Randomness report — Verity draw engine

Generated 4 November 2026, 21:12 GMT by php artisan verity:randomness-report (2.8 min).

Overall: PASS — NIST SP 800-22: PASS · chi-square simulations: PASS

What was tested

Verity picks winners with rejection sampling from an HMAC-SHA256 stream (app/Verity/Picker.php): it reads the first 48 bits of HMAC(draw_seed, "pick|{position}|{counter}"). Instant-win tables use the same sampler keyed from the server seed (InstantWinAllocator). Two kinds of evidence:

1. The raw bits the sampler consumes, run through the full NIST SP 800-22 Rev. 1a battery (15 tests, pure-PHP implementation in tools/randomness/, checked against the spec’s worked examples in backend/tests/Unit/NistTest.php).
2. The outcomes: simulated draws and instant-win tables through the real engine (DrawEngine::run, InstantWinAllocator::allocate), with chi-square goodness-of-fit tests that every ticket is equally likely.

Reproduce exactly: php artisan verity:randomness-report --seed=110f2ed8aa07ad44df88f98ac91bb693a6edae5091ba67a5a524249f983b88a7 (same sizes as below).

1. NIST SP 800-22 on the sampler’s bits

• Sample: 104,857,600 bits = 100 sequences × 1,048,576 bits, the concatenated 48-bit values Picker reads (pick|i|0 for i = 0, 1, 2 …) under a draw seed derived from the master seed.

• Parameters: NIST defaults for 1 Mbit sequences — block frequency M = 128; non-overlapping templates m = 9 (all 148 aperiodic templates), overlapping m = 9; linear complexity M = 500; serial m = 16; approximate entropy m = 10; universal L = 7, Q = 1280.

• Assessment (NIST §4.2): a stream passes if the proportion of sequences with p ≥ 0.01 is at least p̂ − 3√(p̂(1−p̂)/m) (p̂ = 0.99) and its p-values are uniform (10-bin chi-square, P ≥ 0.0001). Random excursions tests only apply to sequences with ≥ 500 cycles, so they run on fewer sequences.

• Battery verdict: there are 188 p-value streams (148 of them templates), so a few chance failures are expected even from a perfect generator — NIST §4.2.1 says to judge the battery as a whole. Expected chance failures: 3.6; allowed (99.9% Poisson bound): 11; observed: 3 → PASS.

Test	Variant	Sequences	Passing p ≥ 0.01	Uniformity P	Result
Frequency	—	100	100/100 (min 96.02)	0.3041	PASS
BlockFrequency	—	100	99/100 (min 96.02)	0.8832	PASS
CumulativeSums	forward	100	100/100 (min 96.02)	0.1296	PASS
CumulativeSums	backward	100	100/100 (min 96.02)	0.7598	PASS
Runs	—	100	100/100 (min 96.02)	0.0401	PASS
LongestRun	—	100	99/100 (min 96.02)	0.7981	PASS
Rank	—	100	100/100 (min 96.02)	0.3669	PASS
FFT	—	100	99/100 (min 96.02)	0.6787	PASS
NonOverlappingTemplate	148 templates	100	147/148 templates pass (worst 96/100; 2.7 failures expected by chance)	—	PASS
OverlappingTemplate	—	100	98/100 (min 96.02)	0.5341	PASS
Universal	—	100	100/100 (min 96.02)	0.6787	PASS
ApproximateEntropy	—	100	99/100 (min 96.02)	0.4190	PASS
RandomExcursions	x = -4	60	60/60 (min 57.09)	0.0079	PASS
RandomExcursions	x = -3	60	56/60 (min 57.09)	0.7399	FAIL
RandomExcursions	x = -2	60	57/60 (min 57.09)	0.5341	FAIL
RandomExcursions	x = -1	60	60/60 (min 57.09)	0.0392	PASS
RandomExcursions	x = 1	60	59/60 (min 57.09)	0.5341	PASS
RandomExcursions	x = 2	60	60/60 (min 57.09)	0.8881	PASS
RandomExcursions	x = 3	60	60/60 (min 57.09)	0.4071	PASS
RandomExcursions	x = 4	60	60/60 (min 57.09)	0.1952	PASS
RandomExcursionsVariant	x = -9	60	60/60 (min 57.09)	0.1783	PASS
RandomExcursionsVariant	x = -8	60	59/60 (min 57.09)	0.2133	PASS
RandomExcursionsVariant	x = -7	60	59/60 (min 57.09)	0.1347	PASS
RandomExcursionsVariant	x = -6	60	59/60 (min 57.09)	0.1005	PASS
RandomExcursionsVariant	x = -5	60	59/60 (min 57.09)	0.9114	PASS
RandomExcursionsVariant	x = -4	60	60/60 (min 57.09)	0.8343	PASS
RandomExcursionsVariant	x = -3	60	59/60 (min 57.09)	0.2536	PASS
RandomExcursionsVariant	x = -2	60	60/60 (min 57.09)	0.2993	PASS
RandomExcursionsVariant	x = -1	60	60/60 (min 57.09)	0.8623	PASS
RandomExcursionsVariant	x = 1	60	59/60 (min 57.09)	0.9496	PASS
RandomExcursionsVariant	x = 2	60	59/60 (min 57.09)	0.7061	PASS
RandomExcursionsVariant	x = 3	60	60/60 (min 57.09)	0.0669	PASS
RandomExcursionsVariant	x = 4	60	60/60 (min 57.09)	0.1481	PASS
RandomExcursionsVariant	x = 5	60	60/60 (min 57.09)	0.6025	PASS
RandomExcursionsVariant	x = 6	60	60/60 (min 57.09)	0.6371	PASS
RandomExcursionsVariant	x = 7	60	60/60 (min 57.09)	0.9643	PASS
RandomExcursionsVariant	x = 8	60	60/60 (min 57.09)	0.9643	PASS
RandomExcursionsVariant	x = 9	60	60/60 (min 57.09)	0.5341	PASS
Serial	∇ψ²	100	99/100 (min 96.02)	0.5955	PASS
Serial	∇²ψ²	100	99/100 (min 96.02)	0.2622	PASS
LinearComplexity	—	100	97/100 (min 96.02)	0.2493	PASS

2. Chi-square tests on outcomes

• Main draws: 1,000,000 simulated draws, each over 1,000 entries with its own seed and beacon value, 3 winners each (without replacement). Expected per ticket per pick: 1,000. Draws with a repeated winner: 0.
• Instant-win tables: 1,000,000 simulated tables of 1,000 tickets with 1 + 3 + 6 prizes. Tables with a repeated ticket: 0.
• Pass if p ≥ 0.001 (a family of ten tests; at 0.01 one chance failure in ten runs would be expected).

Test	Bins	χ²	df	p	Result
Main draw: pick 1 by ticket position	1000	975.1	999	0.7000	PASS
Main draw: pick 2 by ticket position	1000	913.9	999	0.9741	PASS
Main draw: pick 3 by ticket position	1000	980.6	999	0.6549	PASS
Main draw: all picks by ticket position	1000	972.5	999	0.7198	PASS
Main draw: pick 1 × pick 2 (10 × 10 deciles; pick 2 is drawn without replacement, so the diagonal expects slightly fewer)	100	108.6	99	0.2397	PASS
Instant-win table: top prize (1 per table) by ticket number	1000	1,069.1	999	0.0607	PASS
Instant-win table: middle prize (3 per table) by ticket number	1000	987.0	999	0.6008	PASS
Instant-win table: low prize (6 per table) by ticket number	1000	1,042.8	999	0.1635	PASS
Instant-win table: all winning numbers by decile	10	10.0	9	0.3467	PASS

Reading this

A p-value is the chance of a result at least this uneven if the engine were perfectly fair. Under a fair engine p-values are spread evenly between 0 and 1, so a single low value is expected now and then; NIST’s assessment looks at proportions and uniformity across many sequences for that reason. Nothing here can prove randomness — no test can — but a biased sampler (for example x mod n without rejection, or a weak generator) fails these tests quickly. For the full method see NIST SP 800-22 Rev. 1a and docs/SECURITY.md.