4 #include "base/cl_sysdep.h"
7 #include "cln/numtheory.h"
12 #include "integer/cl_I.h"
13 #include "cln/exception.h"
17 #define floor cln_floor
19 // MacOS X does "#define _R 0x00040000L". Grr...
24 // Algorithm 1 (for very small p only):
25 // Try different values.
26 // Assume p is prime and a nonzero square in Z/pZ.
27 static uint32 search_sqrt (uint32 p, uint32 a)
32 // 0 < x <= p/2, x2 = x^2 mod p.
35 x2 += x; x++; x2 += x;
41 // Algorithm 2 (for p > 2 only):
43 // [Beth et al.: Computer Algebra, 1988, Kapitel 5.3.3.]
44 // [Cohen, A Course in Computational Algebraic Number Theory,
45 // Section 3.4.4., Algorithm 3.4.6.]
46 // Input: R = Z/pZ with p>2, and a (nonzero square in R).
47 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a);
48 // Compute in the polynomial ring R[X]/(X^2-a).
50 // A polynomial c0+c1*X mod (X^2-a)
54 pol2 (const cl_MI& _c0, const cl_MI& _c1) : c0 (_c0), c1 (_c1) {}
57 const cl_modint_ring& R;
61 return pol2(R->zero(),R->zero());
65 return pol2(R->one(),R->zero());
67 const pol2 plus (const pol2& u, const pol2& v)
69 return pol2(u.c0+v.c0, u.c1+v.c1);
71 const pol2 minus (const pol2& u, const pol2& v)
73 return pol2(u.c0-v.c0, u.c1-v.c1);
75 const pol2 mul (const pol2& u, const pol2& v)
77 return pol2(u.c0*v.c0+u.c1*v.c1*a, u.c0*v.c1+u.c1*v.c0);
79 const pol2 square (const pol2& u)
81 return pol2(cln::square(u.c0) + cln::square(u.c1)*a, (u.c0*u.c1)<<1);
83 const pol2 expt_pos (const pol2& x, const cl_I& y)
85 // Right-Left Binary, [Cohen, Algorithm 1.2.1.]
88 while (!oddp(b)) { a = square(a); b = b = b >> 1; } // a^b = x^y
101 return pol2(R->random(),R->random());
103 // Computes the degree of gcd(u(X),X^2-a) and, if it is 1,
104 // also the zero if this polynomial of degree 1.
106 cl_composite_condition* condition;
110 gcd_result (cl_composite_condition* c) : condition (c) {}
111 gcd_result (int deg) : condition (NULL), gcd_degree (deg) {}
112 gcd_result (int deg, const cl_MI& sol) : condition (NULL), gcd_degree (deg), solution (sol) {}
114 const gcd_result gcd (const pol2& u)
117 // constant polynomial u(X)
119 return gcd_result(2);
121 return gcd_result(0);
123 // u(X) = c0 + c1*X has zero -c0/c1.
124 var cl_MI_x c1inv = R->recip(u.c1);
126 return c1inv.condition;
127 var cl_MI z = -u.c0*c1inv;
128 if (cln::square(z) == a)
129 return gcd_result(1,z);
131 return gcd_result(0);
134 pol2ring (const cl_modint_ring& _R, const cl_MI& _a) : R (_R), a (_a) {}
136 static const sqrt_mod_p_t cantor_zassenhaus_sqrt (const cl_modint_ring& R, const cl_MI& a)
138 var pol2ring PR = pol2ring(R,a);
139 var cl_I& p = R->modulus;
140 // Assuming p is a prime, then R[X]/(X^2-a) is the direct product of
141 // two rings R[X]/(X-sqrt(a)), each being isomorphic to R. Thus taking
142 // a (p-1)/2-th power in this ring will return one of (0,+1,-1) in
143 // each ring, with independent probabilities (1/p, (p-1)/2p, (p-1)/2p).
144 // For any polynomial u(X), setting v(X) := u(X)^((p-1)/2) yields
145 // gcd(u(X),X^2-a) * gcd(v(X)-1,X^2-a) * gcd(v(X)+1,X^2-a) = X^2-a.
146 // If p is not prime, all of these gcd's are likely to be 1.
147 var cl_I e = (p-1) >> 1;
149 // Choose a random polynomial u(X) in the ring.
150 var pol2 u = PR.random();
151 // Compute v(X) = u(X)^((p-1)/2).
152 var pol2 v = PR.expt_pos(u,e);
153 // Compute the three gcds.
154 var pol2ring::gcd_result g1 = PR.gcd(PR.minus(v,PR.one()));
157 if (g1.gcd_degree == 1)
158 return sqrt_mod_p_t(2,g1.solution,-g1.solution);
159 if (g1.gcd_degree == 2)
161 var pol2ring::gcd_result g2 = PR.gcd(PR.plus(v,PR.one()));
164 if (g2.gcd_degree == 1)
165 return sqrt_mod_p_t(2,g2.solution,-g2.solution);
166 if (g2.gcd_degree == 2)
168 var pol2ring::gcd_result g3 = PR.gcd(u);
171 if (g3.gcd_degree == 1)
172 return sqrt_mod_p_t(2,g3.solution,-g3.solution);
173 if (g1.gcd_degree + g2.gcd_degree + g3.gcd_degree < 2)
174 // If the sum of the degrees of the gcd is != 2,
175 // p cannot be prime.
176 return new cl_composite_condition(p);
180 // Algorithm 3 (for p > 2 only):
182 // [Cohen, A Course in Computational Algebraic Number Theory,
183 // Section 1.5.1., Algorithm 1.5.1.]
184 static const sqrt_mod_p_t tonelli_shanks_sqrt (const cl_modint_ring& R, const cl_MI& a)
187 // Write p-1 = 2^e*m, m odd. G = (Z/pZ)^* (cyclic of order p-1) has
188 // subgroups G_0 < G_1 < ... < G_e, G_j of order 2^j. (G_e is called
189 // the "2-Sylow subgroup" of G.) More precisely
190 // G_j = { x in (Z/pZ)^* : x^(2^j) = 1 },
191 // G/G_j = { x^(2^j) : x in (Z/pZ)^* }.
192 // We compute the square root of a first in G/G_e, then lift it to
193 // G/G_(e-1), etc., up to G/G_0.
194 // Start with b = a^((m+1)/2), then (a^-1*b^2)^(2^e) = 1, i.e.
196 // Lifting from G/G_j to G/G_(j-1) is easy: Assume a = b^2 in G/G_j.
197 // If a = b^2 in G/G_(j-1), then nothing needs to be done. Else
198 // a^-1*b^2 is in G_j \ G_(j-1). If j=e, a^-1*b^2 is a non-square
199 // mod p, hence a is a non-square as well, contradiction. If j<e,
200 // take h in G_(j+1) \ G_j, so that h^2 in G_j \ G_(j-1), and
201 // a^-1*b^2*h^2 is in G_(j-1). So multiply b with h.
202 var cl_I& p = R->modulus;
203 var uintC e = ord2(p-1);
204 var cl_I m = (p-1) >> e;
205 // p-1 = 2^e*m, m odd.
206 // We will have the invariant c = a^-1*b^2 in G/G_j.
208 // Initialize b = a^((m+1)/2), c = a^m, but avoid to divide by a.
209 var cl_MI c = R->expt_pos(a,(m-1)>>1);
210 var cl_MI b = R->mul(a,c);
212 // Find h in G_e \ G_(e-1): h = h'^m, where h' is any non-square.
217 // Since this computation is a bit costly, we cache its result
218 // on the ring's property list.
219 static const cl_symbol key = (cl_symbol)(cl_string)"generator of 2-Sylow subgroup of (Z/pZ)^*";
220 struct cl_sylow2gen_property : public cl_property {
221 SUBCLASS_cl_property();
225 cl_sylow2gen_property (const cl_symbol& k, const cl_MI& h) : cl_property (k), h_rep (h.rep) {}
227 var cl_sylow2gen_property* prop = (cl_sylow2gen_property*) R->get_property(key);
229 h = cl_MI(R,prop->h_rep);
231 do { h = R->random(); }
232 until (jacobi(R->retract(h),p) == -1);
233 h = R->expt_pos(h,m);
234 R->add_property(new cl_sylow2gen_property(key,h));
238 // Now c = a^-1*b^2 in G_j, h in G_j \ G_(j-1).
239 // Determine the smallest i such that c in G_i.
241 var cl_MI ci = c; // c_i = c^(2^i)
242 for ( ; i < j; i++, ci = R->square(ci))
246 // Some problem: if j=e, a non-square, if j<e, the
247 // previous iteration didn't do its job correctly.
248 // Indicates that p is not prime.
249 return new cl_composite_condition(p);
251 for (var uintC count = j-i-1; count > 0; count--)
253 // Now h in G_(i+1) \ G_i.
257 // Now c = a^-1*b^2 in G_(i-1), h in G_i \ G_(i-1).
260 if (R->square(b) != a)
262 return new cl_composite_condition(p);
263 return sqrt_mod_p_t(2,b,-b);
266 // Break-Even-Points (on a i486 with 33 MHz):
267 // Algorithm 1 fastest for p < 1500..2000
268 // Algorithm 3 generally fastest for p > 2000.
269 // But the running time of algorithm 3 is proportional to e^2.
270 // For large e, algorithm 2 becomes faster.
271 // l=50 bits: for e >= 40
272 // l=100 bits: for e >= 55
273 // l=200 bits: for e >= 80
274 // l=400 bits: for e >= 130
275 // in general something like e > l/(log(l)/(2*log(2))-1).
277 const sqrt_mod_p_t sqrt_mod_p (const cl_modint_ring& R, const cl_MI& a)
279 if (!(a.ring() == R)) throw runtime_exception();
280 var cl_I& p = R->modulus;
281 var cl_I aa = R->retract(a);
282 switch (jacobi(aa,p)) {
283 case -1: // no solution
284 return sqrt_mod_p_t(0);
285 case 0: // gcd(aa,p) > 1
288 return sqrt_mod_p_t(1,a);
291 return new cl_composite_condition(p,gcd(aa,p));
292 case 1: // two solutions
297 var cl_I x1 = search_sqrt(cl_I_to_UL(p),cl_I_to_UL(aa));
299 if (x1==x2) // can only happen when p = 2
300 return sqrt_mod_p_t(1,R->canonhom(x1));
302 return sqrt_mod_p_t(2,R->canonhom(x1),R->canonhom(x2));
304 var uintC l = integer_length(p);
305 var uintC e = ord2(p-1);
306 //if (e > 30 && e > l/(::log((double)l)*0.72-1))
307 if (e > 30 && e > l/(::log((double)l)*0.92-2.41))
309 return cantor_zassenhaus_sqrt(R,a);
312 return tonelli_shanks_sqrt(R,a);