/(X^2-a).
45 // A polynomial c0+c1*X mod (X^2-a)
49 pol2 (const cl_MI& _c0, const cl_MI& _c1) : c0 (_c0), c1 (_c1) {}
52 const cl_modint_ring& R;
56 return pol2(R->zero(),R->zero());
60 return pol2(R->one(),R->zero());
62 const pol2 plus (const pol2& u, const pol2& v)
64 return pol2(u.c0+v.c0, u.c1+v.c1);
66 const pol2 minus (const pol2& u, const pol2& v)
68 return pol2(u.c0-v.c0, u.c1-v.c1);
70 const pol2 mul (const pol2& u, const pol2& v)
72 return pol2(u.c0*v.c0+u.c1*v.c1*a, u.c0*v.c1+u.c1*v.c0);
74 const pol2 square (const pol2& u)
76 return pol2(::square(u.c0) + ::square(u.c1)*a, (u.c0*u.c1)<<1);
78 const pol2 expt_pos (const pol2& x, const cl_I& y)
80 // Right-Left Binary, [Cohen, Algorithm 1.2.1.]
83 while (!oddp(b)) { a = square(a); b = b = b >> 1; } // a^b = x^y
96 return pol2(R->random(),R->random());
98 // Computes the degree of gcd(u(X),X^2-a) and, if it is 1,
99 // also the zero if this polynomial of degree 1.
101 cl_composite_condition* condition;
105 gcd_result (cl_composite_condition* c) : condition (c) {}
106 gcd_result (int deg) : condition (NULL), gcd_degree (deg) {}
107 gcd_result (int deg, const cl_MI& sol) : condition (NULL), gcd_degree (deg), solution (sol) {}
109 const gcd_result gcd (const pol2& u)
112 // constant polynomial u(X)
114 return gcd_result(2);
116 return gcd_result(0);
117 // u(X) = c0 + c1*X has zero -c0/c1.
118 var cl_MI_x c1inv = R->recip(u.c1);
120 return c1inv.condition;
121 var cl_MI z = -u.c0*c1inv;
122 if (::square(z) == a)
123 return gcd_result(1,z);
125 return gcd_result(0);
128 pol2ring (const cl_modint_ring& _R, const cl_MI& _a) : R (_R), a (_a) {}
130 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a)
132 var pol2ring PR = pol2ring(R,a);
133 var cl_I& p = R->modulus;
134 // Assuming p is a prime, then R[X]/(X^2-a) is the direct product of
135 // two rings R[X]/(X-sqrt(a)), each being isomorphic to R. Thus taking
136 // a (p-1)/2-th power in this ring will return one of (0,+1,-1) in
137 // each ring, with independent probabilities (1/p, (p-1)/2p, (p-1)/2p).
138 // For any polynomial u(X), setting v(X) := u(X)^((p-1)/2) yields
139 // gcd(u(X),X^2-a) * gcd(v(X)-1,X^2-a) * gcd(v(X)+1,X^2-a) = X^2-a.
140 // If p is not prime, all of these gcd's are likely to be 1.
141 var cl_I e = (p-1) >> 1;
143 // Choose a random polynomial u(X) in the ring.
144 var pol2 u = PR.random();
145 // Compute v(X) = u(X)^((p-1)/2).
146 var pol2 v = PR.expt_pos(u,e);
147 // Compute the three gcds.
148 var pol2ring::gcd_result g1 = PR.gcd(PR.minus(v,PR.one()));
151 if (g1.gcd_degree == 1)
152 return sqrt_mod_p_t(2,g1.solution,-g1.solution);
153 if (g1.gcd_degree == 2)
155 var pol2ring::gcd_result g2 = PR.gcd(PR.plus(v,PR.one()));
158 if (g2.gcd_degree == 1)
159 return sqrt_mod_p_t(2,g2.solution,-g2.solution);
160 if (g2.gcd_degree == 2)
162 var pol2ring::gcd_result g3 = PR.gcd(u);
165 if (g3.gcd_degree == 1)
166 return sqrt_mod_p_t(2,g3.solution,-g3.solution);
167 if (g1.gcd_degree + g2.gcd_degree + g3.gcd_degree < 2)
168 // If the sum of the degrees of the gcd is != 2,
169 // p cannot be prime.
170 return new cl_composite_condition(p);
174 // Algorithm 3 (for p > 2 only):
176 // [Cohen, A Course in Computational Algebraic Number Theory,
177 // Section 1.5.1., Algorithm 1.5.1.]
178 static const sqrt_mod_p_t tonelli_shanks_sqrt (const cl_modint_ring& R, const cl_MI& a)
181 // Write p-1 = 2^e*m, m odd. G = (Z/pZ)^* (cyclic of order p-1) has
182 // subgroups G_0 < G_1 < ... < G_e, G_j of order 2^j. (G_e is called
183 // the "2-Sylow subgroup" of G.) More precisely
184 // G_j = { x in (Z/pZ)^* : x^(2^j) = 1 },
185 // G/G_j = { x^(2^j) : x in (Z/pZ)^* }.
186 // We compute the square root of a first in G/G_e, then lift it to
187 // G/G_(e-1), etc., up to G/G_0.
188 // Start with b = a^((m+1)/2), then (a^-1*b^2)^(2^e) = 1, i.e.
190 // Lifting from G/G_j to G/G_(j-1) is easy: Assume a = b^2 in G/G_j.
191 // If a = b^2 in G/G_(j-1), then nothing needs to be done. Else
192 // a^-1*b^2 is in G_j \ G_(j-1). If j=e, a^-1*b^2 is a non-square
193 // mod p, hence a is a non-square as well, contradiction. If j<e,
194 // take h in G_(j+1) \ G_j, so that h^2 in G_j \ G_(j-1), and
195 // a^-1*b^2*h^2 is in G_(j-1). So multiply b with h.
196 var cl_I& p = R->modulus;
197 var uintL e = ord2(p-1);
198 var cl_I m = (p-1) >> e;
199 // p-1 = 2^e*m, m odd.
200 // We will have the invariant c = a^-1*b^2 in G/G_j.
202 // Initialize b = a^((m+1)/2), c = a^m, but avoid to divide by a.
203 var cl_MI c = R->expt_pos(a,(m-1)>>1);
204 var cl_MI b = R->mul(a,c);
206 // Find h in G_e \ G_(e-1): h = h'^m, where h' is any non-square.
211 // Since this computation is a bit costly, we cache its result
212 // on the ring's property list.
213 static const cl_symbol key = (cl_symbol)(cl_string)"generator of 2-Sylow subgroup of (Z/pZ)^*";
214 struct cl_sylow2gen_property : public cl_property {
215 SUBCLASS_cl_property();
219 cl_sylow2gen_property (const cl_symbol& k, const cl_MI& h) : cl_property (k), h_rep (h.rep) {}
221 var cl_sylow2gen_property* prop = (cl_sylow2gen_property*) R->get_property(key);
223 h = cl_MI(R,prop->h_rep);
225 do { h = R->random(); }
226 until (jacobi(R->retract(h),p) == -1);
227 h = R->expt_pos(h,m);
228 R->add_property(new cl_sylow2gen_property(key,h));
232 // Now c = a^-1*b^2 in G_j, h in G_j \ G_(j-1).
233 // Determine the smallest i such that c in G_i.
235 var cl_MI ci = c; // c_i = c^(2^i)
236 for ( ; i < j; i++, ci = R->square(ci))
240 // Some problem: if j=e, a non-square, if j<e, the
241 // previous iteration didn't do its job correctly.
242 // Indicates that p is not prime.
243 return new cl_composite_condition(p);
245 for (var uintL count = j-i-1; count > 0; count--)
247 // Now h in G_(i+1) \ G_i.
251 // Now c = a^-1*b^2 in G_(i-1), h in G_i \ G_(i-1).
254 if (R->square(b) != a)
256 return new cl_composite_condition(p);
257 return sqrt_mod_p_t(2,b,-b);
260 // Break-Even-Points (on a i486 with 33 MHz):
261 // Algorithm 1 fastest for p < 1500..2000
262 // Algorithm 3 generally fastest for p > 2000.
263 // But the running time of algorithm 3 is proportional to e^2.
264 // For large e, algorithm 2 becomes faster.
265 // l=50 bits: for e >= 40
266 // l=100 bits: for e >= 55
267 // l=200 bits: for e >= 80
268 // l=400 bits: for e >= 130
269 // in general something like e > l/(log(l)/(2*log(2))-1).
271 const sqrt_mod_p_t sqrt_mod_p (const cl_modint_ring& R, const cl_MI& a)
273 if (!(a.ring() == R)) cl_abort();
274 var cl_I& p = R->modulus;
275 var cl_I aa = R->retract(a);
276 switch (jacobi(aa,p)) {
277 case -1: // no solution
278 return sqrt_mod_p_t(0);
279 case 0: // gcd(aa,p) > 1
282 return sqrt_mod_p_t(1,a);
285 return new cl_composite_condition(p,gcd(aa,p));
286 case 1: // two solutions
291 var cl_I x1 = search_sqrt(cl_I_to_UL(p),cl_I_to_UL(aa));
293 if (x1==x2) // can only happen when p = 2
294 return sqrt_mod_p_t(1,R->canonhom(x1));
296 return sqrt_mod_p_t(2,R->canonhom(x1),R->canonhom(x2));
298 var uintL l = integer_length(p);
299 var uintL e = ord2(p-1);
300 //if (e > 30 && e > l/(log((double)l)*0.72-1))
301 if (e > 30 && e > l/(log((double)l)*0.92-2.41))
303 return cantor_zassenhaus_sqrt(R,a);
306 return tonelli_shanks_sqrt(R,a);