monero-double-spend-races.md

Monero double spend attack success probability

Below is a table of the attack success probability for different values of the adversary's hashpower share and the number of blocks that the victim waits for "full confirmation" of a transaction.

The computations are based on Theorem 1 of Grunspan & Perez-Marco (2018). "Double spend races." It is assumed that the attacker has a one-block advantage at the start of the attack.

Theorem 1 is

---

Let $0<q<1/2$, respectively $p=1-q$, be the relative hash power of the group of the attackers, respectively of honest miners. After $z$ blocks have been validated by the honest miners, the probability of success of the attackers is

$$P(z)=I_{4pq}(z,1/2)$$

where $I_{x}(a,b)$ is the regularized incomplete beta function

---

See https://miningpoolstats.stream/monero for current hashpower share of the major mining pools.

Columns are the hashpower share of the adversary. Rows are the number of mined blocks that the victim waits before considering a transaction "confirmed".

	0.05	0.1	0.2	0.3	0.4	0.45
1	10.00000	20.00000	40.00000	60.00000	80.00000	90.00000
2	1.45000	5.60000	20.80000	43.20000	70.40000	85.05000
3	0.23162	1.71200	11.58400	32.61600	63.48800	81.37463
4	0.03872	0.54560	6.66880	25.20720	57.95840	78.34244
5	0.00664	0.17818	3.91629	19.76173	53.31354	75.71581
6	0.00116	0.05914	2.33084	15.64496	49.30037	73.37548
7	0.00021	0.01986	1.40071	12.47504	45.76879	71.25164
8	0.00004	0.00672	0.84795	10.00251	42.62064	69.29921
9	0.00001	0.00229	0.51629	8.05539	39.78730	67.48712
10	0.00000	0.00079	0.31582	6.51067	37.21840	65.79282
11	0.00000	0.00027	0.19394	5.27799	34.87557	64.19932
12	0.00000	0.00009	0.11948	4.28960	32.72869	62.69347
13	0.00000	0.00003	0.07381	3.49395	30.75355	61.26479
14	0.00000	0.00001	0.04571	2.85131	28.93035	59.90480
15	0.00000	0.00000	0.02836	2.33077	27.24259	58.60650
16	0.00000	0.00000	0.01763	1.90809	25.67635	57.36402
17	0.00000	0.00000	0.01098	1.56413	24.21974	56.17240
18	0.00000	0.00000	0.00685	1.28371	22.86252	55.02740
19	0.00000	0.00000	0.00427	1.05470	21.59579	53.92534
20	0.00000	0.00000	0.00267	0.86739	20.41173	52.86301
21	0.00000	0.00000	0.00167	0.71398	19.30344	51.83759
22	0.00000	0.00000	0.00105	0.58819	18.26482	50.84660
23	0.00000	0.00000	0.00066	0.48492	17.29041	49.88782
24	0.00000	0.00000	0.00041	0.40007	16.37531	48.95926
25	0.00000	0.00000	0.00026	0.33027	15.51511	48.05913
26	0.00000	0.00000	0.00016	0.27282	14.70584	47.18583
27	0.00000	0.00000	0.00010	0.22548	13.94388	46.33789
28	0.00000	0.00000	0.00006	0.18646	13.22594	45.51397
29	0.00000	0.00000	0.00004	0.15427	12.54903	44.71286
30	0.00000	0.00000	0.00003	0.12769	11.91040	43.93344

R code to reproduce the table:

# install.packages("zipfR")
# install.packages("knitr")

# Rows: 1 to 30 number of blocks to wait
# Column: hashpower share: 5%, 10%, 20%, 30%, 40%, 45%

# Based on Theorem 1 of Grunspan & Perez-Marco (2018). "Double spend races."

n.blocks <- 30
hashpower.share <- c(0.05, 0.10, 0.20, 0.30, 0.40, 0.45)

blocks <- 1:n.blocks
results <- matrix(0, nrow = n.blocks, ncol = length(hashpower.share))

for (i in seq_along(hashpower.share)) {
  q = hashpower.share[i]
  p = 1 - q
  results[, i] <- zipfR::Rbeta(x = 4*p*q, a = blocks, b = 1/2)
}

results <- 100 * results # Convert to percentage
rownames(results) <- as.character(blocks)

comparison.block <- 10
results.comparison <- results

for (j in seq_len(ncol(results.comparison))) {
  divisor <- results[blocks == comparison.block, j]
  results.comparison[, j] <- results.comparison[, j] / divisor
}


knitr::kable(results, format = "pipe", row.names = TRUE,
  col.names = hashpower.share, digits = 5)