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Signed-digit multi-comb ecmult_gen algorithm

intro

  • we want to compute R = gn*G = (gn - b)*G + b*G where G is the generator point, gn is a scalar and b is a scalar used to blind the multiplication
  • there's a comb(d, G/2) formula to compute (gn - b)*G and to express it as sum of +,- powers of 2 multiplied by G/2
  • supposing we have a precomputed table where possible sums of +, - powers of 2 multiplied by G/2 is stored, we can compute (gn - b)*G by picking the terms we want and then summing them. this is a cheaper operation than multiplication.
  • we can implement R = gn*G with a method to compute and access this precomputed table.
  • definition: block, teeth, spacing refer to group of bits in a scalar and looks like: block, teeth, spacing

secp256k1_ecmult_gen: accessing the precomputed table

comb(d, G/2) = sum((2*d_i-1)*2^i*G/2 for i=0..263)
             = (+- 2^0 +- 2^1 +- 2^2 +- 2^3 ............ +- 2^261 +- 2^262 +- 2^263) * G/2
             = ((+- 2^0 +- 2^4 ..... +- 2^260) + (+- 2^1 +- 2^5 ..... +- 2^261) + (+- 2^2 +- 2^6 ..... +- 2^262) + (+- 2^3 +- 2^7 ..... +- 2^263)) *G/2
  • all possible combinations of (+- 2^0 +- 2^4 ..... +- 2^240) is pre-computed and stored in a table.
  • example: for default configuration with blocks = 11, teeth = 6 when spacing = 0, table would look like:
00000 00001 .... 11111
block0 -2^20 -2^16 -2^12 -2^8 -2^4 -2^0 -2^20 -2^16 -2^12 -2^8 -2^4 +2^0 .... -2^20 +2^16 +2^12 +2^8 +2^4 +2^0
... ... ... ... ...
block10 -2^260 -2^256 -2^252 -2^248 -2^244 -2^240 -2^260 -2^256 -2^252 -2^248 -2^244 +2^240 ... -2^260 +2^256 +2^252 +2^248 +2^244 +2^240

  • since there are 6 teeth here, each block would have 2^6 possibilities of + and -

    • 101010 would mean +2^20 -2^16 +2^12 -2^8 +2^4 -2^0 multiplied by G/2
    • since result of 101010 is negation of result of 010101, the precomputed table only needs to store half the values.
    • precomputed table keeps first bit as 0 in the table -> so there's 2**5 possibilities of + and - now.
    • so we'd look up value for 010101 at index = 10101 and negate the result

  • similarly, another table for spacing = 1 and all possible combinations of (+- 2^1 +- 2^5 ..... +- 2^261). we'd have 4 different tables in total for this example because spacing = 4.

secp256k1_ecmult_gen_compute_table : precomputing the table

  • consider block 0:

    • the index 0 in precomputed table stores -2^20 -2^16 -2^12 -2^8 -2^4 -2^0
    • ds[0..5] stores [2 * 2^0, 2 * 2^4, 2 * 2^8, 2 * 2^12, 2 * 2^16, 2 * 2^20]
    • the rest of the values in the row is this value with different combination of +, - signs.
    they can be computed by adding/not adding the various values in ds[0..5] 
    ds[0] = 2  * (2 ^ (block * teeth * spacing)) * gen/2
ds[1] = 2  * (2 ^ (block * teeth * spacing)) * (2 ^ spacing) * gen/2
ds[2] = 2  * (2 ^ (block * teeth * spacing)) * (2 ^ 2*spacing) gen/2
ds[3] = 2  * (2 ^ (block * teeth * spacing)) * (2 ^ 3*spacing) gen/2
ds[4] = 2  * (2 ^ (block * teeth * spacing)) * (2 ^ 4*spacing) gen/2
ds[5] = 2  * (2 ^ (block * teeth * spacing)) * (2 ^ 5*spacing) gen/2

vs[0] = - (2 ^ (block * teeth * spacing))
- (2 ^ (block * teeth * spacing)) * (2 ^ spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 2*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 3*spacing)
...
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-3)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-2)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-1)*spacing)

vs[1] = vs[0] + ds[0]
vs[1] = + (2 ^ (block * teeth * spacing))
- (2 ^ (block * teeth * spacing)) * (2 ^ spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 2*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 3*spacing)
...
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-3)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-2)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-1)*spacing)

vs[2] = vs[0] + ds[1]
vs[2] = - (2 ^ (block * teeth * spacing))
+ (2 ^ (block * teeth * spacing)) * (2 ^ spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 2*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ 3*spacing)
...
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-3)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-2)*spacing)
- (2 ^ (block * teeth * spacing)) * (2 ^ (teeth-1)*spacing)

    table construction

    • we don't really need to store ds[5] since it's unused (we're only computing half table since the other half is a negation).
      • how much can we save anyways? 10 blocks * 4 tables * sizeof(secp256k1_gej) = 40 * 152 ~ 6 kb

  • see human understandable version of full table for spacing = 0 in a default configuration with blocks = 11, teeth = 6, spacing = 4

00000 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111
block 0 -2^20 -2^16 -2^12 -2^8 -2^4 -2^0 -2^20 -2^16 -2^12 -2^8 -2^4 +2^0 -2^20 -2^16 -2^12 -2^8 +2^4 -2^0 -2^20 -2^16 -2^12 -2^8 +2^4 +2^0 -2^20 -2^16 -2^12 +2^8 -2^4 -2^0 -2^20 -2^16 -2^12 +2^8 -2^4 +2^0 -2^20 -2^16 -2^12 +2^8 +2^4 -2^0 -2^20 -2^16 -2^12 +2^8 +2^4 +2^0 -2^20 -2^16 +2^12 -2^8 -2^4 -2^0 -2^20 -2^16 +2^12 -2^8 -2^4 +2^0 -2^20 -2^16 +2^12 -2^8 +2^4 -2^0 -2^20 -2^16 +2^12 -2^8 +2^4 +2^0 -2^20 -2^16 +2^12 +2^8 -2^4 -2^0 -2^20 -2^16 +2^12 +2^8 -2^4 +2^0 -2^20 -2^16 +2^12 +2^8 +2^4 -2^0 -2^20 -2^16 +2^12 +2^8 +2^4 +2^0 -2^20 +2^16 -2^12 -2^8 -2^4 -2^0 -2^20 +2^16 -2^12 -2^8 -2^4 +2^0 -2^20 +2^16 -2^12 -2^8 +2^4 -2^0 -2^20 +2^16 -2^12 -2^8 +2^4 +2^0 -2^20 +2^16 -2^12 +2^8 -2^4 -2^0 -2^20 +2^16 -2^12 +2^8 -2^4 +2^0 -2^20 +2^16 -2^12 +2^8 +2^4 -2^0 -2^20 +2^16 -2^12 +2^8 +2^4 +2^0 -2^20 +2^16 +2^12 -2^8 -2^4 -2^0 -2^20 +2^16 +2^12 -2^8 -2^4 +2^0 -2^20 +2^16 +2^12 -2^8 +2^4 -2^0 -2^20 +2^16 +2^12 -2^8 +2^4 +2^0 -2^20 +2^16 +2^12 +2^8 -2^4 -2^0 -2^20 +2^16 +2^12 +2^8 -2^4 +2^0 -2^20 +2^16 +2^12 +2^8 +2^4 -2^0 -2^20 +2^16 +2^12 +2^8 +2^4 +2^0
block 1 -2^44 -2^40 -2^36 -2^32 -2^28 -2^24 -2^44 -2^40 -2^36 -2^32 -2^28 +2^24 -2^44 -2^40 -2^36 -2^32 +2^28 -2^24 -2^44 -2^40 -2^36 -2^32 +2^28 +2^24 -2^44 -2^40 -2^36 +2^32 -2^28 -2^24 -2^44 -2^40 -2^36 +2^32 -2^28 +2^24 -2^44 -2^40 -2^36 +2^32 +2^28 -2^24 -2^44 -2^40 -2^36 +2^32 +2^28 +2^24 -2^44 -2^40 +2^36 -2^32 -2^28 -2^24 -2^44 -2^40 +2^36 -2^32 -2^28 +2^24 -2^44 -2^40 +2^36 -2^32 +2^28 -2^24 -2^44 -2^40 +2^36 -2^32 +2^28 +2^24 -2^44 -2^40 +2^36 +2^32 -2^28 -2^24 -2^44 -2^40 +2^36 +2^32 -2^28 +2^24 -2^44 -2^40 +2^36 +2^32 +2^28 -2^24 -2^44 -2^40 +2^36 +2^32 +2^28 +2^24 -2^44 +2^40 -2^36 -2^32 -2^28 -2^24 -2^44 +2^40 -2^36 -2^32 -2^28 +2^24 -2^44 +2^40 -2^36 -2^32 +2^28 -2^24 -2^44 +2^40 -2^36 -2^32 +2^28 +2^24 -2^44 +2^40 -2^36 +2^32 -2^28 -2^24 -2^44 +2^40 -2^36 +2^32 -2^28 +2^24 -2^44 +2^40 -2^36 +2^32 +2^28 -2^24 -2^44 +2^40 -2^36 +2^32 +2^28 +2^24 -2^44 +2^40 +2^36 -2^32 -2^28 -2^24 -2^44 +2^40 +2^36 -2^32 -2^28 +2^24 -2^44 +2^40 +2^36 -2^32 +2^28 -2^24 -2^44 +2^40 +2^36 -2^32 +2^28 +2^24 -2^44 +2^40 +2^36 +2^32 -2^28 -2^24 -2^44 +2^40 +2^36 +2^32 -2^28 +2^24 -2^44 +2^40 +2^36 +2^32 +2^28 -2^24 -2^44 +2^40 +2^36 +2^32 +2^28 +2^24
block 2 -2^68 -2^64 -2^60 -2^56 -2^52 -2^48 -2^68 -2^64 -2^60 -2^56 -2^52 +2^48 -2^68 -2^64 -2^60 -2^56 +2^52 -2^48 -2^68 -2^64 -2^60 -2^56 +2^52 +2^48 -2^68 -2^64 -2^60 +2^56 -2^52 -2^48 -2^68 -2^64 -2^60 +2^56 -2^52 +2^48 -2^68 -2^64 -2^60 +2^56 +2^52 -2^48 -2^68 -2^64 -2^60 +2^56 +2^52 +2^48 -2^68 -2^64 +2^60 -2^56 -2^52 -2^48 -2^68 -2^64 +2^60 -2^56 -2^52 +2^48 -2^68 -2^64 +2^60 -2^56 +2^52 -2^48 -2^68 -2^64 +2^60 -2^56 +2^52 +2^48 -2^68 -2^64 +2^60 +2^56 -2^52 -2^48 -2^68 -2^64 +2^60 +2^56 -2^52 +2^48 -2^68 -2^64 +2^60 +2^56 +2^52 -2^48 -2^68 -2^64 +2^60 +2^56 +2^52 +2^48 -2^68 +2^64 -2^60 -2^56 -2^52 -2^48 -2^68 +2^64 -2^60 -2^56 -2^52 +2^48 -2^68 +2^64 -2^60 -2^56 +2^52 -2^48 -2^68 +2^64 -2^60 -2^56 +2^52 +2^48 -2^68 +2^64 -2^60 +2^56 -2^52 -2^48 -2^68 +2^64 -2^60 +2^56 -2^52 +2^48 -2^68 +2^64 -2^60 +2^56 +2^52 -2^48 -2^68 +2^64 -2^60 +2^56 +2^52 +2^48 -2^68 +2^64 +2^60 -2^56 -2^52 -2^48 -2^68 +2^64 +2^60 -2^56 -2^52 +2^48 -2^68 +2^64 +2^60 -2^56 +2^52 -2^48 -2^68 +2^64 +2^60 -2^56 +2^52 +2^48 -2^68 +2^64 +2^60 +2^56 -2^52 -2^48 -2^68 +2^64 +2^60 +2^56 -2^52 +2^48 -2^68 +2^64 +2^60 +2^56 +2^52 -2^48 -2^68 +2^64 +2^60 +2^56 +2^52 +2^48
block 3 -2^92 -2^88 -2^84 -2^80 -2^76 -2^72 -2^92 -2^88 -2^84 -2^80 -2^76 +2^72 -2^92 -2^88 -2^84 -2^80 +2^76 -2^72 -2^92 -2^88 -2^84 -2^80 +2^76 +2^72 -2^92 -2^88 -2^84 +2^80 -2^76 -2^72 -2^92 -2^88 -2^84 +2^80 -2^76 +2^72 -2^92 -2^88 -2^84 +2^80 +2^76 -2^72 -2^92 -2^88 -2^84 +2^80 +2^76 +2^72 -2^92 -2^88 +2^84 -2^80 -2^76 -2^72 -2^92 -2^88 +2^84 -2^80 -2^76 +2^72 -2^92 -2^88 +2^84 -2^80 +2^76 -2^72 -2^92 -2^88 +2^84 -2^80 +2^76 +2^72 -2^92 -2^88 +2^84 +2^80 -2^76 -2^72 -2^92 -2^88 +2^84 +2^80 -2^76 +2^72 -2^92 -2^88 +2^84 +2^80 +2^76 -2^72 -2^92 -2^88 +2^84 +2^80 +2^76 +2^72 -2^92 +2^88 -2^84 -2^80 -2^76 -2^72 -2^92 +2^88 -2^84 -2^80 -2^76 +2^72 -2^92 +2^88 -2^84 -2^80 +2^76 -2^72 -2^92 +2^88 -2^84 -2^80 +2^76 +2^72 -2^92 +2^88 -2^84 +2^80 -2^76 -2^72 -2^92 +2^88 -2^84 +2^80 -2^76 +2^72 -2^92 +2^88 -2^84 +2^80 +2^76 -2^72 -2^92 +2^88 -2^84 +2^80 +2^76 +2^72 -2^92 +2^88 +2^84 -2^80 -2^76 -2^72 -2^92 +2^88 +2^84 -2^80 -2^76 +2^72 -2^92 +2^88 +2^84 -2^80 +2^76 -2^72 -2^92 +2^88 +2^84 -2^80 +2^76 +2^72 -2^92 +2^88 +2^84 +2^80 -2^76 -2^72 -2^92 +2^88 +2^84 +2^80 -2^76 +2^72 -2^92 +2^88 +2^84 +2^80 +2^76 -2^72 -2^92 +2^88 +2^84 +2^80 +2^76 +2^72
block 4 -2^116 -2^112 -2^108 -2^104 -2^100 -2^96 -2^116 -2^112 -2^108 -2^104 -2^100 +2^96 -2^116 -2^112 -2^108 -2^104 +2^100 -2^96 -2^116 -2^112 -2^108 -2^104 +2^100 +2^96 -2^116 -2^112 -2^108 +2^104 -2^100 -2^96 -2^116 -2^112 -2^108 +2^104 -2^100 +2^96 -2^116 -2^112 -2^108 +2^104 +2^100 -2^96 -2^116 -2^112 -2^108 +2^104 +2^100 +2^96 -2^116 -2^112 +2^108 -2^104 -2^100 -2^96 -2^116 -2^112 +2^108 -2^104 -2^100 +2^96 -2^116 -2^112 +2^108 -2^104 +2^100 -2^96 -2^116 -2^112 +2^108 -2^104 +2^100 +2^96 -2^116 -2^112 +2^108 +2^104 -2^100 -2^96 -2^116 -2^112 +2^108 +2^104 -2^100 +2^96 -2^116 -2^112 +2^108 +2^104 +2^100 -2^96 -2^116 -2^112 +2^108 +2^104 +2^100 +2^96 -2^116 +2^112 -2^108 -2^104 -2^100 -2^96 -2^116 +2^112 -2^108 -2^104 -2^100 +2^96 -2^116 +2^112 -2^108 -2^104 +2^100 -2^96 -2^116 +2^112 -2^108 -2^104 +2^100 +2^96 -2^116 +2^112 -2^108 +2^104 -2^100 -2^96 -2^116 +2^112 -2^108 +2^104 -2^100 +2^96 -2^116 +2^112 -2^108 +2^104 +2^100 -2^96 -2^116 +2^112 -2^108 +2^104 +2^100 +2^96 -2^116 +2^112 +2^108 -2^104 -2^100 -2^96 -2^116 +2^112 +2^108 -2^104 -2^100 +2^96 -2^116 +2^112 +2^108 -2^104 +2^100 -2^96 -2^116 +2^112 +2^108 -2^104 +2^100 +2^96 -2^116 +2^112 +2^108 +2^104 -2^100 -2^96 -2^116 +2^112 +2^108 +2^104 -2^100 +2^96 -2^116 +2^112 +2^108 +2^104 +2^100 -2^96 -2^116 +2^112 +2^108 +2^104 +2^100 +2^96
block 5 -2^140 -2^136 -2^132 -2^128 -2^124 -2^120 -2^140 -2^136 -2^132 -2^128 -2^124 +2^120 -2^140 -2^136 -2^132 -2^128 +2^124 -2^120 -2^140 -2^136 -2^132 -2^128 +2^124 +2^120 -2^140 -2^136 -2^132 +2^128 -2^124 -2^120 -2^140 -2^136 -2^132 +2^128 -2^124 +2^120 -2^140 -2^136 -2^132 +2^128 +2^124 -2^120 -2^140 -2^136 -2^132 +2^128 +2^124 +2^120 -2^140 -2^136 +2^132 -2^128 -2^124 -2^120 -2^140 -2^136 +2^132 -2^128 -2^124 +2^120 -2^140 -2^136 +2^132 -2^128 +2^124 -2^120 -2^140 -2^136 +2^132 -2^128 +2^124 +2^120 -2^140 -2^136 +2^132 +2^128 -2^124 -2^120 -2^140 -2^136 +2^132 +2^128 -2^124 +2^120 -2^140 -2^136 +2^132 +2^128 +2^124 -2^120 -2^140 -2^136 +2^132 +2^128 +2^124 +2^120 -2^140 +2^136 -2^132 -2^128 -2^124 -2^120 -2^140 +2^136 -2^132 -2^128 -2^124 +2^120 -2^140 +2^136 -2^132 -2^128 +2^124 -2^120 -2^140 +2^136 -2^132 -2^128 +2^124 +2^120 -2^140 +2^136 -2^132 +2^128 -2^124 -2^120 -2^140 +2^136 -2^132 +2^128 -2^124 +2^120 -2^140 +2^136 -2^132 +2^128 +2^124 -2^120 -2^140 +2^136 -2^132 +2^128 +2^124 +2^120 -2^140 +2^136 +2^132 -2^128 -2^124 -2^120 -2^140 +2^136 +2^132 -2^128 -2^124 +2^120 -2^140 +2^136 +2^132 -2^128 +2^124 -2^120 -2^140 +2^136 +2^132 -2^128 +2^124 +2^120 -2^140 +2^136 +2^132 +2^128 -2^124 -2^120 -2^140 +2^136 +2^132 +2^128 -2^124 +2^120 -2^140 +2^136 +2^132 +2^128 +2^124 -2^120 -2^140 +2^136 +2^132 +2^128 +2^124 +2^120
block 6 -2^164 -2^160 -2^156 -2^152 -2^148 -2^144 -2^164 -2^160 -2^156 -2^152 -2^148 +2^144 -2^164 -2^160 -2^156 -2^152 +2^148 -2^144 -2^164 -2^160 -2^156 -2^152 +2^148 +2^144 -2^164 -2^160 -2^156 +2^152 -2^148 -2^144 -2^164 -2^160 -2^156 +2^152 -2^148 +2^144 -2^164 -2^160 -2^156 +2^152 +2^148 -2^144 -2^164 -2^160 -2^156 +2^152 +2^148 +2^144 -2^164 -2^160 +2^156 -2^152 -2^148 -2^144 -2^164 -2^160 +2^156 -2^152 -2^148 +2^144 -2^164 -2^160 +2^156 -2^152 +2^148 -2^144 -2^164 -2^160 +2^156 -2^152 +2^148 +2^144 -2^164 -2^160 +2^156 +2^152 -2^148 -2^144 -2^164 -2^160 +2^156 +2^152 -2^148 +2^144 -2^164 -2^160 +2^156 +2^152 +2^148 -2^144 -2^164 -2^160 +2^156 +2^152 +2^148 +2^144 -2^164 +2^160 -2^156 -2^152 -2^148 -2^144 -2^164 +2^160 -2^156 -2^152 -2^148 +2^144 -2^164 +2^160 -2^156 -2^152 +2^148 -2^144 -2^164 +2^160 -2^156 -2^152 +2^148 +2^144 -2^164 +2^160 -2^156 +2^152 -2^148 -2^144 -2^164 +2^160 -2^156 +2^152 -2^148 +2^144 -2^164 +2^160 -2^156 +2^152 +2^148 -2^144 -2^164 +2^160 -2^156 +2^152 +2^148 +2^144 -2^164 +2^160 +2^156 -2^152 -2^148 -2^144 -2^164 +2^160 +2^156 -2^152 -2^148 +2^144 -2^164 +2^160 +2^156 -2^152 +2^148 -2^144 -2^164 +2^160 +2^156 -2^152 +2^148 +2^144 -2^164 +2^160 +2^156 +2^152 -2^148 -2^144 -2^164 +2^160 +2^156 +2^152 -2^148 +2^144 -2^164 +2^160 +2^156 +2^152 +2^148 -2^144 -2^164 +2^160 +2^156 +2^152 +2^148 +2^144
block 7 -2^188 -2^184 -2^180 -2^176 -2^172 -2^168 -2^188 -2^184 -2^180 -2^176 -2^172 +2^168 -2^188 -2^184 -2^180 -2^176 +2^172 -2^168 -2^188 -2^184 -2^180 -2^176 +2^172 +2^168 -2^188 -2^184 -2^180 +2^176 -2^172 -2^168 -2^188 -2^184 -2^180 +2^176 -2^172 +2^168 -2^188 -2^184 -2^180 +2^176 +2^172 -2^168 -2^188 -2^184 -2^180 +2^176 +2^172 +2^168 -2^188 -2^184 +2^180 -2^176 -2^172 -2^168 -2^188 -2^184 +2^180 -2^176 -2^172 +2^168 -2^188 -2^184 +2^180 -2^176 +2^172 -2^168 -2^188 -2^184 +2^180 -2^176 +2^172 +2^168 -2^188 -2^184 +2^180 +2^176 -2^172 -2^168 -2^188 -2^184 +2^180 +2^176 -2^172 +2^168 -2^188 -2^184 +2^180 +2^176 +2^172 -2^168 -2^188 -2^184 +2^180 +2^176 +2^172 +2^168 -2^188 +2^184 -2^180 -2^176 -2^172 -2^168 -2^188 +2^184 -2^180 -2^176 -2^172 +2^168 -2^188 +2^184 -2^180 -2^176 +2^172 -2^168 -2^188 +2^184 -2^180 -2^176 +2^172 +2^168 -2^188 +2^184 -2^180 +2^176 -2^172 -2^168 -2^188 +2^184 -2^180 +2^176 -2^172 +2^168 -2^188 +2^184 -2^180 +2^176 +2^172 -2^168 -2^188 +2^184 -2^180 +2^176 +2^172 +2^168 -2^188 +2^184 +2^180 -2^176 -2^172 -2^168 -2^188 +2^184 +2^180 -2^176 -2^172 +2^168 -2^188 +2^184 +2^180 -2^176 +2^172 -2^168 -2^188 +2^184 +2^180 -2^176 +2^172 +2^168 -2^188 +2^184 +2^180 +2^176 -2^172 -2^168 -2^188 +2^184 +2^180 +2^176 -2^172 +2^168 -2^188 +2^184 +2^180 +2^176 +2^172 -2^168 -2^188 +2^184 +2^180 +2^176 +2^172 +2^168
block 8 -2^212 -2^208 -2^204 -2^200 -2^196 -2^192 -2^212 -2^208 -2^204 -2^200 -2^196 +2^192 -2^212 -2^208 -2^204 -2^200 +2^196 -2^192 -2^212 -2^208 -2^204 -2^200 +2^196 +2^192 -2^212 -2^208 -2^204 +2^200 -2^196 -2^192 -2^212 -2^208 -2^204 +2^200 -2^196 +2^192 -2^212 -2^208 -2^204 +2^200 +2^196 -2^192 -2^212 -2^208 -2^204 +2^200 +2^196 +2^192 -2^212 -2^208 +2^204 -2^200 -2^196 -2^192 -2^212 -2^208 +2^204 -2^200 -2^196 +2^192 -2^212 -2^208 +2^204 -2^200 +2^196 -2^192 -2^212 -2^208 +2^204 -2^200 +2^196 +2^192 -2^212 -2^208 +2^204 +2^200 -2^196 -2^192 -2^212 -2^208 +2^204 +2^200 -2^196 +2^192 -2^212 -2^208 +2^204 +2^200 +2^196 -2^192 -2^212 -2^208 +2^204 +2^200 +2^196 +2^192 -2^212 +2^208 -2^204 -2^200 -2^196 -2^192 -2^212 +2^208 -2^204 -2^200 -2^196 +2^192 -2^212 +2^208 -2^204 -2^200 +2^196 -2^192 -2^212 +2^208 -2^204 -2^200 +2^196 +2^192 -2^212 +2^208 -2^204 +2^200 -2^196 -2^192 -2^212 +2^208 -2^204 +2^200 -2^196 +2^192 -2^212 +2^208 -2^204 +2^200 +2^196 -2^192 -2^212 +2^208 -2^204 +2^200 +2^196 +2^192 -2^212 +2^208 +2^204 -2^200 -2^196 -2^192 -2^212 +2^208 +2^204 -2^200 -2^196 +2^192 -2^212 +2^208 +2^204 -2^200 +2^196 -2^192 -2^212 +2^208 +2^204 -2^200 +2^196 +2^192 -2^212 +2^208 +2^204 +2^200 -2^196 -2^192 -2^212 +2^208 +2^204 +2^200 -2^196 +2^192 -2^212 +2^208 +2^204 +2^200 +2^196 -2^192 -2^212 +2^208 +2^204 +2^200 +2^196 +2^192
block 9 -2^236 -2^232 -2^228 -2^224 -2^220 -2^216 -2^236 -2^232 -2^228 -2^224 -2^220 +2^216 -2^236 -2^232 -2^228 -2^224 +2^220 -2^216 -2^236 -2^232 -2^228 -2^224 +2^220 +2^216 -2^236 -2^232 -2^228 +2^224 -2^220 -2^216 -2^236 -2^232 -2^228 +2^224 -2^220 +2^216 -2^236 -2^232 -2^228 +2^224 +2^220 -2^216 -2^236 -2^232 -2^228 +2^224 +2^220 +2^216 -2^236 -2^232 +2^228 -2^224 -2^220 -2^216 -2^236 -2^232 +2^228 -2^224 -2^220 +2^216 -2^236 -2^232 +2^228 -2^224 +2^220 -2^216 -2^236 -2^232 +2^228 -2^224 +2^220 +2^216 -2^236 -2^232 +2^228 +2^224 -2^220 -2^216 -2^236 -2^232 +2^228 +2^224 -2^220 +2^216 -2^236 -2^232 +2^228 +2^224 +2^220 -2^216 -2^236 -2^232 +2^228 +2^224 +2^220 +2^216 -2^236 +2^232 -2^228 -2^224 -2^220 -2^216 -2^236 +2^232 -2^228 -2^224 -2^220 +2^216 -2^236 +2^232 -2^228 -2^224 +2^220 -2^216 -2^236 +2^232 -2^228 -2^224 +2^220 +2^216 -2^236 +2^232 -2^228 +2^224 -2^220 -2^216 -2^236 +2^232 -2^228 +2^224 -2^220 +2^216 -2^236 +2^232 -2^228 +2^224 +2^220 -2^216 -2^236 +2^232 -2^228 +2^224 +2^220 +2^216 -2^236 +2^232 +2^228 -2^224 -2^220 -2^216 -2^236 +2^232 +2^228 -2^224 -2^220 +2^216 -2^236 +2^232 +2^228 -2^224 +2^220 -2^216 -2^236 +2^232 +2^228 -2^224 +2^220 +2^216 -2^236 +2^232 +2^228 +2^224 -2^220 -2^216 -2^236 +2^232 +2^228 +2^224 -2^220 +2^216 -2^236 +2^232 +2^228 +2^224 +2^220 -2^216 -2^236 +2^232 +2^228 +2^224 +2^220 +2^216
block 10 -2^260 -2^256 -2^252 -2^248 -2^244 -2^240 -2^260 -2^256 -2^252 -2^248 -2^244 +2^240 -2^260 -2^256 -2^252 -2^248 +2^244 -2^240 -2^260 -2^256 -2^252 -2^248 +2^244 +2^240 -2^260 -2^256 -2^252 +2^248 -2^244 -2^240 -2^260 -2^256 -2^252 +2^248 -2^244 +2^240 -2^260 -2^256 -2^252 +2^248 +2^244 -2^240 -2^260 -2^256 -2^252 +2^248 +2^244 +2^240 -2^260 -2^256 +2^252 -2^248 -2^244 -2^240 -2^260 -2^256 +2^252 -2^248 -2^244 +2^240 -2^260 -2^256 +2^252 -2^248 +2^244 -2^240 -2^260 -2^256 +2^252 -2^248 +2^244 +2^240 -2^260 -2^256 +2^252 +2^248 -2^244 -2^240 -2^260 -2^256 +2^252 +2^248 -2^244 +2^240 -2^260 -2^256 +2^252 +2^248 +2^244 -2^240 -2^260 -2^256 +2^252 +2^248 +2^244 +2^240 -2^260 +2^256 -2^252 -2^248 -2^244 -2^240 -2^260 +2^256 -2^252 -2^248 -2^244 +2^240 -2^260 +2^256 -2^252 -2^248 +2^244 -2^240 -2^260 +2^256 -2^252 -2^248 +2^244 +2^240 -2^260 +2^256 -2^252 +2^248 -2^244 -2^240 -2^260 +2^256 -2^252 +2^248 -2^244 +2^240 -2^260 +2^256 -2^252 +2^248 +2^244 -2^240 -2^260 +2^256 -2^252 +2^248 +2^244 +2^240 -2^260 +2^256 +2^252 -2^248 -2^244 -2^240 -2^260 +2^256 +2^252 -2^248 -2^244 +2^240 -2^260 +2^256 +2^252 -2^248 +2^244 -2^240 -2^260 +2^256 +2^252 -2^248 +2^244 +2^240 -2^260 +2^256 +2^252 +2^248 -2^244 -2^240 -2^260 +2^256 +2^252 +2^248 -2^244 +2^240 -2^260 +2^256 +2^252 +2^248 +2^244 -2^240 -2^260 +2^256 +2^252 +2^248 +2^244 +2^240