mdw@git.distorted.org.uk Git - catacomb/blob - utils/gcm-ref

   1 #! /usr/bin/python
   2 ### -*- coding: utf-8 -*-
   3
   4 from sys import argv, exit
   5
   6 import catacomb as C
   7
   8 ###--------------------------------------------------------------------------
   9 ### Random utilities.
  10
  11 def words(s):
  12   """Split S into 32-bit pieces and report their values as hex."""
  13   return ' '.join('%08x' % C.MP.loadb(s[i:i + 4])
  14                   for i in xrange(0, len(s), 4))
  15
  16 def words_64(s):
  17   """Split S into 64-bit pieces and report their values as hex."""
  18   return ' '.join('%016x' % C.MP.loadb(s[i:i + 8])
  19                   for i in xrange(0, len(s), 8))
  20
  21 def repmask(val, wd, n):
  22   """Return a mask consisting of N copies of the WD-bit value VAL."""
  23   v = C.GF(val)
  24   a = C.GF(0)
  25   for i in xrange(n): a = (a << wd) | v
  26   return a
  27
  28 def combs(things, k):
  29   """Iterate over all possible combinations of K of the THINGS."""
  30   ii = range(k)
  31   n = len(things)
  32   while True:
  33     yield [things[i] for i in ii]
  34     for j in xrange(k):
  35       if j == k - 1: lim = n
  36       else: lim = ii[j + 1]
  37       i = ii[j] + 1
  38       if i < lim:
  39         ii[j] = i
  40         break
  41       ii[j] = j
  42     else:
  43       return
  44
  45 POLYMAP = {}
  46
  47 def poly(nbits):
  48   """
  49   Return the lexically first irreducible polynomial of degree NBITS of lowest
  50   weight.
  51   """
  52   try: return POLYMAP[nbits]
  53   except KeyError: pass
  54   base = C.GF(0).setbit(nbits).setbit(0)
  55   for k in xrange(1, nbits, 2):
  56     for cc in combs(range(1, nbits), k):
  57       p = base + sum((C.GF(0).setbit(c) for c in cc), C.GF(0))
  58       if p.irreduciblep(): POLYMAP[nbits] = p; return p
  59   raise ValueError, nbits
  60
  61 def gcm_mangle(x):
  62   """Flip the bits within each byte according to GCM's insane convention."""
  63   y = C.WriteBuffer()
  64   for b in x:
  65     b = ord(b)
  66     bb = 0
  67     for i in xrange(8):
  68       bb <<= 1
  69       if b&1: bb |= 1
  70       b >>= 1
  71     y.putu8(bb)
  72   return y.contents
  73
  74 def endswap_words_32(x):
  75   """End-swap each 32-bit word of X."""
  76   x = C.ReadBuffer(x)
  77   y = C.WriteBuffer()
  78   while x.left: y.putu32l(x.getu32b())
  79   return y.contents
  80
  81 def endswap_words_64(x):
  82   """End-swap each 64-bit word of X."""
  83   x = C.ReadBuffer(x)
  84   y = C.WriteBuffer()
  85   while x.left: y.putu64l(x.getu64b())
  86   return y.contents
  87
  88 def endswap_bytes(x):
  89   """End-swap X by bytes."""
  90   y = C.WriteBuffer()
  91   for ch in reversed(x): y.put(ch)
  92   return y.contents
  93
  94 def gfmask(n):
  95   return C.GF(C.MP(0).setbit(n) - 1)
  96
  97 def gcm_mul(x, y):
  98   """Multiply X and Y according to the GCM rules."""
  99   w = len(x)
 100   p = poly(8*w)
 101   u, v = C.GF.loadl(gcm_mangle(x)), C.GF.loadl(gcm_mangle(y))
 102   z = (u*v)%p
 103   return gcm_mangle(z.storel(w))
 104
 105 DEMOMAP = {}
 106 def demo(func):
 107   name = func.func_name
 108   assert(name.startswith('demo_'))
 109   DEMOMAP[name[5:].replace('_', '-')] = func
 110   return func
 111
 112 def iota(i = 0):
 113   vi = [i]
 114   def next(): vi[0] += 1; return vi[0] - 1
 115   return next
 116
 117 ###--------------------------------------------------------------------------
 118 ### Portable table-driven implementation.
 119
 120 def shift_left(x):
 121   """Given a field element X (in external format), return X t."""
 122   w = len(x)
 123   p = poly(8*w)
 124   return gcm_mangle(C.GF.storel((C.GF.loadl(gcm_mangle(x)) << 1)%p))
 125
 126 def shift_right(x):
 127   """Given a field element X (in external format), return X/t."""
 128   w = len(x)
 129   p = poly(8*w)
 130   return gcm_mangle(C.GF.storel((C.GF.loadl(gcm_mangle(x))*p.modinv(2))%p))
 131
 132 def table_common(u, v, flip, getword, ixmask):
 133   """
 134   Multiply U by V using table lookup; common for `table-b' and `table-l'.
 135
 136   This matches the `simple_mulk_...' implementation in `gcm.c'.  One entry
 137   per bit is the best we can manage if we want a constant-time
 138   implementation: processing n bits at a time means we need to scan
 139   (2^n - 1)/n times as much memory.
 140
 141     * FLIP is a function (assumed to be an involution) on one argument X to
 142       convert X from external format to table-entry format or back again.
 143
 144     * GETWORD is a function on one argument B to retrieve the next 32-bit
 145       chunk of a field element held in a `ReadBuffer'.  Bits within a word
 146       are processed most-significant first.
 147
 148     * IXMASK is a mask XORed into table indices to permute the table so that
 149       its order matches that induced by GETWORD.
 150
 151   The table is built such that tab[i XOR IXMASK] = U t^i.
 152   """
 153   w = len(u); assert(w == len(v))
 154   a = C.ByteString.zero(w)
 155   tab = [None]*(8*w)
 156   for i in xrange(8*w):
 157     print ';; %9s = %7s = %s' % ('utab[%d]' % i, 'u t^%d' % i, words(u))
 158     tab[i ^ ixmask] = flip(u)
 159     u = shift_left(u)
 160   v = C.ReadBuffer(v)
 161   i = 0
 162   while v.left:
 163     t = getword(v)
 164     for j in xrange(32):
 165       bit = (t >> 31)&1
 166       if bit: a ^= tab[i]
 167       print ';; %6s = %d: a <- %s [%9s = %s]' % \
 168         ('v[%d]' % (i ^ ixmask), bit, words(a),
 169          'utab[%d]' % (i ^ ixmask), words(tab[i]))
 170       i += 1; t <<= 1
 171   return flip(a)
 172
 173 @demo
 174 def demo_table_b(u, v):
 175   """Big-endian table lookup."""
 176   return table_common(u, v, lambda x: x, lambda b: b.getu32b(), 0)
 177
 178 @demo
 179 def demo_table_l(u, v):
 180   """Little-endian table lookup."""
 181   return table_common(u, v, endswap_words_32, lambda b: b.getu32l(), 0x18)
 182
 183 ###--------------------------------------------------------------------------
 184 ### Implementation using 64×64->128-bit binary polynomial multiplication.
 185
 186 _i = iota()
 187 TAG_INPUT_U = _i()
 188 TAG_INPUT_V = _i()
 189 TAG_SHIFTED_V = _i()
 190 TAG_KPIECE_U = _i()
 191 TAG_KPIECE_V = _i()
 192 TAG_PRODPIECE = _i()
 193 TAG_PRODSUM = _i()
 194 TAG_PRODUCT = _i()
 195 TAG_REDCBITS = _i()
 196 TAG_REDCFULL = _i()
 197 TAG_REDCMIX = _i()
 198 TAG_OUTPUT = _i()
 199
 200 def split_gf(x, n):
 201   n /= 8
 202   return [C.GF.loadb(x[i:i + n]) for i in xrange(0, len(x), n)]
 203
 204 def join_gf(xx, n):
 205   x = C.GF(0)
 206   for i in xrange(len(xx)): x = (x << n) | xx[i]
 207   return x
 208
 209 def present_gf(x, w, n, what):
 210   firstp = True
 211   m = gfmask(n)
 212   for i in xrange(0, w, 128):
 213     print ';; %12s%c         =%s' % \
 214       (firstp and what or '',
 215        firstp and ':' or ' ',
 216        ''.join([j < w
 217                 and '          0x%s' % hex(((x >> j)&m).storeb(n/8))
 218                 or ''
 219                 for j in xrange(i, i + 128, n)]))
 220     firstp = False
 221
 222 def present_gf_pclmul(tag, wd, x, w, n, what):
 223   if tag != TAG_PRODPIECE: present_gf(x, w, n, what)
 224
 225 def reverse(x, w):
 226   return C.GF.loadl(x.storeb(w/8))
 227
 228 def rev32(x):
 229   w = x.noctets
 230   m_ffff = repmask(0xffff, 32, w/4)
 231   m_ff = repmask(0xff, 16, w/2)
 232   x = ((x&m_ffff) << 16) | ((x >> 16)&m_ffff)
 233   x = ((x&m_ff) << 8) | ((x >> 8)&m_ff)
 234   return x
 235
 236 def rev8(x):
 237   w = x.noctets
 238   m_0f = repmask(0x0f, 8, w)
 239   m_33 = repmask(0x33, 8, w)
 240   m_55 = repmask(0x55, 8, w)
 241   x = ((x&m_0f) << 4) | ((x >> 4)&m_0f)
 242   x = ((x&m_33) << 2) | ((x >> 2)&m_33)
 243   x = ((x&m_55) << 1) | ((x >> 1)&m_55)
 244   return x
 245
 246 def present_gf_vmullp64(tag, wd, x, w, n, what):
 247   if tag == TAG_PRODPIECE or tag == TAG_REDCFULL:
 248     return
 249   elif (wd == 128 or wd == 64) and TAG_PRODSUM <= tag <= TAG_PRODUCT:
 250     y = x
 251   elif (wd == 96 or wd == 192 or wd == 256) and \
 252        TAG_PRODSUM <= tag < TAG_OUTPUT:
 253     y = x
 254   else:
 255     xx = x.storeb(w/8)
 256     extra = len(xx)%8
 257     if extra: xx += C.ByteString.zero(8 - extra)
 258     yb = C.WriteBuffer()
 259     for i in xrange(len(xx), 0, -8): yb.put(xx[i - 8:i])
 260     y = C.GF.loadb(yb.contents)
 261   present_gf(y, (w + 63)&~63, n, what)
 262
 263 def present_gf_pmull(tag, wd, x, w, n, what):
 264   if tag == TAG_PRODPIECE or tag == TAG_REDCFULL:
 265     return
 266   elif tag == TAG_INPUT_V or tag == TAG_SHIFTED_V or tag == TAG_KPIECE_V:
 267     w = (w + 63)&~63
 268     bx = C.ReadBuffer(x.storeb(w/8))
 269     by = C.WriteBuffer()
 270     while bx.left: chunk = bx.get(8); by.put(chunk).put(chunk)
 271     x = C.GF.loadb(by.contents)
 272     w *= 2
 273   elif TAG_PRODSUM <= tag <= TAG_PRODUCT:
 274     x <<= 1
 275   y = reverse(rev8(x), w)
 276   present_gf(y, w, n, what)
 277
 278 def poly64_mul_simple(u, v, presfn, wd, dispwd, mulwd, uwhat, vwhat):
 279   """
 280   Multiply U by V, returning the product.
 281
 282   This is the fallback long multiplication.
 283   """
 284
 285   uw, vw = 8*len(u), 8*len(v)
 286
 287   ## We start by carving the operands into 64-bit pieces.  This is
 288   ## straightforward except for the 96-bit case, where we end up with two
 289   ## short pieces which we pad at the beginning.
 290   upad = (-uw)%mulwd; u += C.ByteString.zero(upad); uw += upad
 291   vpad = (-vw)%mulwd; v += C.ByteString.zero(vpad); vw += vpad
 292   uu = split_gf(u, mulwd); vv = split_gf(v, mulwd)
 293
 294   ## Report and accumulate the individual product pieces.
 295   x = C.GF(0)
 296   ulim, vlim = uw/mulwd, vw/mulwd
 297   for i in xrange(ulim + vlim - 2, -1, -1):
 298     t = C.GF(0)
 299     for j in xrange(max(0, i - vlim + 1), min(vlim, i + 1)):
 300       s = uu[ulim - 1 - i + j]*vv[vlim - 1 - j]
 301       presfn(TAG_PRODPIECE, wd, s, 2*mulwd, dispwd,
 302              '%s_%d %s_%d' % (uwhat, i - j, vwhat, j))
 303       t += s
 304     presfn(TAG_PRODSUM, wd, t, 2*mulwd, dispwd,
 305            '(%s %s)_%d' % (uwhat, vwhat, ulim + vlim - 2 - i))
 306     x += t << (mulwd*i)
 307   presfn(TAG_PRODUCT, wd, x, uw + vw, dispwd, '%s %s' % (uwhat, vwhat))
 308
 309   return x >> (upad + vpad)
 310
 311 def poly64_mul_karatsuba(u, v, klimit, presfn, wd,
 312                          dispwd, mulwd, uwhat, vwhat):
 313   """
 314   Multiply U by V, returning the product.
 315
 316   If the length of U and V is at least KLIMIT, and the operands are otherwise
 317   suitable, then do Karatsuba--Ofman multiplication; otherwise, delegate to
 318   `poly64_mul_simple'.
 319   """
 320   w = 8*len(u)
 321
 322   if w < klimit or w != 8*len(v) or w%(2*mulwd) != 0:
 323     return poly64_mul_simple(u, v, presfn, wd, dispwd, mulwd, uwhat, vwhat)
 324
 325   hw = w/2
 326   u0, u1 = u[:hw/8], u[hw/8:]
 327   v0, v1 = v[:hw/8], v[hw/8:]
 328   uu, vv = u0 ^ u1, v0 ^ v1
 329
 330   presfn(TAG_KPIECE_U, wd, C.GF.loadb(uu), hw, dispwd, '%s*' % uwhat)
 331   presfn(TAG_KPIECE_V, wd, C.GF.loadb(vv), hw, dispwd, '%s*' % vwhat)
 332   uuvv = poly64_mul_karatsuba(uu, vv, klimit, presfn, wd, dispwd, mulwd,
 333                               '%s*' % uwhat, '%s*' % vwhat)
 334
 335   presfn(TAG_KPIECE_U, wd, C.GF.loadb(u0), hw, dispwd, '%s0' % uwhat)
 336   presfn(TAG_KPIECE_V, wd, C.GF.loadb(v0), hw, dispwd, '%s0' % vwhat)
 337   u0v0 = poly64_mul_karatsuba(u0, v0, klimit, presfn, wd, dispwd, mulwd,
 338                               '%s0' % uwhat, '%s0' % vwhat)
 339
 340   presfn(TAG_KPIECE_U, wd, C.GF.loadb(u1), hw, dispwd, '%s1' % uwhat)
 341   presfn(TAG_KPIECE_V, wd, C.GF.loadb(v1), hw, dispwd, '%s1' % vwhat)
 342   u1v1 = poly64_mul_karatsuba(u1, v1, klimit, presfn, wd, dispwd, mulwd,
 343                               '%s1' % uwhat, '%s1' % vwhat)
 344
 345   uvuv = uuvv + u0v0 + u1v1
 346   presfn(TAG_PRODSUM, wd, uvuv, w, dispwd, '%s!%s' % (uwhat, vwhat))
 347
 348   x = u1v1 + (uvuv << hw) + (u0v0 << w)
 349   presfn(TAG_PRODUCT, wd, x, 2*w, dispwd, '%s %s' % (uwhat, vwhat))
 350   return x
 351
 352 def poly64_mul(u, v, presfn, dispwd, mulwd, klimit, uwhat, vwhat):
 353   """
 354   Multiply U by V using a primitive 64-bit binary polynomial mutliplier.
 355
 356   Such a multiplier exists as the appallingly-named `pclmul[lh]q[lh]qdq' on
 357   x86, and as `vmull.p64'/`pmull' on ARM.
 358
 359   Operands arrive in a `register format', which is a byte-swapped variant of
 360   the external format.  Implementations differ on the precise details,
 361   though.  Returns the double-precision product.
 362   """
 363
 364   w = 8*len(u); assert(w == 8*len(v))
 365   x = poly64_mul_karatsuba(u, v, klimit, presfn,
 366                            w, dispwd, mulwd, uwhat, vwhat)
 367
 368   return x.storeb(w/4)
 369
 370 def poly64_redc(y, presfn, dispwd, redcwd):
 371   """
 372   Reduce a double-precision product X modulo the appropriate polynomial.
 373
 374   The operand arrives in a `register format', which is a byte-swapped variant
 375   of the external format.  Implementations differ on the precise details,
 376   though.  Returns the single-precision reduced value.
 377   """
 378
 379   w = 4*len(y)
 380   p = poly(w)
 381
 382   ## Our polynomial has the form p = t^d + r where r = SUM_{0<=i<d} r_i t^i,
 383   ## with each r_i either 0 or 1.  Because we choose the lexically earliest
 384   ## irreducible polynomial with the necessary degree, r_i = 1 happens only
 385   ## for a small number of tiny i.  In our field, we have t^d = r.
 386   ##
 387   ## We carve the product into convenient n-bit pieces, for some n dividing d
 388   ## -- typically n = 32 or 64.  Let d = m n, and write y = SUM_{0<=i<2m} y_i
 389   ## t^{ni}.  The upper portion, the y_i with i >= m, needs reduction; but
 390   ## y_i t^{ni} = y_i r t^{n(i-m)}, so we just multiply the top half by r and
 391   ## add it to the bottom half.  This all depends on r_i = 0 for all i >=
 392   ## n/2.  We process each nonzero coefficient of r separately, in two
 393   ## passes.
 394   ##
 395   ## Multiplying a chunk y_i by some t^j is the same as shifting it left by j
 396   ## bits (or would be if GCM weren't backwards, but let's not worry about
 397   ## that right now).  The high j bits will spill over into the next chunk,
 398   ## while the low n - j bits will stay where they are.  It's these high bits
 399   ## which cause trouble -- particularly the high bits of the top chunk,
 400   ## since we'll add them on to y_m, which will need further reduction.  But
 401   ## only the topmost j bits will do this.
 402   ##
 403   ## The trick is that we do all of the bits which spill over first -- all of
 404   ## the top j bits in each chunk, for each j -- in one pass, and then a
 405   ## second pass of all the bits which don't.  Because j, j' < n/2 for any
 406   ## two nonzero coefficient degrees j and j', we have j + j' < n whence j <
 407   ## n - j' -- so all of the bits contributed to y_m will be handled in the
 408   ## second pass when we handle the bits that don't spill over.
 409   rr = [i for i in xrange(1, w) if p.testbit(i)]
 410   m = gfmask(redcwd)
 411
 412   ## Handle the spilling bits.
 413   yy = split_gf(y, redcwd)
 414   b = C.GF(0)
 415   for rj in rr:
 416     br = [(yi << (redcwd - rj))&m for yi in yy[w/redcwd:]]
 417     presfn(TAG_REDCBITS, w, join_gf(br, redcwd), w, dispwd, 'b(%d)' % rj)
 418     b += join_gf(br, redcwd) << (w - redcwd)
 419   presfn(TAG_REDCFULL, w, b, 2*w, dispwd, 'b')
 420   s = C.GF.loadb(y) + b
 421   presfn(TAG_REDCMIX, w, s, 2*w, dispwd, 's')
 422
 423   ## Handle the nonspilling bits.
 424   ss = split_gf(s.storeb(w/4), redcwd)
 425   a = C.GF(0)
 426   for rj in rr:
 427     ar = [si >> rj for si in ss[w/redcwd:]]
 428     presfn(TAG_REDCBITS, w, join_gf(ar, redcwd), w, dispwd, 'a(%d)' % rj)
 429     a += join_gf(ar, redcwd)
 430   presfn(TAG_REDCFULL, w, a, w, dispwd, 'a')
 431
 432   ## Mix everything together.
 433   m = gfmask(w)
 434   z = (s&m) + (s >> w) + a
 435   presfn(TAG_OUTPUT, w, z, w, dispwd, 'z')
 436
 437   ## And we're done.
 438   return z.storeb(w/8)
 439
 440 def poly64_shiftcommon(u, v, presfn, dispwd = 32, mulwd = 64,
 441                        redcwd = 32, klimit = 256):
 442   w = 8*len(u)
 443   presfn(TAG_INPUT_U, w, C.GF.loadb(u), w, dispwd, 'u')
 444   presfn(TAG_INPUT_V, w, C.GF.loadb(v), w, dispwd, 'v')
 445   vv = shift_right(v)
 446   presfn(TAG_SHIFTED_V, w, C.GF.loadb(vv), w, dispwd, "v'")
 447   y = poly64_mul(u, vv, presfn, dispwd, mulwd, klimit, "u", "v'")
 448   z = poly64_redc(y, presfn, dispwd, redcwd)
 449   return z
 450
 451 def poly64_directcommon(u, v, presfn, dispwd = 32, mulwd = 64,
 452                         redcwd = 32, klimit = 256):
 453   w = 8*len(u)
 454   presfn(TAG_INPUT_U, w, C.GF.loadb(u), w, dispwd, 'u')
 455   presfn(TAG_INPUT_V, w, C.GF.loadb(v), w, dispwd, 'v')
 456   y = poly64_mul(u, v, presfn, dispwd, mulwd, klimit, "u", "v")
 457   y = (C.GF.loadb(y) << 1).storeb(w/4)
 458   z = poly64_redc(y, presfn, dispwd, redcwd)
 459   return z
 460
 461 @demo
 462 def demo_pclmul(u, v):
 463   return poly64_shiftcommon(u, v, presfn = present_gf_pclmul)
 464
 465 @demo
 466 def demo_vmullp64(u, v):
 467   w = 8*len(u)
 468   return poly64_shiftcommon(u, v, presfn = present_gf_vmullp64,
 469                             redcwd = w%64 == 32 and 32 or 64)
 470
 471 @demo
 472 def demo_pmull(u, v):
 473   w = 8*len(u)
 474   return poly64_directcommon(u, v, presfn = present_gf_pmull,
 475                              redcwd = w%64 == 32 and 32 or 64)
 476
 477 ###--------------------------------------------------------------------------
 478 ### @@@ Random debris to be deleted. @@@
 479
 480 def cutting_room_floor():
 481
 482   x = C.bytes('cde4bef260d7bcda163547d348b7551195e77022907dd1df')
 483   y = C.bytes('f7dac5c9941d26d0c6eb14ad568f86edd1dc9268eeee5332')
 484
 485   u, v = C.GF.loadb(x), C.GF.loadb(y)
 486
 487   g = u*v << 1
 488   print 'y = %s' % words(g.storeb(48))
 489   b1 = (g&repmask(0x01, 32, 6)) << 191
 490   b2 = (g&repmask(0x03, 32, 6)) << 190
 491   b7 = (g&repmask(0x7f, 32, 6)) << 185
 492   b = b1 + b2 + b7
 493   print 'b = %s' % words(b.storeb(48)[0:28])
 494   h = g + b
 495   print 'w = %s' % words(h.storeb(48))
 496
 497   a0 = (h&repmask(0xffffffff, 32, 6)) << 192
 498   a1 = (h&repmask(0xfffffffe, 32, 6)) << 191
 499   a2 = (h&repmask(0xfffffffc, 32, 6)) << 190
 500   a7 = (h&repmask(0xffffff80, 32, 6)) << 185
 501   a = a0 + a1 + a2 + a7
 502
 503   print '     a_1 = %s' % words(a1.storeb(48)[0:24])
 504   print '     a_2 = %s' % words(a2.storeb(48)[0:24])
 505   print '     a_7 = %s' % words(a7.storeb(48)[0:24])
 506
 507   print 'low+unit = %s' % words((h + a0).storeb(48)[0:24])
 508   print ' low+0,2 = %s' % words((h + a0 + a2).storeb(48)[0:24])
 509   print '     1,7 = %s' % words((a1 + a7).storeb(48)[0:24])
 510
 511   print 'a = %s' % words(a.storeb(48)[0:24])
 512   z = h + a
 513   print 'z = %s' % words(z.storeb(48))
 514
 515   z = gcm_mul(x, y)
 516   print 'u v mod p = %s' % words(z)
 517
 518 ###--------------------------------------------------------------------------
 519 ### Main program.
 520
 521 style = argv[1]
 522 u = C.bytes(argv[2])
 523 v = C.bytes(argv[3])
 524 zz = DEMOMAP[style](u, v)
 525 assert zz == gcm_mul(u, v)
 526
 527 ###----- That's all, folks --------------------------------------------------