Canonically encode powers of primitive element e #64
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meshcollider
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For review, the tables can be generated with the following sage code: GF1024_LOG = [-1] + [0] * 1023
GF1024_EXP = [1] * 1024
v = 1
for i in range(1, 1023):
v0 = v & 31
v1 = v >> 5
v0n = (F.fetch_int(23)*F.fetch_int(v1)).integer_representation()
v1n = (F.fetch_int(9)*F.fetch_int(v1) + F.fetch_int(v0)).integer_representation()
v = v1n << 5 | v0n
GF1024_EXP[i] = v
GF1024_LOG[v] = iAnd the syndrome constants can be generated using: for k in range(1, 6):
for b in [1,2,4,8,16]:
c0 = GF1024_EXP[(997*k + GF1024_LOG[b]) % 1023]
c1 = GF1024_EXP[(998*k + GF1024_LOG[b]) % 1023]
c2 = GF1024_EXP[(999*k + GF1024_LOG[b]) % 1023]
c = c2 << 20 | c1 << 10 | c0
print("0x%x" % c)The canonical encoding of every unit of GF(1024) can be checked with the following code: for i in range(1023):
v = GF1024_EXP[i]
v0 = F.fetch_int(v & 31)
v1 = F.fetch_int(v >> 5)
if (e^i).list() != [v0, v1]:
print("Error:", i) |
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looks good to me :) |
Owner
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I've recreated the same tables using the following Sage code (which makes use of Sage's ability to do extension field arithmetic): # Binary field
B.<b> = GF(2)
# Polynomials over the binary field
BP.<bp> = B[]
# Encoder field GF(32)
F.<f> = GF(32, modulus=bp^5 + bp^3 + 1, repr='int')
# Polynomials over the encoder field
FP.<fp> = F[]
# Decoder field GF(1024)
E.<e> = F.extension(modulus=fp^2 + F.fetch_int(9)*fp + F.fetch_int(23))
# Convert an E field element to an integer 0..1023.
def gf_to_int(v):
return v[0].integer_representation() + 32*v[1].integer_representation()
# Convert an integer 0..1023 to an E field element.
def int_to_gf(v):
return F.fetch_int(v & 31) + e*F.fetch_int(v >> 5)
GF1024_EXP = [gf_to_int(e**i) for i in range(1024)]
GF1024_LOG = [-1] + [discrete_log(int_to_gf(i), e, ord=1023, operation="*") for i in range(1, 1024)]
print(GF1024_EXP)
print(GF1024_LOG)
a1, a2, a3 = e**997, e**998, e**999
for k in range(1, 6):
for b in [1,2,4,8,16]:
c = (gf_to_int(a1**k * int_to_gf(b)) |
gf_to_int(a2**k * int_to_gf(b)) << 10 |
gf_to_int(a3**k * int_to_gf(b)) << 20)
print("0x%08x" % c) |
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ACK 42b570d |
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There is no behaviour change in this PR.
Currently, the primitive element
egenerating the multiplicative group of GF(1024) is encoded as an integer inGF1024_EXP[]as({9}, {15}), or303. Because of this choice, the table is generated in the following way:For an element
(v1, v0), we computee*(v1, v0) = (v1', v0')as:(from https://github.com/sipa/ezbase32/blob/master/bech32.cpp#L90-L122)
After asking sipa about this choice of encoding, he couldn't recall why the
({9},{15})was chosen. It has no effect on the checksum, as we are not modifying the choice of primitive element itself, only the encoding of it in theGF1024_EXP[]table. For clarity, this PR instead uses the clearer choice of({1},{0})to encodee. This encoding is simpler because elementsa*e + bin GF(1024) are then directly encoded as(a, b). The multiplication byethen becomes:Where the
{9}and{23}come directly from the choice of modulus for the extension fieldx^2 + 9*x + 23.We then update the syndrome computation values accordingly, which just encode powers of two * powers of
e.