cost_pairing.py 21.5 KB
 MASSON Simon committed Mar 29, 2019 1 2 3 4 from sage.all_cmdline import * from CocksPinchVariant import * import sage.rings.integer from BLS12 import * Emmanuel Thomé committed Apr 08, 2019 5 # from BLS24 import * MASSON Simon committed Mar 29, 2019 6 from KSS16 import * Emmanuel Thomé committed Apr 08, 2019 7 # from KSS18 import * MASSON Simon committed Mar 29, 2019 8 9 10 from BN import * from MNT6 import * from final_expo_k57 import * GUILLEVIC Aurore committed May 27, 2019 11 from enumerate_sparse_T import bit_positions_2naf, bit_positions MASSON Simon committed Mar 29, 2019 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 mystery_201903151748_simon_a_raison=False # TODO: take into account h_t not always being 0 in the k=6 or k=8 # cases... [WIP for k=8 -- want to automate a bit] Qmsi = QQ['m,s,inv'] m,s,inv = Qmsi.gens() #F_{p^k} arithmetic cost def cost_m(k) : # return the cost of a multiplication over F_{p^k} if k%2 == 0 : return 3*cost_m(k//2) elif k%3 == 0 : return 6*cost_m(k//3) elif k == 1: return m elif k == 5 : return 13*m elif k == 7 : return 22*m else : return 'not done for this embedding degree' # special cases def densexsparse_m6(k): assert k%6==0 return 13*cost_m(k//6) # aurore thesis def densexsparse_m8(k): assert k%8==0 return 8*cost_m(k//4) # ??????????????????????BarDuq18??? sparse_m12 = 13*cost_m(2) # BarDuq 39 = 3*13 and 3 is the cost of Mp2 = 3 Mp sparse_m16 = 8*cost_m(4) def cost_s(k) : # return the cost of a square over F_{p^k} if k%2 == 0 : return 2*cost_m(k//2) elif k%3 == 0 : return 2*cost_m(k//3) + 3 * cost_s(k//3) elif k == 1: return m elif k == 5 : return 13*m elif k == 7 : return 22*m else : return 'not done for this embedding degree' # special cases cyclo_s6 = 3*cost_s(2) # eprint 2009/565 cyclo_s8 = 2*cost_s(4) # eprint 2009/565 section 3.1 compr_s12 = 12*m # BarDuq cyclo_s12 = 18*m cyclo_s16 = 2*cost_s(8) Emmanuel Thomé committed Sep 11, 2019 70 71 72 def cost_prenorm(k,ell): # returns the cost of computing # prenorm_{j,\ell}(a)=a^(1+p^j+...+p^{j*(ell-1)}) Emmanuel Thomé committed Sep 11, 2019 73 74 # in F_p^k. This is used only for k odd. # (we have norm(a) = a * prenorm_{1,k-1}(a^p) ) Emmanuel Thomé committed Sep 11, 2019 75 76 77 78 79 80 # Note that the cost is independent of j. if ell == 1: return 0 elif is_prime(ell): return cost_m(k) + cost_f(k) + cost_prenorm(k, ell-1) else: Emmanuel Thomé committed Sep 11, 2019 81 82 83 84 85 # It's not necessarily the best strategy. E.g. for k=67, we're # led to compute prenorm for ell=33 (ell=66 at first, but let's # focus on the sub call). That is in fact best done by going to # 32 first (cost F+M, then 5F+5M). In contrast, the factoring # approach below costs 7F+7M... Emmanuel Thomé committed Sep 11, 2019 86 87 88 89 90 mu = factor(ell)[0][0] return cost_prenorm(k, mu) + cost_prenorm(k, ell // mu) # prenorm_{j,u*v}(a) = prenorm_{j*v,u}(prenorm_{j,v}(a)) # and raising to the power p^{j*v} costs exacly one frobenius. MASSON Simon committed Mar 29, 2019 91 92 93 94 95 96 97 98 def cost_i(k) : # return the cost of an inversion over F_{p^k} if k%2 == 0 : return 2*cost_s(k//2) + 2*cost_m(k//2) + cost_i(k//2) elif k%3 == 0 : return 3*cost_s(k//3) + 9*cost_m(k//3) + cost_i(k//3) elif k == 1: return inv Emmanuel Thomé committed Sep 11, 2019 99 100 101 102 103 104 105 106 107 108 elif k == 5 : # u1 = frob(a) # u3 = frob(frob(u1)) # costs only 1 frob # v = u1 * u3 # v = a^(p+p^3) # w = frob(v) # v = a^(p^2+p^4) # b = v * w # n = coeff(a,0)*coeff(b,0) + alpha*sum([coeff(a,i)*coeff(b,k-i) for i in range(1,k)]) # ni = inv(n) # ai = ni * a return 3*cost_f(k) + 2*cost_m(k) + inv + 2*k*m MASSON Simon committed Mar 29, 2019 109 110 # elif k == 7 : # # u1 = frob(a) Emmanuel Thomé committed Sep 11, 2019 111 # # u4 = frob(frob(frob(u1))) MASSON Simon committed Mar 29, 2019 112 113 114 115 116 117 118 # # v = u1 * u4 # v = a^(p+p^4) # # w = frob(v) # v = a^(p^2+p^5) # # z = frob(w) # v = a^(p^3+p^6) # # b = v * w * z # # n = coeff(a,0)*coeff(b,0) + alpha*sum([coeff(a,i)*coeff(b,k-i) for i in range(1,k)]) # # ni = inv(n) # # ai = ni * a Emmanuel Thomé committed Sep 11, 2019 119 return 4*cost_f(k) + 3*cost_m(k) + inv + 2*k*m MASSON Simon committed Mar 29, 2019 120 121 122 123 elif k%2 == 1: # generalization of the above. # Note that we can go further. If (k-1)/2 >= 4, then we may apply # the same trick to save some more multiplications. Emmanuel Thomé committed Sep 11, 2019 124 return cost_f(k) + cost_prenorm(k, k-1) + inv + 2*k*m MASSON Simon committed Mar 29, 2019 125 126 127 128 129 130 131 132 # elif k == 5 or k == 7 : # return (k-1)*cost_f(k) + (k-2)*cost_m(k) + inv + 2*k*m else : return 'not done for this embedding degree' def cost_f(k, d=1) : # return the cost of a d-Frobenius over F_{p^k} assert k % d == 0 MASSON Simon committed May 27, 2019 133 if (k//d) % 2 == 0 : MASSON Simon committed Mar 29, 2019 134 135 136 137 138 139 140 141 142 143 # for F_{p^{k/d}} a tower defined by binomials, the multipliers in # the Frobenius (p^d-th power) expressions are all powers of a # k/d-th root of unity. If k/d is even, one of them is -1. At any # rate, this root of unity boils down to a scalar, therefore we # don't need cost_m(d) but really d * cost_m(1) return (k//d-2) * d * cost_m(1) else: return (k//d-1) * d * cost_m(1) def cost_i_and_f(k) : MASSON Simon committed May 27, 2019 144 if k % 2 == 0 or k % 3 == 0: MASSON Simon committed Mar 29, 2019 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 return cost_i(k) + cost_f(k) elif k == 5 or k == 7: # Then we know that the inversion computes the Frobenius anyway. return cost_i(k) else : return 'not done for this embedding degree' def table_costFpk(k_list): K = ''.join(["&%d"%k for k in k_list]) M = ''.join(["&%s"%cost_m(k)(m=1)+r"\bfm" for k in k_list]) S = ''.join(["&%s"%cost_s(k)(m=1)+r"\bfm" for k in k_list]) F = ''.join(["&%s"%cost_f(k)(m=1)+r"\bfm" for k in k_list]) sc_dict = {6:cyclo_s6, 8:cyclo_s8, 12:cyclo_s12, 16:cyclo_s16} SC = ''.join(["&%s"%sc_dict[k](m=1)+r"\bfm" if k in sc_dict else "&" for k in k_list]) I0 = ''.join(["&%s"%cost_i(k)(m=1,inv=0)+r"\bfm" for k in k_list]) I1 = ''.join(["&%s"%cost_i(k)(m=1,inv=25)+r"\bfm" for k in k_list]) Emmanuel Thomé committed Sep 11, 2019 161 F = F.replace(r"&1\bfm",r"&\bfm") MASSON Simon committed Mar 29, 2019 162 163 164 M = M.replace(r"&1\bfm",r"&\bfm") S = S.replace(r"&1\bfm",r"&\bfm") I0 = I0.replace(r"&0\bfm",r"&0") Emmanuel Thomé committed Sep 11, 2019 165 F = F.replace(r"&0\bfm",r"&0") MASSON Simon committed Mar 29, 2019 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 contents = [ r"$$\begin{array}{|c|" + "c|" * len(k_list) + "}", r"\hline", "k" + K + r"\\", r"\hline", r"\bfm_k" + M + r"\\", r"\bfs_k" + S + r"\\", r"\bff_k" + F + r"\\", r"\bfs_k^{\text{cyclo}}" + SC + r"\\", r"\bfi_k-\bfi_1" + I0 + r"\\", r"\text{\bfi_k, with \bfi_1=25\bfm}" + I1 + r"\\", r"\hline", r"\end{array}$$", ] print "% This table is generated by:" Emmanuel Thomé committed Apr 08, 2019 181 print "%% PYTHONPATH=cocks-pinch-variant/ sage -c 'load(\"cocks-pinch-variant/cost_pairing.py\"); table_costFpk(%s)'" % (k_list) MASSON Simon committed Mar 29, 2019 182 183 184 185 186 187 188 189 for s in contents: print s def Hw(x) : return len(bit_positions_2naf(x)) proof.arithmetic(False) 190 C5=CocksPinchVariantResult(5,10000000147,0xe000000000008000,1,ht=3,hy=0x11e36418c7c8b454,max_B1=600) MASSON Simon committed Mar 29, 2019 191 192 C6=CocksPinchVariantResult(6,3,0xefffffffffffffe00000000000000000,1,ht=-1,hy=0xffbbffffffffffffc020,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600) C7=CocksPinchVariantResult(7,20,0x5fffb820248,6,ht=-2,allowed_cofactor=1232,allowed_size_cofactor=10,max_B1=600) MASSON Simon committed Sep 05, 2019 193 #C8=CocksPinchVariantResult(8,4,0xffffffffeff7c200,5,ht=5,hy=-0xd700,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600) 194 C8=CocksPinchVariantResult(8,4,0xffc00020fffffffc,1,ht=1,hy=0xdc04,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600) MASSON Simon committed Mar 29, 2019 195 196 GUILLEVIC Aurore committed May 27, 2019 197 CMNT6=MNT6(u=873723667900031396506414143162332159382674816702805606206979732381600254701804231398281169537138620,a=209307816050232262803672282154940341360062431838092388077917610639183322072827259682607127795420474686833003315766797546568469776750651773087882545447646552119008299040167030969895802846139484415144,b=2319663192174958547181026340141410918530227127674793888869119262391240421488942353013995765010333162065568990954578077256489549792305772041454141172011940607053889955897003759289947924385489341215143,D=8317003,c=1) GUILLEVIC Aurore committed May 27, 2019 198 199 CBN446=BN(u=2**110+2**36+1,b=2**8+1) CBLS446=BLS12(u=-(2**74+2**73+2**63+2**57+2**50+2**17+1),b=1) MASSON Simon committed Mar 29, 2019 200 201 202 203 204 205 206 207 208 CBN12=BN(eval(preparse("2^114+2^101-2^14-1"))) CBLS12=BLS12(eval(preparse("-2^77+2^50+2^33"))) CKSS16=KSS16(eval(preparse("2^35-2^32-2^18+2^8+1"))) C1=Integer(3072) def finite_field_cost(logp): #time_m words = ceil(RR(logp)/64) if words == 5 : MASSON Simon committed Sep 06, 2019 209 time_m = 35 # relic benchmark MASSON Simon committed Mar 29, 2019 210 if words == 6 : MASSON Simon committed Sep 06, 2019 211 time_m = 65 # relic benchmark GUILLEVIC Aurore committed May 27, 2019 212 if words == 7 : MASSON Simon committed Sep 06, 2019 213 time_m = 85 # relic benchmark MASSON Simon committed Mar 29, 2019 214 if words == 8 : MASSON Simon committed Sep 06, 2019 215 time_m = 106 # relic benchmark MASSON Simon committed Mar 29, 2019 216 elif words == 9 : MASSON Simon committed Sep 06, 2019 217 time_m = 129 # relic benchmark MASSON Simon committed Mar 29, 2019 218 elif words == 10 : MASSON Simon committed Sep 06, 2019 219 time_m = 154 # relic benchmark MASSON Simon committed Mar 29, 2019 220 elif words == 11 : MASSON Simon committed Sep 06, 2019 221 time_m = 1.5*11**2 MASSON Simon committed Mar 29, 2019 222 elif words == 48 : MASSON Simon committed Sep 06, 2019 223 time_m = 4882 # gmp benchmark MASSON Simon committed Mar 29, 2019 224 225 226 227 228 229 230 231 return time_m def is_one_of_our_known_pairing_friendly_curves(C): return isinstance(C, BN) or \ isinstance(C, BLS12) or \ isinstance(C, KSS16) or \ isinstance(C, MNT6) or \ False; Emmanuel Thomé committed Apr 08, 2019 232 233 # isinstance(C, BLS24) or \ # isinstance(C, KSS18) or \ MASSON Simon committed Mar 29, 2019 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 def polymorphic_get_logp(C): if is_one_of_our_known_pairing_friendly_curves(C): return C.p().nbits() elif isinstance(C, CocksPinchVariantResult): return ZZ(C.p).nbits() elif isinstance(C, Integer): return C else: raise ValueError("not implemented") def polymorphic_get_logr(C): if is_one_of_our_known_pairing_friendly_curves(C): return C.r().nbits() elif isinstance(C, CocksPinchVariantResult): return ZZ(C.r).nbits() elif isinstance(C, Integer): return 256 else: raise ValueError("not implemented") def polymorphic_get_name(C): if isinstance(C, CocksPinchVariantResult): return "$k=%s$" % C.k elif isinstance(C, BN): return 'BN' elif isinstance(C, MNT6): return 'MNT6' elif isinstance(C, BLS12): return 'BLS12' Emmanuel Thomé committed Apr 08, 2019 264 265 # elif isinstance(C, BLS24): # return 'BLS24' MASSON Simon committed Mar 29, 2019 266 267 elif isinstance(C, KSS16): return 'KSS16' Emmanuel Thomé committed Apr 08, 2019 268 269 # elif isinstance(C, KSS18): # return 'KSS18' MASSON Simon committed Mar 29, 2019 270 271 272 273 274 275 276 277 278 279 elif isinstance(C, Integer): return '$k=1$' else: raise ValueError("not implemented") def polymorphic_get_miller_loop_length(C): if isinstance(C, CocksPinchVariantResult): return C.T elif isinstance(C, BN): return 6*C.u()+2 Emmanuel Thomé committed Apr 08, 2019 280 elif isinstance(C, BLS12): # or isinstance(C, BLS24): GUILLEVIC Aurore committed May 27, 2019 281 return C.u() Emmanuel Thomé committed Apr 08, 2019 282 elif isinstance(C, KSS16): # or isinstance(C, KSS18): MASSON Simon committed Mar 29, 2019 283 284 return C.u() elif isinstance(C, MNT6): GUILLEVIC Aurore committed May 27, 2019 285 return C.tr() - 1 MASSON Simon committed Mar 29, 2019 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 elif isinstance(C, Integer): raise ValueError("not implemented") else: raise ValueError("not implemented") def polymorphic_get_embedding_degree(C): if isinstance(C, CocksPinchVariantResult): return C.k elif is_one_of_our_known_pairing_friendly_curves(C): return C.k() elif isinstance(C, Integer): return 1 else: raise ValueError("not implemented") def polymorphic_get_fD(C): if isinstance(C, CocksPinchVariantResult): return C.fD elif is_one_of_our_known_pairing_friendly_curves(C): GUILLEVIC Aurore committed May 27, 2019 305 return C.D() MASSON Simon committed Mar 29, 2019 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 elif isinstance(C, Integer): return 1 else: raise ValueError("not implemented") def millerLoopCost(C): k = polymorphic_get_embedding_degree(C) D = polymorphic_get_fD(C) name = polymorphic_get_name(C) # k, D, name, logp, logT, HwT) : #extra computations for BN, BLS and KSS (see BarDuq Table2) miller_fixup = 0 if k == 5 or k == 7 : cost_addline = 10*cost_m(k) + 3*cost_s(k) cost_doubleline = 6*cost_m(k) + 4*cost_s(k) + 2*k*cost_m(1) cost_verticalline = k*cost_m(1) cost_update1 = 4*cost_m(k) + 2*cost_s(k) cost_update2 = 4*cost_m(k) elif k == 6 and D == 3 : cost_addline = 10*cost_m(k//6) + 2*cost_s(k//6) + (k//3)*cost_m(1) cost_doubleline = 2*cost_m(k//6) + 7*cost_s(k//6) + (k//3)*cost_m(1) cost_verticalline = 0 cost_update1 = cost_s(k)+densexsparse_m6(k) cost_update2 = densexsparse_m6(k) elif k%4 == 0 and D == 4 : cost_addline = 9*cost_m(k//4) + 5*cost_s(k//4) + (k//2)*cost_m(1) cost_doubleline = 2*cost_m(k//4) + 8*cost_s(k//4) + (k//2)*cost_m(1) cost_verticalline = 0 cost_update1 = cost_s(k)+densexsparse_m8(k) cost_update2 = densexsparse_m8(k) if name == 'KSS16' : # extra partial add and partial double + 3 frob and 2 multiplications MASSON Simon committed May 27, 2019 341 if mystery_201903151748_simon_a_raison: MASSON Simon committed Mar 29, 2019 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 miller_fixup = (cost_m(k//4) + 5 *cost_s(k//4) + k//2 * cost_m(1)) + \ + (5*cost_m(k//4) + 2*cost_s(k//4) + k//2 * cost_m(1)) else: miller_fixup = 5*cost_m(k//4) + cost_s(k//4) + k * cost_m(1) miller_fixup += 3*cost_f(k) + 2*sparse_m16 elif k == 12 and D == 3: cost_doubleline = 3*cost_m(k//6) + 6*cost_s(k//6) + (k//3)*cost_m(1) cost_addline = 11*cost_m(k//6) + 2*cost_s(k//6) + (k//3)*cost_m(1) cost_verticalline = 0 cost_update1 = cost_s(k)+densexsparse_m6(k) cost_update2 = densexsparse_m6(k) if name == 'BN': # for Q1 = [p]Q and Q2=[p]Q1 # first extra add # second light add # 2 multiplications miller_fixup = 4*cost_f(k) + \ + cost_addline + \ + 4*cost_m(k//6) + 4*cost_m(1) + \ + 2*sparse_m12 elif k == 1: cost_addline=None cost_doubleline=None cost_verticalline=None cost_update1=None cost_update2=None tot_miller = 4626 * m + cost_i(k) if k!= 1 : T = polymorphic_get_miller_loop_length(C) logT = T.nbits() HwT = Hw(T) tot_miller = (logT-1) * (cost_doubleline + cost_verticalline) \ + (logT-2) * cost_update1 \ + (HwT-1) * (cost_addline + cost_verticalline + cost_update2) \ + (cost_i(k) if k%2==1 else 0) \ + miller_fixup return [cost_addline, cost_doubleline, cost_verticalline, cost_update1, cost_update2, ZZ(tot_miller(m=1,s=1,inv=25))] def cost_firstexp(k): assert k in [5,6,7,8] if Integer(k).is_prime(): return cost_i_and_f(k)+cost_m(k) if k == 6 : # a <- a^(1+p) c0 = cost_f(k) + cost_m(k) # a = a0 + a1y ; a^(p^3) = a0-a1y ; norm_{6/3}(a) = a0^2-y^2a1^2 # with cheap multiplication by y^2; then a^(p^3-1) = # 1/norm*(a0^2+y^2a1^2-2ya0a1) c1 = 2*cost_s(k//2) c1 += cost_m(k//2) c1 += cost_i(k//2) c1 += 2*cost_m(k//2) return c0 + c1 if k == 8 : return cost_i(k) + cost_m(k) def finalExpoCost(C): k = polymorphic_get_embedding_degree(C) name = polymorphic_get_name(C) if isinstance(C, Integer): tot_expo = 4100 * cost_m(1) return tot_expo(m=1,s=1,inv=25) elif name == 'BN': GUILLEVIC Aurore committed May 27, 2019 410 411 412 T = C.u() logT = T.nbits() HwT = Hw(T) GUILLEVIC Aurore committed May 27, 2019 413 414 #BN_expo_z = 4*(114 - 1)*cost_m(2) + (6*3 - 3)*cost_m(2) + 3*cost_m(12) + 3*3*cost_s(2) + cost_i(2) BN_expo_z = 4*(logT - 1)*cost_m(2) + (6*(HwT-1) - 3)*cost_m(2) + (HwT-1)*cost_m(12) + 3*(HwT-1)*cost_s(2) + cost_i(2) MASSON Simon committed Mar 29, 2019 415 416 417 418 #BarDuq says 114*compr_s12 + 3* cost_m(12) + (i + (24*4 - 5)*cost_m(1)) tot_expo = cost_i(12) + 12*cost_m(12) + 3*cyclo_s12 + 4* cost_f(12) + 3*BN_expo_z return tot_expo(m=1,s=1,inv=25) elif name == 'BLS12': GUILLEVIC Aurore committed May 27, 2019 419 420 421 T = C.u() logT = T.nbits() HwT = Hw(T) GUILLEVIC Aurore committed May 27, 2019 422 423 424 #BLS_expo_z = 4*(77 - 1)*cost_m(2)+ (6*2 - 3)*cost_m(2) + 2*cost_m(12) + 3*2*cost_s(2) + cost_i(2) BLS_expo_z = 4*(logT - 1)*cost_m(2)+ (6*(HwT-1) - 3)*cost_m(2) + (HwT-1)*cost_m(12) + 3*(HwT-1)*cost_s(2) + cost_i(2) #4*(log(u)-1)*m_i + (6*hw(u) - 3) *m_i + hw(u) * m_{6i} + 3*hw(u) * s_i + 1*I_i MASSON Simon committed Mar 29, 2019 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 #BarDuq says 77*compr_s12 + 2*cost_m(12) + (i + (24*3 - 5)*cost_m(1)) tot_expo = cost_i(12) + 12*cost_m(12) + 2*cyclo_s12 + 4*cost_f(12) + 5*BLS_expo_z return tot_expo(m=1,s=1,inv=25) elif name == 'KSS16': KSS16_expo_z = 34*cyclo_s16 + 4*cost_m(16) tot_expo = 34*cyclo_s16 + 32*cost_m(16)+24*cost_m(4)+8*cost_f(16)+cost_i(16) + 9*KSS16_expo_z return tot_expo(m=1,s=1,inv=25) if not isinstance(C, CocksPinchVariantResult): # FE for MNT6 BLS24 KSS18 not done raise ValueError("not implemented") logp = C.p.nbits() logr = C.r.nbits() T = C.T D = C.fD i = C.i logT = T.nbits() HwT = Hw(T) HwCofr=Hw(C.twist(0)['card']//C.r) hy = ZZ(C.hy) ht = ZZ(C.ht) loghy=hy.nbits(); Hwhy=Hw(hy) loght=ht.nbits(); Hwht=Hw(ht) c1 = cost_firstexp(k) # Now compute c2 (second part of FE) if k==5 or k==7: assert isinstance(C, CocksPinchVariantResult) cost_T = (logT-1)*cost_s(k) + (HwT-1)*cost_m(k) MASSON Simon committed May 27, 2019 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 # see final-expo-k57.sage c2 = (k-2)*(cost_f(k) + cost_T + 2*cost_m(k)) + cost_m(k) logc = logp - logr c2 += (logc -1)*cost_s(k) + (HwCofr-1)*cost_m(k) # one inversion costs (k-1) Frobenius since norm == 1 cost_inv_torus = (k-1) * cost_f(k) if i > 1: c2 += cost_inv_torus if i < k-1 and 2*i > k: # this just happens to match the formulas that we have. c2 += cost_inv_torus # cost for k=5 i=1: 1c + 3T + 7M + 3p # cost for k=5 i=2: 1I + 1c + 3T + 7M + 3p # cost for k=5 i=3: 2I + 1c + 3T + 7M + 3p # cost for k=5 i=4: 1I + 1c + 3T + 7M + 3p # cost for k=7 i=1: 1c + 5T + 11M + 5p # cost for k=7 i=2: 1I + 1c + 5T + 11M + 5p # cost for k=7 i=3: 1I + 1c + 5T + 11M + 5p # cost for k=7 i=4: 2I + 1c + 5T + 11M + 5p # cost for k=7 i=5: 2I + 1c + 5T + 11M + 5p # cost for k=7 i=6: 1I + 1c + 5T + 11M + 5p MASSON Simon committed Mar 29, 2019 479 480 481 482 483 484 485 486 elif k == 6 : # See: # sage: attach("formules-familles-CocksPinch.sage") # sage: formulas(6) assert D==3 MASSON Simon committed May 27, 2019 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 # start with this: c_exp_T = (logT-1)*cyclo_s6 + (HwT-1)*cost_m(k) c2 = c_exp_T + cost_f(k) + cyclo_s6 + 4*cost_m(k) if i == 5: # extra cost for raising to the power p+t0 = p+2-T: we # need a square... c2 += cyclo_s6 # Then, see code/formules-familles-CocksPinch.sage # it's a slight mess, to be honest. assert (1 + ht + hy) % 2 == 0 if ht % 2 == 0: hu = ht//2 hz = -1 if T%3 == 0 else 1 hw = (hy-hz)//2 else: hu = (ht+1)//2 MASSON Simon committed Mar 29, 2019 506 hw = hy//2 MASSON Simon committed May 27, 2019 507 508 509 510 511 512 513 514 515 U = T - (T % 3) logU=U.nbits(); HwU=Hw(U) loghu=hu.nbits(); Hwhu=Hw(hu) loghw=hw.nbits(); Hwhw=Hw(hw) c_exp_U = (logU-1)*cyclo_s6 + (HwU-1)*cost_m(k) c_exp_hu = (loghu-1)*cyclo_s6 + (Hwhu-1)*cost_m(k) c_exp_hw = (loghw-1)*cyclo_s6 + (Hwhw-1)*cost_m(k) c2 += 12*cost_m(k) + 2*cost_s(k) + 2*(c_exp_U + c_exp_hu + c_exp_hw) MASSON Simon committed Mar 29, 2019 516 517 518 519 520 521 522 523 524 525 526 527 528 529 elif k == 8 : # See: # sage: attach("formules-familles-CocksPinch.sage") # sage: formulas(8) assert D==4 assert T%2==0 assert ht % 2 == 1 # t0 is odd, so that ht must be odd too. Raising to the power # ht+1 or ht-1 costs at most as much as raising to the power ht, # and maybe one multiplication less. Note also that ht+1 is # necessarily even. MASSON Simon committed May 27, 2019 530 531 532 c_exp_T = (logT-1)*cyclo_s8 + (HwT-1)*cost_m(k) # first, this: (because phi_8(p)/r = 1 + c(T^2+p^2)(p-T)) MASSON Simon committed Sep 05, 2019 533 534 535 # recall that a p^2-power cost is the same as a p-power # for the reason explained in remark \ref{frob}. # What happen if i != 1 when the expo is to the power T^i ? MASSON Simon committed May 27, 2019 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 c2 = 3*c_exp_T + 2*cost_f(8) + 3*cost_m(k) # Then, see code/formules-familles-CocksPinch.sage # for raising to the power c, we get: # 11M + 2u + 4T + 2y # with one of the multiplies that (for i=3 and i=7) can be # elided if T=0 mod 4. This is with u = multiplication by # (h_t+1)//2. hu = (ht+1)//2; loghu=hu.nbits(); Hwhu=Hw(hu) c_exp_hy = (loghy-1)*cyclo_s8 + (Hwhy-1)*cost_m(k) c_exp_hu = (loghu-1)*cyclo_s8 + (Hwhu-1)*cost_m(k) c2 += 4 * c_exp_T + 2 * c_exp_hu + 2 * c_exp_hy + 10 * cost_m(k) MASSON Simon committed Sep 05, 2019 551 # Do we miss some inversions ? MASSON Simon committed May 27, 2019 552 553 554 if (T%4 == 2 or i == 1 or i == 5): c2 += cost_m(k) MASSON Simon committed Mar 29, 2019 555 556 557 558 559 560 561 562 563 tot_expo = c1 + c2 return tot_expo(m=1,s=1,inv=25) def pairingCost(C): costMiller = millerLoopCost(C) costFinalExp = finalExpoCost(C) logp = polymorphic_get_logp(C) MASSON Simon committed May 27, 2019 564 MASSON Simon committed Mar 29, 2019 565 566 567 time_m = finite_field_cost(logp) tot_miller = costMiller[-1] GUILLEVIC Aurore committed May 27, 2019 568 time_miller = round(tot_miller * time_m/1000000, 2) MASSON Simon committed Mar 29, 2019 569 tot_expo = costFinalExp GUILLEVIC Aurore committed May 27, 2019 570 time_expo = round(tot_expo * time_m/1000000, 2) MASSON Simon committed Mar 29, 2019 571 572 573 574 575 576 577 578 579 580 581 return dict( k=polymorphic_get_embedding_degree(C), D=polymorphic_get_fD(C), name=polymorphic_get_name(C), logp=polymorphic_get_logp(C), time_m=time_m, tot_miller=tot_miller, time_miller=time_miller, tot_expo=tot_expo, time_expo=time_expo, tot_pairing = tot_miller + tot_expo, GUILLEVIC Aurore committed May 27, 2019 582 time_pairing = round(time_miller+time_expo, 2), MASSON Simon committed Mar 29, 2019 583 584 585 586 587 ) def table_cost_pairing() : timing_recap = [] GUILLEVIC Aurore committed May 27, 2019 588 for C in [C5,C6,C7,C8,CBN446,CBLS446,CBN12,CBLS12,CKSS16,C1]: MASSON Simon committed Mar 29, 2019 589 590 591 592 593 594 595 L=pairingCost(C) timing_recap.append(L) #timing recap is generated print "% This table is generated by:" Emmanuel Thomé committed Apr 08, 2019 596 print "% PYTHONPATH=cocks-pinch-variant sage -c 'load(\"cocks-pinch-variant/cost_pairing.py\"); table_cost_pairing()'" MASSON Simon committed Mar 29, 2019 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 def wrap_cell(cell): return "\\begin{tabular}{@{}c@{}} %s \\end{tabular}" % cell for L in timing_recap : cell0 = "%s & %d-bit" % (L['name'], L['logp']) cell_miller = "%d\\bfm \\\\ %sms" % (L['tot_miller'], L['time_miller']) cell_miller = wrap_cell(cell_miller) cell_expo = "%d\\bfm \\\\ %sms" % (L['tot_expo'], L['time_expo']) cell_expo = wrap_cell(cell_expo) cell_pairing = " %d\\bfm & %sms" % (L['tot_pairing'], L['time_pairing']) print cell0 + " & " + cell_miller + " &" print " " + cell_expo + " & " print " " + cell_pairing + " \\\\" print " \\hline "