#!/usr/bin/env sage
r"""
Generate finite field parameters for minisketch.
This script selects the finite fields used by minisketch
for various sizes and generates the required tables for
the implementation.
The output (after formatting) can be found in src/fields/*.cpp.
"""
B.<b> = GF(2)
P.<p> = B[]
def apply_map(m, v):
r = 0
i = 0
while v != 0:
if (v & 1):
r ^^= m[i]
i += 1
v >>= 1
return r
def recurse_moduli(acc, maxweight, maxdegree):
for pos in range(maxweight, maxdegree + 1, 1):
poly = acc + p^pos
if maxweight == 1:
if poly.is_irreducible():
return (pos, poly)
else:
(deg, ret) = recurse_moduli(poly, maxweight - 1, pos - 1)
if ret is not None:
return (pos, ret)
return (None, None)
def compute_moduli(bits):
# Return all optimal irreducible polynomials for GF(2^bits)
# The result is a list of tuples (weight, degree of second-highest nonzero coefficient, polynomial)
maxdegree = bits - 1
result = []
for weight in range(1, bits, 2):
deg, res = None, None
while True:
ret = recurse_moduli(p^bits + 1, weight, maxdegree)
if ret[0] is not None:
(deg, res) = ret
maxdegree = deg - 1
else:
break
if res is not None:
result.append((weight + 2, deg, res))
return result
def bits_to_int(vals):
ret = 0
base = 1
for val in vals:
ret += Integer(val) * base
base *= 2
return ret
def sqr_table(f, bits, n=1):
ret = []
for i in range(bits):
ret.append((f^(2^n*i)).integer_representation())
return ret
# Compute x**(2**n)
def pow2(x, n):
for i in range(n):
x = x**2
return x
def qrt_table(F, f, bits):
# Table for solving x2 + x = a
# This implements the technique from https://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37927/40412, Lemma 1
for i in range(bits):
if (f**i).trace() != 0:
u = f**i
ret = []
for i in range(0, bits):
d = f^i
y = sum(pow2(d, j) * sum(pow2(u, k) for k in range(j)) for j in range(1, bits))
ret.append(y.integer_representation() ^^ (y.integer_representation() & 1))
return ret
def conv_tables(F, NF, bits):
# Generate a F(2) linear projection that maps elements from one field
# to an isomorphic field with a different modulus.
f = F.gen()
fp = f.minimal_polynomial()
assert(fp == F.modulus())
nfp = fp.change_ring(NF)
nf = sorted(nfp.roots(multiplicities=False))[0]
ret = []
matrepr = [[B(0) for x in range(bits)] for y in range(bits)]
for i in range(bits):
val = (nf**i).integer_representation()
ret.append(val)
for j in range(bits):
matrepr[j][i] = B((val >> j) & 1)
mat = Matrix(matrepr).inverse().transpose()
ret2 = []
for i in range(bits):
ret2.append(bits_to_int(mat[i]))
for t in range(100):
f1a = F.random_element()
f1b = F.random_element()
f1r = f1a * f1b
f2a = NF.fetch_int(apply_map(ret, f1a.integer_representation()))
f2b = NF.fetch_int(apply_map(ret, f1b.integer_representation()))
f2r = NF.fetch_int(apply_map(ret, f1r.integer_representation()))
f2s = f2a * f2b
assert(f2r == f2s)
for t in range(100):
f2a = NF.random_element()
f2b = NF.random_element()
f2r = f2a * f2b
f1a = F.fetch_int(apply_map(ret2, f2a.integer_representation()))
f1b = F.fetch_int(apply_map(ret2, f2b.integer_representation()))
f1r = F.fetch_int(apply_map(ret2, f2r.integer_representation()))
f1s = f1a * f1b
assert(f1r == f1s)
return (ret, ret2)
def fmt(i,typ):
if i == 0:
return "0"
else:
return "0x%x" % i
def lintranstype(typ, bits, maxtbl):
gsize = min(maxtbl, bits)
array_size = (bits + gsize - 1) // gsize
bits_list = []
total = 0
for i in range(array_size):
rsize = (bits - total + array_size - i - 1) // (array_size - i)
total += rsize
bits_list.append(rsize)
return "RecLinTrans<%s, %s>" % (typ, ", ".join("%i" % x for x in bits_list))
INT=0
CLMUL=1
CLMUL_TRI=2
MD=3
def print_modulus_md(mod):
ret = ""
pos = mod.degree()
for c in reversed(list(mod)):
if c:
if ret:
ret += " + "
if pos == 0:
ret += "1"
elif pos == 1:
ret += "x"
else:
ret += "x<sup>%i</sup>" % pos
pos -= 1
return ret
def pick_modulus(bits, style):
# Choose the lexicographicly-first lowest-weight modulus
# optionally subject to implementation specific constraints.
moduli = compute_moduli(bits)
if style == INT or style == MD:
multi_sqr = False
need_trans = False
elif style == CLMUL:
# Fast CLMUL reduction requires that bits + the highest
# set bit are less than 66.
moduli = filter(lambda x: bits+x[1] <= 66, moduli) + moduli
multi_sqr = True
need_trans = True
if not moduli or moduli[0][2].change_ring(ZZ)(2) == 3 + 2**bits:
# For modulus 3, CLMUL_TRI is obviously better.
return None
elif style == CLMUL_TRI:
moduli = filter(lambda x: bits+x[1] <= 66, moduli) + moduli
moduli = filter(lambda x: x[0] == 3, moduli)
multi_sqr = True
need_trans = True
else:
assert(False)
if not moduli:
return None
return moduli[0][2]
def print_result(bits, style):
if style == INT:
multi_sqr = False
need_trans = False
table_id = "%i" % bits
elif style == MD:
pass
elif style == CLMUL:
multi_sqr = True
need_trans = True
table_id = "%i" % bits
elif style == CLMUL_TRI:
multi_sqr = True
need_trans = True
table_id = "TRI%i" % bits
else:
assert(False)
nmodulus = pick_modulus(bits, INT)
modulus = pick_modulus(bits, style)
if modulus is None:
return
if style == MD:
print("* *%s*" % print_modulus_md(modulus))
return
if bits > 32:
typ = "uint64_t"
elif bits > 16:
typ = "uint32_t"
elif bits > 8:
typ = "uint16_t"
else:
typ = "uint8_t"
ttyp = lintranstype(typ, bits, 4)
rtyp = lintranstype(typ, bits, 6)
F.<f> = GF(2**bits, modulus=modulus)
include_table = True
if style != INT and style != CLMUL:
cmodulus = pick_modulus(bits, CLMUL)
if cmodulus == modulus:
include_table = False
table_id = "%i" % bits
if include_table:
print("typedef %s StatTable%s;" % (rtyp, table_id))
rtyp = "StatTable%s" % table_id
if (style == INT):
print("typedef %s DynTable%s;" % (ttyp, table_id))
ttyp = "DynTable%s" % table_id
if need_trans:
if modulus != nmodulus:
# If the bitstream modulus is not the best modulus for
# this implementation a conversion table will be needed.
ctyp = rtyp
NF.<nf> = GF(2**bits, modulus=nmodulus)
ctables = conv_tables(NF, F, bits)
loadtbl = "&LOAD_TABLE_%s" % table_id
savetbl = "&SAVE_TABLE_%s" % table_id
if include_table:
print("constexpr %s LOAD_TABLE_%s({%s});" % (ctyp, table_id, ", ".join([fmt(x,typ) for x in ctables[0]])))
print("constexpr %s SAVE_TABLE_%s({%s});" % (ctyp, table_id, ", ".join([fmt(x,typ) for x in ctables[1]])))
else:
ctyp = "IdTrans"
loadtbl = "&ID_TRANS"
savetbl = "&ID_TRANS"
else:
assert(modulus == nmodulus)
if include_table:
print("constexpr %s SQR_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in sqr_table(f, bits, 1)])))
if multi_sqr:
# Repeated squaring is a linearised polynomial so in F(2^n) it is
# F(2) linear and can be computed by a simple bit-matrix.
# Repeated squaring is especially useful in powering ladders such as
# for inversion.
sqr2 = "nullptr"
sqr4 = "nullptr"
sqr8 = "nullptr"
sqr16 = "nullptr"
if ((bits - 1) >= 4):
if include_table:
print("constexpr %s SQR2_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in sqr_table(f, bits, 2)])))
sqr2 = "&SQR2_TABLE_%s" % table_id
if ((bits - 1) >= 8):
if include_table:
print("constexpr %s SQR4_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in sqr_table(f, bits, 4)])))
sqr4 = "&SQR4_TABLE_%s" % table_id
if ((bits - 1) >= 16):
if include_table:
print("constexpr %s SQR8_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in sqr_table(f, bits, 8)])))
sqr8 = "&SQR8_TABLE_%s" % table_id
if ((bits - 1) >= 32):
if include_table:
print("constexpr %s SQR16_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in sqr_table(f, bits, 16)])))
sqr16 = "&SQR16_TABLE_%s" % table_id
if include_table:
print("constexpr %s QRT_TABLE_%s({%s});" % (rtyp, table_id, ", ".join([fmt(x,typ) for x in qrt_table(F, f, bits)])))
modulus_weight = modulus.hamming_weight()
modulus_degree = (modulus - p**bits).degree()
modulus_int = (modulus - p**bits).change_ring(ZZ)(2)
lfsr = ""
if style == INT:
print("typedef Field<%s, %i, %i, %s, %s, &SQR_TABLE_%s, &QRT_TABLE_%s%s> Field%i;" % (typ, bits, modulus_int, rtyp, ttyp, table_id, table_id, lfsr, bits))
elif style == CLMUL:
print("typedef Field<%s, %i, %i, %s, &SQR_TABLE_%s, %s, %s, %s, %s, &QRT_TABLE_%s, %s, %s, %s%s> Field%i;" % (typ, bits, modulus_int, rtyp, table_id, sqr2, sqr4, sqr8, sqr16, table_id, ctyp, loadtbl, savetbl, lfsr, bits))
elif style == CLMUL_TRI:
print("typedef FieldTri<%s, %i, %i, %s, &SQR_TABLE_%s, %s, %s, %s, %s, &QRT_TABLE_%s, %s, %s, %s> FieldTri%i;" % (typ, bits, modulus_degree, rtyp, table_id, sqr2, sqr4, sqr8, sqr16, table_id, ctyp, loadtbl, savetbl, bits))
else:
assert(False)
for bits in range(2, 65):
print("// %i bit field" % bits)
print_result(bits, INT)
print("")
for bits in range(2, 65):
print("// %i bit field" % bits)
print_result(bits, CLMUL)
print_result(bits, CLMUL_TRI)
print("")
for bits in range(2, 65):
print_result(bits, MD)
