1# Tables Pre-calculation
 2The most time consuming part of a ZKSnark proof calculation is the scalar multiplication of elliptic curve points. Direct mechanism accumulates each multiplication. However, prover only needs the total accumulation.
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 4There are two potential improvements to the naive approach:
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 61. Apply Strauss-Shamir method (https://stackoverflow.com/questions/50993471/ec-scalar-multiplication-with-strauss-shamir-method).
 72. Leave the doubling operation for the last step
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 9Both options can be combined. 
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11In the following table, we show the results of using the naive method, Srauss-Shamir and Strauss-Shamir + No doubling. These last two options are repeated for different table grouping order.
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13There are 50000 G1 Elliptical Curve Points, and the scalars are 254 bits (BN256 curve). 
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15There may be some concern on the additional size of the tables since they need to be loaded into a smartphone during the proof, and the time required to load these tables may exceed the benefits. If this is a problem, another althernative is to compute the tables during the proof itself. Depending on the Group Size, timing may be better than the naive approach.
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18| Algorithm (G1) | GS 2 | GS 3 | GS 4 | GS 5 | GS 6 | GS 7 | GS 8 | GS 9 | 
19|---|---|---|--|---|---|---|---|---|
20| Naive | 6.63s | - | - | - | - | - |  - | - |
21| Strauss |  13.16s | 9.03s | 6.95s | 5.61s | 4.91s | 4.26s | 3.88s | 3.54 s |
22| Strauss + Table Computation | 16.13s | 11.32s | 8.47s | 7.10s | 6.2s | 5.94s | 6.01s | 6.69s |
23| No Doubling | 3.74s | 3.00s | 2.38s | 1.96s | 1.79s | 1.54s | 1.50s | 1.44s|
24| No Doubling + Table Computation  | 6.83s | 5.1s | 4.16s | 3.52s| 3.22s | 3.21s | 3.57s | 4.56s |
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26There are 5000 G2 Elliptical Curve Points, and the scalars are 254 bits (BN256 curve). 
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28| Algorithm (G2) | GS 2 | GS 3 | GS 4 | GS 5 | GS 6 | GS 7 | GS 8 | GS 9 | 
29|---|---|---|--|---|---|---|---|---|
30| Naive | 3.55s | | | | | | | |
31| Strauss |  3.55s | 2.54s | 1.96s | 1.58s |  1.38s | 1.20s | 1.03s | 937ms |
32| Strauss + Table Computation | 3.59s | 2.58s | 2.04s | 1.71s | 1.51s | 1.46s | 1.51s | 1.82s |
33| No Doubling | 1.49s | 1.16s | 952ms | 719ms | 661ms | 548ms | 506ms| 444ms |
34| No Doubling + Table Computation  | 1.55s |  1.21s | 984ms | 841ms | 826ms | 847ms | 1.03s | 1.39s |
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36| GS | Extra Disk Space per Constraint (G1)|
37|----|--------|
38| 2  |  64 B  |
39| 3  |  106 B |
40| 4  |  192 B |
41| 5  |  346 B |
42| 6  |  618 B |
43| 7  |  1106 B |
44| 8  |  1984 B |
45| 9  |  3577 B |
46| N  |  2^(N+6)/N - 64 B |
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48Extra disk space per constraint in G2 is twice the requirements for G1
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