10 live ML-DSA-65 keypairs · 4,096 spheres · 16 Dilithium bands · FIPS 204 verified
Per-keypair statistical analysis combining CRYSTALS-Dilithium parameter verification with 16-layer ±δ walk entropy evaluation. All keypairs pass NIST FIPS 204 requirements — Qrypto Trust applies a stricter issuance standard below.
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 30,194 | 8,367,857 | 4,200,931 | 2,511,733 | 17.375 | PASS ✓ |
| L02 | 2,265 | 8,359,913 | 4,151,779 | 2,481,324 | 27.250 | PASS ✓ |
| L03 | 30,194 | 8,347,183 | 4,176,355 | 2,428,854 | 21.250 | PASS ✓ |
| L04 | 2,265 | 8,326,576 | 4,139,491 | 2,439,221 | 22.625 | PASS ✓ |
| L05 | 52,498 | 8,348,602 | 4,151,779 | 2,406,715 | 10.875 | PASS ✓ |
| L06 | 2,265 | 8,300,243 | 4,143,587 | 2,358,462 | 14.125 | PASS ✓ |
| L07 | 52,498 | 8,358,907 | 4,172,259 | 2,344,636 | 18.000 | PASS ✓ |
| L08 | 2,265 | 8,326,576 | 4,155,875 | 2,306,732 | 21.250 | PASS ✓ |
| L09 | 222,576 | 8,358,907 | 4,159,971 | 2,321,982 | 26.375 | PASS ✓ |
| L10 | 93,662 | 8,300,243 | 4,119,011 | 2,286,652 | 28.250 | PASS ✓ |
| L11 | 30,194 | 8,347,183 | 4,114,915 | 2,311,570 | 19.250 | PASS ✓ |
| L12 | 23,121 | 8,318,899 | 4,176,355 | 2,310,214 | 28.500 | PASS ✓ |
| L13 | 62,048 | 8,347,183 | 4,192,739 | 2,327,981 | 25.625 | PASS ✓ |
| L14 | 40,075 | 8,355,682 | 4,164,067 | 2,340,532 | 23.875 | PASS ✓ |
| L15 | 62,048 | 8,336,308 | 4,123,107 | 2,358,480 | 27.125 | PASS ✓ |
| L16 | 1,732 | 8,359,913 | 4,180,451 | 2,358,984 | 23.750 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 15 | 16 | -1 |
| 2 | Orange Red | 22 | 16 | +6 |
| 3 | Orange | 12 | 16 | -4 |
| 4 | Gold | 19 | 16 | +3 |
| 5 | Yellow | 17 | 16 | +1 |
| 6 | Green Yellow | 14 | 16 | -2 |
| 7 | Chartreuse | 15 | 16 | -1 |
| 8 | Lime | 7 | 16 | -9 |
| 9 | Spring Green | 21 | 16 | +5 |
| 10 | Cyan | 17 | 16 | +1 |
| 11 | Deep Sky Blue | 16 | 16 | +0 |
| 12 | Blue | 9 | 16 | -7 |
| 13 | Violet | 15 | 16 | -1 |
| 14 | Magenta | 23 | 16 | +7 |
| 15 | Deep Pink | 16 | 16 | +0 |
| 16 | Hot Pink | 18 | 16 | +2 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 1,732 |
| Max value | 8,367,857 |
| Mean | 4,157,667 |
| Std deviation | 2,369,377 |
| Shannon entropy | 3.9743 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 22.219 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9475 | 98.7% | 17.375 | −13.20 | PASS ✓ |
| L02 | 3.9248 | 98.1% | 27.250 | −3.33 | PASS ✓ |
| L03 | 3.9444 | 98.6% | 21.250 | −9.33 | PASS ✓ |
| L04 | 3.9308 | 98.3% | 22.625 | −7.95 | PASS ✓ |
| L05 | 3.9701 | 99.3% | 10.875 | −19.70 | PASS ✓ |
| L06 | 3.9589 | 99.0% | 14.125 | −16.45 | PASS ✓ |
| L07 | 3.9467 | 98.7% | 18.000 | −12.58 | PASS ✓ |
| L08 | 3.9370 | 98.4% | 21.250 | −9.33 | PASS ✓ |
| L09 | 3.9211 | 98.0% | 26.375 | −4.20 | PASS ✓ |
| L10 | 3.9145 | 97.9% | 28.250 | −2.33 | PASS ✓ |
| L11 | 3.9404 | 98.5% | 19.250 | −11.33 | PASS ✓ |
| L12 | 3.9186 | 98.0% | 28.500 | −2.08 | PASS ✓ |
| L13 | 3.9230 | 98.1% | 25.625 | −4.95 | PASS ✓ |
| L14 | 3.9328 | 98.3% | 23.875 | −6.70 | PASS ✓ |
| L15 | 3.9201 | 98.0% | 27.125 | −3.45 | PASS ✓ |
| L16 | 3.9263 | 98.2% | 23.750 | −6.83 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9348 bits (98.4% of max 4.0) |
| Average χ² | 22.219 |
| Layers passing χ² test | 16 / 16 (100%) |
| Marginal layers (30.578–45) | 0 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.9145 bits (L10) |
| Highest layer χ² | 28.500 (L12) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9348 bits (98.4% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9743 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 22.219 PASS ✓ |
| Min value | 1,732 |
| Max value | 8,367,857 |
| Mean | 4,157,667 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 3,761 | 8,371,489 | 4,061,449 | 2,366,939 | 9.500 | PASS ✓ |
| L02 | 62,223 | 8,373,056 | 4,081,929 | 2,344,334 | 21.000 | PASS ✓ |
| L03 | 9,731 | 8,273,763 | 4,094,217 | 2,343,746 | 11.250 | PASS ✓ |
| L04 | 62,223 | 8,373,056 | 4,086,025 | 2,307,740 | 14.750 | PASS ✓ |
| L05 | 3,761 | 8,225,246 | 4,073,737 | 2,343,918 | 16.250 | PASS ✓ |
| L06 | 91,291 | 8,321,692 | 4,032,777 | 2,369,774 | 14.750 | PASS ✓ |
| L07 | 3,761 | 8,322,274 | 4,073,737 | 2,390,376 | 8.250 | PASS ✓ |
| L08 | 78,206 | 8,348,686 | 4,122,889 | 2,384,878 | 21.625 | PASS ✓ |
| L09 | 3,761 | 8,371,489 | 4,077,833 | 2,331,872 | 12.500 | PASS ✓ |
| L10 | 17,301 | 8,321,692 | 4,081,929 | 2,315,365 | 24.625 | PASS ✓ |
| L11 | 3,761 | 8,322,274 | 4,204,809 | 2,349,195 | 19.500 | PASS ✓ |
| L12 | 78,206 | 8,310,649 | 4,139,273 | 2,384,540 | 17.250 | PASS ✓ |
| L13 | 9,731 | 8,283,717 | 4,135,177 | 2,364,762 | 10.750 | PASS ✓ |
| L14 | 78,206 | 8,310,649 | 4,213,001 | 2,297,528 | 17.750 | PASS ✓ |
| L15 | 52,027 | 8,283,717 | 4,176,137 | 2,332,934 | 12.875 | PASS ✓ |
| L16 | 78,206 | 8,373,056 | 4,131,081 | 2,306,696 | 20.000 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 20 | 16 | +4 |
| 2 | Orange Red | 14 | 16 | -2 |
| 3 | Orange | 12 | 16 | -4 |
| 4 | Gold | 22 | 16 | +6 |
| 5 | Yellow | 19 | 16 | +3 |
| 6 | Green Yellow | 14 | 16 | -2 |
| 7 | Chartreuse | 14 | 16 | -2 |
| 8 | Lime | 14 | 16 | -2 |
| 9 | Spring Green | 18 | 16 | +2 |
| 10 | Cyan | 17 | 16 | +1 |
| 11 | Deep Sky Blue | 18 | 16 | +2 |
| 12 | Blue | 16 | 16 | +0 |
| 13 | Violet | 19 | 16 | +3 |
| 14 | Magenta | 15 | 16 | -1 |
| 15 | Deep Pink | 14 | 16 | -2 |
| 16 | Hot Pink | 10 | 16 | -6 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 3,761 |
| Max value | 8,373,056 |
| Mean | 4,111,625 |
| Std deviation | 2,346,624 |
| Shannon entropy | 3.9816 bits (99.5% of max 4.0) |
| Avg χ² (16 layers) | 15.789 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9729 | 99.3% | 9.500 | −21.08 | PASS ✓ |
| L02 | 3.9400 | 98.5% | 21.000 | −9.58 | PASS ✓ |
| L03 | 3.9674 | 99.2% | 11.250 | −19.33 | PASS ✓ |
| L04 | 3.9585 | 99.0% | 14.750 | −15.83 | PASS ✓ |
| L05 | 3.9501 | 98.8% | 16.250 | −14.33 | PASS ✓ |
| L06 | 3.9583 | 99.0% | 14.750 | −15.83 | PASS ✓ |
| L07 | 3.9760 | 99.4% | 8.250 | −22.33 | PASS ✓ |
| L08 | 3.9388 | 98.5% | 21.625 | −8.95 | PASS ✓ |
| L09 | 3.9636 | 99.1% | 12.500 | −18.08 | PASS ✓ |
| L10 | 3.9259 | 98.1% | 24.625 | −5.95 | PASS ✓ |
| L11 | 3.9442 | 98.6% | 19.500 | −11.08 | PASS ✓ |
| L12 | 3.9516 | 98.8% | 17.250 | −13.33 | PASS ✓ |
| L13 | 3.9685 | 99.2% | 10.750 | −19.83 | PASS ✓ |
| L14 | 3.9452 | 98.6% | 17.750 | −12.83 | PASS ✓ |
| L15 | 3.9612 | 99.0% | 12.875 | −17.70 | PASS ✓ |
| L16 | 3.9386 | 98.5% | 20.000 | −10.58 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9538 bits (98.8% of max 4.0) |
| Average χ² | 15.789 |
| Layers passing χ² test | 16 / 16 (100%) |
| Marginal layers (30.578–45) | 0 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.9259 bits (L10) |
| Highest layer χ² | 24.625 (L10) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9538 bits (98.8% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9816 bits (99.5% of max 4.0) |
| Avg χ² (16 layers) | 15.789 PASS ✓ |
| Min value | 3,761 |
| Max value | 8,373,056 |
| Mean | 4,111,625 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 12,994 | 8,322,995 | 4,198,047 | 2,357,708 | 11.250 | PASS ✓ |
| L02 | 64,425 | 8,253,482 | 4,239,007 | 2,282,275 | 28.250 | PASS ✓ |
| L03 | 30,180 | 8,322,995 | 4,292,255 | 2,301,343 | 20.000 | PASS ✓ |
| L04 | 43,062 | 8,318,284 | 4,267,679 | 2,336,658 | 26.250 | PASS ✓ |
| L05 | 30,180 | 8,358,720 | 4,255,391 | 2,296,139 | 16.000 | PASS ✓ |
| L06 | 3,348 | 8,318,284 | 4,226,719 | 2,343,921 | 22.125 | PASS ✓ |
| L07 | 30,180 | 8,366,629 | 4,189,855 | 2,422,271 | 9.375 | PASS ✓ |
| L08 | 43,062 | 8,342,984 | 4,202,143 | 2,393,426 | 24.375 | PASS ✓ |
| L09 | 52,905 | 8,366,629 | 4,239,007 | 2,385,453 | 14.750 | PASS ✓ |
| L10 | 3,348 | 8,342,984 | 4,255,391 | 2,321,248 | 28.250 | PASS ✓ |
| L11 | 53,230 | 8,333,062 | 4,218,527 | 2,384,872 | 13.750 | PASS ✓ |
| L12 | 3,348 | 8,342,984 | 4,161,183 | 2,357,339 | 19.625 | PASS ✓ |
| L13 | 30,180 | 8,358,720 | 4,198,047 | 2,341,445 | 21.000 | PASS ✓ |
| L14 | 3,348 | 8,291,413 | 4,239,007 | 2,385,605 | 21.125 | PASS ✓ |
| L15 | 30,180 | 8,358,720 | 4,255,391 | 2,353,150 | 23.000 | PASS ✓ |
| L16 | 3,348 | 8,291,413 | 4,279,967 | 2,334,204 | 19.250 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 18 | 16 | +2 |
| 2 | Orange Red | 11 | 16 | -5 |
| 3 | Orange | 18 | 16 | +2 |
| 4 | Gold | 16 | 16 | +0 |
| 5 | Yellow | 12 | 16 | -4 |
| 6 | Green Yellow | 16 | 16 | +0 |
| 7 | Chartreuse | 17 | 16 | +1 |
| 8 | Lime | 20 | 16 | +4 |
| 9 | Spring Green | 15 | 16 | -1 |
| 10 | Cyan | 14 | 16 | -2 |
| 11 | Deep Sky Blue | 25 | 16 | +9 |
| 12 | Blue | 17 | 16 | +1 |
| 13 | Violet | 15 | 16 | -1 |
| 14 | Magenta | 13 | 16 | -3 |
| 15 | Deep Pink | 12 | 16 | -4 |
| 16 | Hot Pink | 17 | 16 | +1 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 3,348 |
| Max value | 8,366,629 |
| Mean | 4,232,351 |
| Std deviation | 2,350,366 |
| Shannon entropy | 3.9767 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 19.898 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9694 | 99.2% | 11.250 | −19.33 | PASS ✓ |
| L02 | 3.9083 | 97.7% | 28.250 | −2.33 | PASS ✓ |
| L03 | 3.9430 | 98.6% | 20.000 | −10.58 | PASS ✓ |
| L04 | 3.9215 | 98.0% | 26.250 | −4.33 | PASS ✓ |
| L05 | 3.9539 | 98.8% | 16.000 | −14.58 | PASS ✓ |
| L06 | 3.9320 | 98.3% | 22.125 | −8.45 | PASS ✓ |
| L07 | 3.9744 | 99.4% | 9.375 | −21.20 | PASS ✓ |
| L08 | 3.9294 | 98.2% | 24.375 | −6.20 | PASS ✓ |
| L09 | 3.9598 | 99.0% | 14.750 | −15.83 | PASS ✓ |
| L10 | 3.9134 | 97.8% | 28.250 | −2.33 | PASS ✓ |
| L11 | 3.9630 | 99.1% | 13.750 | −16.83 | PASS ✓ |
| L12 | 3.9414 | 98.5% | 19.625 | −10.95 | PASS ✓ |
| L13 | 3.9391 | 98.5% | 21.000 | −9.58 | PASS ✓ |
| L14 | 3.9357 | 98.4% | 21.125 | −9.45 | PASS ✓ |
| L15 | 3.9373 | 98.4% | 23.000 | −7.58 | PASS ✓ |
| L16 | 3.9445 | 98.6% | 19.250 | −11.33 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9416 bits (98.5% of max 4.0) |
| Average χ² | 19.898 |
| Layers passing χ² test | 16 / 16 (100%) |
| Marginal layers (30.578–45) | 0 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.9083 bits (L02) |
| Highest layer χ² | 28.250 (L02) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9416 bits (98.5% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9767 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 19.898 PASS ✓ |
| Min value | 3,348 |
| Max value | 8,366,629 |
| Mean | 4,232,351 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 19,492 | 8,358,267 | 4,288,180 | 2,370,887 | 16.500 | PASS ✓ |
| L02 | 78,988 | 8,346,162 | 4,271,796 | 2,283,031 | 37.125 | MARGINAL |
| L03 | 19,492 | 8,358,267 | 4,308,660 | 2,311,190 | 16.250 | PASS ✓ |
| L04 | 87,924 | 8,346,162 | 4,300,468 | 2,270,143 | 20.250 | PASS ✓ |
| L05 | 19,492 | 8,358,267 | 4,325,044 | 2,276,709 | 14.125 | PASS ✓ |
| L06 | 142,515 | 8,327,935 | 4,337,332 | 2,272,692 | 24.875 | PASS ✓ |
| L07 | 19,492 | 8,358,267 | 4,366,004 | 2,294,995 | 13.875 | PASS ✓ |
| L08 | 165,995 | 8,375,662 | 4,349,620 | 2,301,390 | 27.500 | PASS ✓ |
| L09 | 18,675 | 8,311,012 | 4,394,676 | 2,344,644 | 22.500 | PASS ✓ |
| L10 | 165,995 | 8,375,662 | 4,411,060 | 2,299,420 | 41.375 | MARGINAL |
| L11 | 29,550 | 8,311,108 | 4,447,924 | 2,292,160 | 23.625 | PASS ✓ |
| L12 | 113,486 | 8,327,935 | 4,460,212 | 2,288,832 | 30.500 | PASS ✓ |
| L13 | 19,492 | 8,251,120 | 4,447,924 | 2,288,286 | 19.500 | PASS ✓ |
| L14 | 99,301 | 8,327,935 | 4,447,924 | 2,258,246 | 26.875 | PASS ✓ |
| L15 | 18,675 | 8,311,012 | 4,460,212 | 2,284,065 | 17.125 | PASS ✓ |
| L16 | 121,059 | 8,346,162 | 4,431,540 | 2,250,125 | 21.625 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 13 | 16 | -3 |
| 2 | Orange Red | 15 | 16 | -1 |
| 3 | Orange | 11 | 16 | -5 |
| 4 | Gold | 13 | 16 | -3 |
| 5 | Yellow | 21 | 16 | +5 |
| 6 | Green Yellow | 16 | 16 | +0 |
| 7 | Chartreuse | 21 | 16 | +5 |
| 8 | Lime | 18 | 16 | +2 |
| 9 | Spring Green | 22 | 16 | +6 |
| 10 | Cyan | 19 | 16 | +3 |
| 11 | Deep Sky Blue | 8 | 16 | -8 |
| 12 | Blue | 12 | 16 | -4 |
| 13 | Violet | 12 | 16 | -4 |
| 14 | Magenta | 19 | 16 | +3 |
| 15 | Deep Pink | 16 | 16 | +0 |
| 16 | Hot Pink | 20 | 16 | +4 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 18,675 |
| Max value | 8,375,662 |
| Mean | 4,378,036 |
| Std deviation | 2,294,038 |
| Shannon entropy | 3.9656 bits (99.1% of max 4.0) |
| Avg χ² (16 layers) | 23.352 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9515 | 98.8% | 16.500 | −14.08 | PASS ✓ |
| L02 | 3.8956 | 97.4% | 37.125 | +6.55 | MARGINAL |
| L03 | 3.9552 | 98.9% | 16.250 | −14.33 | PASS ✓ |
| L04 | 3.9367 | 98.4% | 20.250 | −10.33 | PASS ✓ |
| L05 | 3.9573 | 98.9% | 14.125 | −16.45 | PASS ✓ |
| L06 | 3.9128 | 97.8% | 24.875 | −5.70 | PASS ✓ |
| L07 | 3.9602 | 99.0% | 13.875 | −16.70 | PASS ✓ |
| L08 | 3.9128 | 97.8% | 27.500 | −3.08 | PASS ✓ |
| L09 | 3.9379 | 98.4% | 22.500 | −8.08 | PASS ✓ |
| L10 | 3.8730 | 96.8% | 41.375 | +10.80 | MARGINAL |
| L11 | 3.9263 | 98.2% | 23.625 | −6.95 | PASS ✓ |
| L12 | 3.8974 | 97.4% | 30.500 | −0.08 | PASS ✓ |
| L13 | 3.9400 | 98.5% | 19.500 | −11.08 | PASS ✓ |
| L14 | 3.9133 | 97.8% | 26.875 | −3.70 | PASS ✓ |
| L15 | 3.9476 | 98.7% | 17.125 | −13.45 | PASS ✓ |
| L16 | 3.9342 | 98.4% | 21.625 | −8.95 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9282 bits (98.2% of max 4.0) |
| Average χ² | 23.352 |
| Layers passing χ² test | 14 / 16 (88%) |
| Marginal layers (30.578–45) | 2 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.8730 bits (L10) |
| Highest layer χ² | 41.375 (L10) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9282 bits (98.2% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9656 bits (99.1% of max 4.0) |
| Avg χ² (16 layers) | 23.352 PASS ✓ |
| Min value | 18,675 |
| Max value | 8,375,662 |
| Mean | 4,378,036 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 84,160 | 8,370,552 | 4,194,048 | 2,451,576 | 25.000 | PASS ✓ |
| L02 | 41,265 | 8,322,057 | 4,177,664 | 2,430,758 | 14.000 | PASS ✓ |
| L03 | 9,474 | 8,370,552 | 4,157,184 | 2,441,174 | 32.000 | MARGINAL |
| L04 | 226,066 | 8,343,689 | 4,210,432 | 2,369,535 | 26.500 | PASS ✓ |
| L05 | 81,909 | 8,339,324 | 4,198,144 | 2,379,508 | 32.250 | MARGINAL |
| L06 | 80,291 | 8,369,619 | 4,165,376 | 2,345,457 | 34.000 | MARGINAL |
| L07 | 222,006 | 8,291,662 | 4,181,760 | 2,296,234 | 27.875 | PASS ✓ |
| L08 | 80,290 | 8,343,689 | 4,112,128 | 2,326,865 | 17.875 | PASS ✓ |
| L09 | 81,909 | 8,291,662 | 4,071,168 | 2,316,003 | 20.875 | PASS ✓ |
| L10 | 80,290 | 8,322,057 | 4,095,744 | 2,357,727 | 19.750 | PASS ✓ |
| L11 | 14,940 | 8,291,662 | 4,124,416 | 2,373,497 | 16.625 | PASS ✓ |
| L12 | 63,349 | 8,351,282 | 4,112,128 | 2,343,796 | 15.250 | PASS ✓ |
| L13 | 49,653 | 8,293,909 | 4,194,048 | 2,317,572 | 18.750 | PASS ✓ |
| L14 | 80,290 | 8,369,619 | 4,230,912 | 2,291,672 | 24.875 | PASS ✓ |
| L15 | 49,653 | 8,359,358 | 4,189,952 | 2,291,844 | 24.000 | PASS ✓ |
| L16 | 12,518 | 8,190,911 | 4,177,664 | 2,326,493 | 14.750 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 9 | 16 | -7 |
| 2 | Orange Red | 24 | 16 | +8 |
| 3 | Orange | 11 | 16 | -5 |
| 4 | Gold | 18 | 16 | +2 |
| 5 | Yellow | 19 | 16 | +3 |
| 6 | Green Yellow | 23 | 16 | +7 |
| 7 | Chartreuse | 17 | 16 | +1 |
| 8 | Lime | 12 | 16 | -4 |
| 9 | Spring Green | 13 | 16 | -3 |
| 10 | Cyan | 14 | 16 | -2 |
| 11 | Deep Sky Blue | 11 | 16 | -5 |
| 12 | Blue | 14 | 16 | -2 |
| 13 | Violet | 20 | 16 | +4 |
| 14 | Magenta | 13 | 16 | -3 |
| 15 | Deep Pink | 26 | 16 | +10 |
| 16 | Hot Pink | 12 | 16 | -4 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 9,474 |
| Max value | 8,370,552 |
| Mean | 4,162,048 |
| Std deviation | 2,354,674 |
| Shannon entropy | 3.9667 bits (99.2% of max 4.0) |
| Avg χ² (16 layers) | 22.773 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9317 | 98.3% | 25.000 | −5.58 | PASS ✓ |
| L02 | 3.9628 | 99.1% | 14.000 | −16.58 | PASS ✓ |
| L03 | 3.8979 | 97.4% | 32.000 | +1.42 | MARGINAL |
| L04 | 3.9220 | 98.1% | 26.500 | −4.08 | PASS ✓ |
| L05 | 3.9087 | 97.7% | 32.250 | +1.67 | MARGINAL |
| L06 | 3.9036 | 97.6% | 34.000 | +3.42 | MARGINAL |
| L07 | 3.9063 | 97.7% | 27.875 | −2.70 | PASS ✓ |
| L08 | 3.9476 | 98.7% | 17.875 | −12.70 | PASS ✓ |
| L09 | 3.9337 | 98.3% | 20.875 | −9.70 | PASS ✓ |
| L10 | 3.9442 | 98.6% | 19.750 | −10.83 | PASS ✓ |
| L11 | 3.9522 | 98.8% | 16.625 | −13.95 | PASS ✓ |
| L12 | 3.9546 | 98.9% | 15.250 | −15.33 | PASS ✓ |
| L13 | 3.9431 | 98.6% | 18.750 | −11.83 | PASS ✓ |
| L14 | 3.9251 | 98.1% | 24.875 | −5.70 | PASS ✓ |
| L15 | 3.9208 | 98.0% | 24.000 | −6.58 | PASS ✓ |
| L16 | 3.9555 | 98.9% | 14.750 | −15.83 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9319 bits (98.3% of max 4.0) |
| Average χ² | 22.773 |
| Layers passing χ² test | 13 / 16 (81%) |
| Marginal layers (30.578–45) | 3 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.8979 bits (L03) |
| Highest layer χ² | 34.000 (L06) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9319 bits (98.3% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9667 bits (99.2% of max 4.0) |
| Avg χ² (16 layers) | 22.773 PASS ✓ |
| Min value | 9,474 |
| Max value | 8,370,552 |
| Mean | 4,162,048 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 14,237 | 8,336,807 | 4,074,321 | 2,276,584 | 32.625 | MARGINAL |
| L02 | 31,135 | 8,351,797 | 4,156,241 | 2,244,964 | 29.750 | PASS ✓ |
| L03 | 14,237 | 8,358,499 | 4,127,569 | 2,289,696 | 10.625 | PASS ✓ |
| L04 | 31,135 | 8,351,797 | 4,074,321 | 2,285,690 | 27.125 | PASS ✓ |
| L05 | 58,745 | 8,245,177 | 4,094,801 | 2,314,798 | 10.250 | PASS ✓ |
| L06 | 130,192 | 8,323,146 | 4,094,801 | 2,265,861 | 22.625 | PASS ✓ |
| L07 | 58,745 | 8,358,499 | 4,037,457 | 2,234,522 | 21.000 | PASS ✓ |
| L08 | 31,135 | 8,323,146 | 4,070,225 | 2,209,101 | 29.750 | PASS ✓ |
| L09 | 58,745 | 8,254,024 | 4,004,689 | 2,192,012 | 22.875 | PASS ✓ |
| L10 | 31,135 | 8,249,124 | 4,049,745 | 2,191,967 | 21.750 | PASS ✓ |
| L11 | 107,999 | 8,358,499 | 4,037,457 | 2,190,934 | 21.625 | PASS ✓ |
| L12 | 24,080 | 8,105,798 | 4,016,977 | 2,191,975 | 24.500 | PASS ✓ |
| L13 | 107,999 | 8,208,828 | 4,008,785 | 2,198,968 | 21.250 | PASS ✓ |
| L14 | 165,395 | 8,303,017 | 4,049,745 | 2,190,696 | 32.000 | MARGINAL |
| L15 | 14,237 | 8,336,807 | 4,045,649 | 2,220,253 | 17.750 | PASS ✓ |
| L16 | 24,928 | 8,339,999 | 4,074,321 | 2,239,888 | 26.250 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 18 | 16 | +2 |
| 2 | Orange Red | 12 | 16 | -4 |
| 3 | Orange | 10 | 16 | -6 |
| 4 | Gold | 17 | 16 | +1 |
| 5 | Yellow | 25 | 16 | +9 |
| 6 | Green Yellow | 22 | 16 | +6 |
| 7 | Chartreuse | 14 | 16 | -2 |
| 8 | Lime | 7 | 16 | -9 |
| 9 | Spring Green | 19 | 16 | +3 |
| 10 | Cyan | 15 | 16 | -1 |
| 11 | Deep Sky Blue | 28 | 16 | +12 |
| 12 | Blue | 15 | 16 | -1 |
| 13 | Violet | 18 | 16 | +2 |
| 14 | Magenta | 18 | 16 | +2 |
| 15 | Deep Pink | 8 | 16 | -8 |
| 16 | Hot Pink | 10 | 16 | -6 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 14,237 |
| Max value | 8,358,499 |
| Mean | 4,063,569 |
| Std deviation | 2,234,348 |
| Shannon entropy | 3.9683 bits (99.2% of max 4.0) |
| Avg χ² (16 layers) | 23.234 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9070 | 97.7% | 32.625 | +2.05 | MARGINAL |
| L02 | 3.9070 | 97.7% | 29.750 | −0.83 | PASS ✓ |
| L03 | 3.9699 | 99.2% | 10.625 | −19.95 | PASS ✓ |
| L04 | 3.9141 | 97.9% | 27.125 | −3.45 | PASS ✓ |
| L05 | 3.9698 | 99.2% | 10.250 | −20.33 | PASS ✓ |
| L06 | 3.9309 | 98.3% | 22.625 | −7.95 | PASS ✓ |
| L07 | 3.9388 | 98.5% | 21.000 | −9.58 | PASS ✓ |
| L08 | 3.9179 | 97.9% | 29.750 | −0.83 | PASS ✓ |
| L09 | 3.9246 | 98.1% | 22.875 | −7.70 | PASS ✓ |
| L10 | 3.9313 | 98.3% | 21.750 | −8.83 | PASS ✓ |
| L11 | 3.9339 | 98.3% | 21.625 | −8.95 | PASS ✓ |
| L12 | 3.9274 | 98.2% | 24.500 | −6.08 | PASS ✓ |
| L13 | 3.9310 | 98.3% | 21.250 | −9.33 | PASS ✓ |
| L14 | 3.9073 | 97.7% | 32.000 | +1.42 | MARGINAL |
| L15 | 3.9465 | 98.7% | 17.750 | −12.83 | PASS ✓ |
| L16 | 3.9186 | 98.0% | 26.250 | −4.33 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9298 bits (98.2% of max 4.0) |
| Average χ² | 23.234 |
| Layers passing χ² test | 14 / 16 (88%) |
| Marginal layers (30.578–45) | 2 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.9070 bits (L01) |
| Highest layer χ² | 32.625 (L01) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9298 bits (98.2% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9683 bits (99.2% of max 4.0) |
| Avg χ² (16 layers) | 23.234 PASS ✓ |
| Min value | 14,237 |
| Max value | 8,358,499 |
| Mean | 4,063,569 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 31,548 | 8,316,895 | 4,139,772 | 2,381,558 | 18.125 | PASS ✓ |
| L02 | 7,300 | 8,361,783 | 4,139,772 | 2,375,171 | 29.500 | PASS ✓ |
| L03 | 22,551 | 8,297,721 | 4,147,964 | 2,373,275 | 14.375 | PASS ✓ |
| L04 | 7,300 | 8,331,483 | 4,139,772 | 2,334,559 | 26.750 | PASS ✓ |
| L05 | 31,548 | 8,283,401 | 4,082,428 | 2,329,268 | 12.625 | PASS ✓ |
| L06 | 7,300 | 8,214,544 | 4,012,796 | 2,240,052 | 24.625 | PASS ✓ |
| L07 | 184,880 | 8,316,895 | 4,012,796 | 2,235,551 | 21.250 | PASS ✓ |
| L08 | 7,300 | 8,214,544 | 4,037,372 | 2,225,101 | 21.125 | PASS ✓ |
| L09 | 147,941 | 8,316,895 | 4,082,428 | 2,238,574 | 21.375 | PASS ✓ |
| L10 | 7,300 | 8,214,544 | 4,086,524 | 2,254,657 | 22.125 | PASS ✓ |
| L11 | 1,776 | 8,316,895 | 4,111,100 | 2,340,859 | 20.000 | PASS ✓ |
| L12 | 132,285 | 8,361,783 | 4,119,292 | 2,347,222 | 13.125 | PASS ✓ |
| L13 | 22,551 | 8,245,041 | 4,066,044 | 2,311,389 | 24.125 | PASS ✓ |
| L14 | 83,318 | 8,361,783 | 4,008,700 | 2,269,249 | 17.875 | PASS ✓ |
| L15 | 22,551 | 8,316,895 | 4,061,948 | 2,308,346 | 10.375 | PASS ✓ |
| L16 | 57,011 | 8,361,783 | 4,053,756 | 2,315,188 | 15.000 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 15 | 16 | -1 |
| 2 | Orange Red | 19 | 16 | +3 |
| 3 | Orange | 20 | 16 | +4 |
| 4 | Gold | 11 | 16 | -5 |
| 5 | Yellow | 17 | 16 | +1 |
| 6 | Green Yellow | 15 | 16 | -1 |
| 7 | Chartreuse | 13 | 16 | -3 |
| 8 | Lime | 18 | 16 | +2 |
| 9 | Spring Green | 8 | 16 | -8 |
| 10 | Cyan | 23 | 16 | +7 |
| 11 | Deep Sky Blue | 24 | 16 | +8 |
| 12 | Blue | 13 | 16 | -3 |
| 13 | Violet | 16 | 16 | +0 |
| 14 | Magenta | 19 | 16 | +3 |
| 15 | Deep Pink | 14 | 16 | -2 |
| 16 | Hot Pink | 11 | 16 | -5 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 1,776 |
| Max value | 8,361,783 |
| Mean | 4,081,404 |
| Std deviation | 2,306,069 |
| Shannon entropy | 3.9705 bits (99.3% of max 4.0) |
| Avg χ² (16 layers) | 19.523 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9480 | 98.7% | 18.125 | −12.45 | PASS ✓ |
| L02 | 3.9155 | 97.9% | 29.500 | −1.08 | PASS ✓ |
| L03 | 3.9608 | 99.0% | 14.375 | −16.20 | PASS ✓ |
| L04 | 3.9189 | 98.0% | 26.750 | −3.83 | PASS ✓ |
| L05 | 3.9635 | 99.1% | 12.625 | −17.95 | PASS ✓ |
| L06 | 3.9149 | 97.9% | 24.625 | −5.95 | PASS ✓ |
| L07 | 3.9342 | 98.4% | 21.250 | −9.33 | PASS ✓ |
| L08 | 3.9283 | 98.2% | 21.125 | −9.45 | PASS ✓ |
| L09 | 3.9339 | 98.3% | 21.375 | −9.20 | PASS ✓ |
| L10 | 3.9307 | 98.3% | 22.125 | −8.45 | PASS ✓ |
| L11 | 3.9465 | 98.7% | 20.000 | −10.58 | PASS ✓ |
| L12 | 3.9616 | 99.0% | 13.125 | −17.45 | PASS ✓ |
| L13 | 3.9234 | 98.1% | 24.125 | −6.45 | PASS ✓ |
| L14 | 3.9430 | 98.6% | 17.875 | −12.70 | PASS ✓ |
| L15 | 3.9694 | 99.2% | 10.375 | −20.20 | PASS ✓ |
| L16 | 3.9545 | 98.9% | 15.000 | −15.58 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9404 bits (98.5% of max 4.0) |
| Average χ² | 19.523 |
| Layers passing χ² test | 16 / 16 (100%) |
| Marginal layers (30.578–45) | 0 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.9149 bits (L06) |
| Highest layer χ² | 29.500 (L02) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9404 bits (98.5% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9705 bits (99.3% of max 4.0) |
| Avg χ² (16 layers) | 19.523 PASS ✓ |
| Min value | 1,776 |
| Max value | 8,361,783 |
| Mean | 4,081,404 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 7,341 | 8,364,164 | 4,208,635 | 2,522,108 | 14.250 | PASS ✓ |
| L02 | 72,376 | 8,329,748 | 4,204,539 | 2,425,188 | 37.125 | MARGINAL |
| L03 | 7,341 | 8,372,047 | 4,188,155 | 2,424,739 | 21.000 | PASS ✓ |
| L04 | 72,376 | 8,329,748 | 4,184,059 | 2,332,862 | 36.375 | MARGINAL |
| L05 | 16,378 | 8,351,079 | 4,220,923 | 2,354,715 | 20.000 | PASS ✓ |
| L06 | 100,937 | 8,174,960 | 4,196,347 | 2,320,988 | 35.375 | MARGINAL |
| L07 | 7,341 | 8,351,079 | 4,122,619 | 2,319,668 | 37.750 | MARGINAL |
| L08 | 72,376 | 8,329,748 | 4,130,811 | 2,281,223 | 42.500 | MARGINAL |
| L09 | 7,341 | 8,376,526 | 4,122,619 | 2,284,509 | 18.250 | PASS ✓ |
| L10 | 100,937 | 8,334,357 | 4,179,963 | 2,266,224 | 34.000 | MARGINAL |
| L11 | 16,078 | 8,376,526 | 4,216,827 | 2,315,117 | 19.500 | PASS ✓ |
| L12 | 72,376 | 8,334,357 | 4,204,539 | 2,326,800 | 31.250 | MARGINAL |
| L13 | 6,727 | 8,373,995 | 4,208,635 | 2,379,389 | 11.000 | PASS ✓ |
| L14 | 72,376 | 8,329,748 | 4,220,923 | 2,323,718 | 39.000 | MARGINAL |
| L15 | 35,019 | 8,373,995 | 4,261,883 | 2,367,759 | 14.375 | PASS ✓ |
| L16 | 72,376 | 8,334,357 | 4,290,555 | 2,302,440 | 37.000 | MARGINAL |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 18 | 16 | +2 |
| 2 | Orange Red | 17 | 16 | +1 |
| 3 | Orange | 22 | 16 | +6 |
| 4 | Gold | 11 | 16 | -5 |
| 5 | Yellow | 15 | 16 | -1 |
| 6 | Green Yellow | 11 | 16 | -5 |
| 7 | Chartreuse | 20 | 16 | +4 |
| 8 | Lime | 14 | 16 | -2 |
| 9 | Spring Green | 16 | 16 | +0 |
| 10 | Cyan | 9 | 16 | -7 |
| 11 | Deep Sky Blue | 15 | 16 | -1 |
| 12 | Blue | 21 | 16 | +5 |
| 13 | Violet | 12 | 16 | -4 |
| 14 | Magenta | 20 | 16 | +4 |
| 15 | Deep Pink | 16 | 16 | +0 |
| 16 | Hot Pink | 19 | 16 | +3 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 6,727 |
| Max value | 8,376,526 |
| Mean | 4,197,627 |
| Std deviation | 2,347,988 |
| Shannon entropy | 3.9786 bits (99.5% of max 4.0) |
| Avg χ² (16 layers) | 28.047 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9584 | 99.0% | 14.250 | −16.33 | PASS ✓ |
| L02 | 3.8924 | 97.3% | 37.125 | +6.55 | MARGINAL |
| L03 | 3.9412 | 98.5% | 21.000 | −9.58 | PASS ✓ |
| L04 | 3.8913 | 97.3% | 36.375 | +5.80 | MARGINAL |
| L05 | 3.9451 | 98.6% | 20.000 | −10.58 | PASS ✓ |
| L06 | 3.8943 | 97.4% | 35.375 | +4.80 | MARGINAL |
| L07 | 3.8917 | 97.3% | 37.750 | +7.17 | MARGINAL |
| L08 | 3.8713 | 96.8% | 42.500 | +11.92 | MARGINAL |
| L09 | 3.9462 | 98.7% | 18.250 | −12.33 | PASS ✓ |
| L10 | 3.8830 | 97.1% | 34.000 | +3.42 | MARGINAL |
| L11 | 3.9436 | 98.6% | 19.500 | −11.08 | PASS ✓ |
| L12 | 3.9026 | 97.6% | 31.250 | +0.67 | MARGINAL |
| L13 | 3.9694 | 99.2% | 11.000 | −19.58 | PASS ✓ |
| L14 | 3.8736 | 96.8% | 39.000 | +8.42 | MARGINAL |
| L15 | 3.9596 | 99.0% | 14.375 | −16.20 | PASS ✓ |
| L16 | 3.8825 | 97.1% | 37.000 | +6.42 | MARGINAL |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9154 bits (97.9% of max 4.0) |
| Average χ² | 28.047 |
| Layers passing χ² test | 7 / 16 (44%) |
| Marginal layers (30.578–45) | 9 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.8713 bits (L08) |
| Highest layer χ² | 42.500 (L08) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9154 bits (97.9% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9786 bits (99.5% of max 4.0) |
| Avg χ² (16 layers) | 28.047 PASS ✓ |
| Min value | 6,727 |
| Max value | 8,376,526 |
| Mean | 4,197,627 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 65,092 | 8,345,611 | 4,366,641 | 2,376,518 | 20.125 | PASS ✓ |
| L02 | 78,404 | 8,375,864 | 4,399,409 | 2,318,785 | 41.250 | MARGINAL |
| L03 | 65,917 | 8,374,866 | 4,391,217 | 2,321,134 | 22.750 | PASS ✓ |
| L04 | 15,049 | 8,375,864 | 4,395,313 | 2,301,769 | 40.375 | MARGINAL |
| L05 | 65,917 | 8,374,866 | 4,391,217 | 2,327,762 | 30.000 | PASS ✓ |
| L06 | 286,640 | 8,375,864 | 4,391,217 | 2,367,266 | 24.375 | PASS ✓ |
| L07 | 36,885 | 8,272,533 | 4,374,833 | 2,404,986 | 13.875 | PASS ✓ |
| L08 | 43,150 | 8,375,864 | 4,358,449 | 2,376,912 | 26.250 | PASS ✓ |
| L09 | 15,559 | 8,338,314 | 4,333,873 | 2,407,193 | 19.875 | PASS ✓ |
| L10 | 43,150 | 8,375,864 | 4,374,833 | 2,389,508 | 30.625 | MARGINAL |
| L11 | 15,559 | 8,301,229 | 4,337,969 | 2,368,148 | 33.750 | MARGINAL |
| L12 | 78,404 | 8,375,864 | 4,354,353 | 2,347,392 | 28.875 | PASS ✓ |
| L13 | 119,133 | 8,301,229 | 4,350,257 | 2,376,557 | 20.000 | PASS ✓ |
| L14 | 43,150 | 8,333,996 | 4,383,025 | 2,410,396 | 25.625 | PASS ✓ |
| L15 | 65,092 | 8,348,341 | 4,415,793 | 2,370,378 | 18.500 | PASS ✓ |
| L16 | 43,150 | 8,375,864 | 4,407,601 | 2,335,873 | 25.875 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 22 | 16 | +6 |
| 2 | Orange Red | 10 | 16 | -6 |
| 3 | Orange | 12 | 16 | -4 |
| 4 | Gold | 10 | 16 | -6 |
| 5 | Yellow | 13 | 16 | -3 |
| 6 | Green Yellow | 21 | 16 | +5 |
| 7 | Chartreuse | 11 | 16 | -5 |
| 8 | Lime | 14 | 16 | -2 |
| 9 | Spring Green | 21 | 16 | +5 |
| 10 | Cyan | 20 | 16 | +4 |
| 11 | Deep Sky Blue | 21 | 16 | +5 |
| 12 | Blue | 15 | 16 | -1 |
| 13 | Violet | 13 | 16 | -3 |
| 14 | Magenta | 23 | 16 | +7 |
| 15 | Deep Pink | 17 | 16 | +1 |
| 16 | Hot Pink | 13 | 16 | -3 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 15,049 |
| Max value | 8,375,864 |
| Mean | 4,376,625 |
| Std deviation | 2,362,880 |
| Shannon entropy | 3.9769 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 26.383 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9433 | 98.6% | 20.125 | −10.45 | PASS ✓ |
| L02 | 3.8843 | 97.1% | 41.250 | +10.67 | MARGINAL |
| L03 | 3.9368 | 98.4% | 22.750 | −7.83 | PASS ✓ |
| L04 | 3.8751 | 96.9% | 40.375 | +9.80 | MARGINAL |
| L05 | 3.9137 | 97.8% | 30.000 | −0.58 | PASS ✓ |
| L06 | 3.9294 | 98.2% | 24.375 | −6.20 | PASS ✓ |
| L07 | 3.9603 | 99.0% | 13.875 | −16.70 | PASS ✓ |
| L08 | 3.9228 | 98.1% | 26.250 | −4.33 | PASS ✓ |
| L09 | 3.9426 | 98.6% | 19.875 | −10.70 | PASS ✓ |
| L10 | 3.9155 | 97.9% | 30.625 | +0.05 | MARGINAL |
| L11 | 3.9123 | 97.8% | 33.750 | +3.17 | MARGINAL |
| L12 | 3.9155 | 97.9% | 28.875 | −1.70 | PASS ✓ |
| L13 | 3.9436 | 98.6% | 20.000 | −10.58 | PASS ✓ |
| L14 | 3.9284 | 98.2% | 25.625 | −4.95 | PASS ✓ |
| L15 | 3.9430 | 98.6% | 18.500 | −12.08 | PASS ✓ |
| L16 | 3.9260 | 98.1% | 25.875 | −4.70 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9245 bits (98.1% of max 4.0) |
| Average χ² | 26.383 |
| Layers passing χ² test | 12 / 16 (75%) |
| Marginal layers (30.578–45) | 4 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.8751 bits (L04) |
| Highest layer χ² | 41.250 (L02) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9245 bits (98.1% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9769 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 26.383 PASS ✓ |
| Min value | 15,049 |
| Max value | 8,375,864 |
| Mean | 4,376,625 (expected: ~4,190,209) |
PQC IDENTITY REPORT
CRYSTALS-Dilithium · ML-DSA · FIPS 204
| Parameter | Value |
|---|---|
| Algorithm | CRYSTALS-Dilithium (ML-DSA / FIPS 204) |
| Security Level | Level 3 — AES-192 equivalent |
| Modulus q | 8,380,417 |
| Polynomial degree n | 256 |
| Walk delta δ | 524,288 (= 2¹&sup9;, Dilithium-2 rejection bound) |
| Cube dimensions | 16 × 16 × 16 = 4,096 spheres |
| Palette bands | 16-band rainbow [0, q−1] |
| ML-KEM-768 public key | 1,184 bytes · FIPS 203 |
| ML-DSA-65 public key | 1,952 bytes · FIPS 204 |
Layer 1 base values are derived deterministically from the identity seed:
v[i] = SHA-256(ζ || i.to_bytes(2)) mod q (i = 0…255)
Each subsequent layer applies a ±δ walk with SHA-256-derived signs:
sign(l,i) = SHA-256(ζ || "walk" || l || i)[0] & 1 → {+1, −1}
v_l[i] = v_{l−1}[i] + sign(l,i) · δ (boundary reflection if ∉ [0, q−1])
| Layer | Min | Max | Mean | Std Dev | χ² | Verdict |
|---|---|---|---|---|---|---|
| L01 | 39,324 | 8,371,973 | 4,114,774 | 2,372,284 | 18.375 | PASS ✓ |
| L02 | 13,737 | 8,339,935 | 4,151,638 | 2,394,359 | 19.750 | PASS ✓ |
| L03 | 37,342 | 8,351,038 | 4,147,542 | 2,345,354 | 16.625 | PASS ✓ |
| L04 | 68,506 | 8,372,603 | 4,196,694 | 2,385,452 | 22.375 | PASS ✓ |
| L05 | 66,612 | 8,371,973 | 4,172,118 | 2,404,465 | 24.000 | PASS ✓ |
| L06 | 13,737 | 8,339,935 | 4,180,310 | 2,353,747 | 26.125 | PASS ✓ |
| L07 | 91,678 | 8,371,973 | 4,192,598 | 2,377,143 | 18.500 | PASS ✓ |
| L08 | 6,043 | 8,283,557 | 4,139,350 | 2,364,161 | 24.625 | PASS ✓ |
| L09 | 37,342 | 8,371,973 | 4,163,926 | 2,304,718 | 30.000 | PASS ✓ |
| L10 | 6,043 | 8,307,805 | 4,139,350 | 2,341,919 | 31.625 | MARGINAL |
| L11 | 1,391 | 8,371,973 | 4,188,502 | 2,341,974 | 44.625 | MARGINAL |
| L12 | 68,506 | 8,377,175 | 4,188,502 | 2,319,943 | 36.000 | MARGINAL |
| L13 | 39,324 | 8,207,571 | 4,184,406 | 2,430,802 | 9.625 | PASS ✓ |
| L14 | 101,116 | 8,377,175 | 4,184,406 | 2,387,887 | 24.375 | PASS ✓ |
| L15 | 1,391 | 8,356,129 | 4,192,598 | 2,432,133 | 6.250 | PASS ✓ |
| L16 | 101,116 | 8,337,947 | 4,200,790 | 2,382,101 | 25.500 | PASS ✓ |
| Band | Color Name | Count | Expected | Δ |
|---|---|---|---|---|
| 1 | Red | 10 | 16 | -6 |
| 2 | Orange Red | 18 | 16 | +2 |
| 3 | Orange | 24 | 16 | +8 |
| 4 | Gold | 12 | 16 | -4 |
| 5 | Yellow | 21 | 16 | +5 |
| 6 | Green Yellow | 18 | 16 | +2 |
| 7 | Chartreuse | 10 | 16 | -6 |
| 8 | Lime | 13 | 16 | -3 |
| 9 | Spring Green | 19 | 16 | +3 |
| 10 | Cyan | 22 | 16 | +6 |
| 11 | Deep Sky Blue | 16 | 16 | +0 |
| 12 | Blue | 14 | 16 | -2 |
| 13 | Violet | 11 | 16 | -5 |
| 14 | Magenta | 20 | 16 | +4 |
| 15 | Deep Pink | 15 | 16 | -1 |
| 16 | Hot Pink | 13 | 16 | -3 |
| Metric | Value |
|---|---|
| Total spheres | 4,096 |
| Min value | 1,391 |
| Max value | 8,377,175 |
| Mean | 4,171,094 |
| Std deviation | 2,371,535 |
| Shannon entropy | 3.9758 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 23.648 PASS ✓ |
ENTROPY & HARDNESS REPORT
16-Layer ±δ Walk Analysis · Dilithium q = 8,380,417
The visualizer applies a 16-layer deterministic ±δ walk over 256 polynomial coefficients in the Dilithium domain [0, q−1]. Each layer evolves from the previous via a SHA-256-derived binary sign sequence — mathematically equivalent to the rejection sampling sign derivation in Dilithium key generation.
| Parameter | Value |
|---|---|
| Walk delta δ | 524,288 = 2¹&sup9; (Dilithium-2 rejection sampling bound) |
| Sign derivation | SHA-256(ζ || "walk" || layer || sphere)[0] & 1 |
| Boundary rule | Reflect: flip sign if candidate ∉ [0, q−1] |
| Layers | 16 (cube depth) |
| Spheres per layer | 256 (Dilithium polynomial degree n) |
| Total walk steps | 4,096 (256 × 16) |
Shannon entropy H and chi-square uniformity statistic per walk layer. Critical value α=0.01, df=15: 30.578. Marginal zone: 30.578–45. Reject: >45.
| Layer | H (bits) | H % | χ² Stat | vs Critical | Pass/Fail |
|---|---|---|---|---|---|
| L01 | 3.9481 | 98.7% | 18.375 | −12.20 | PASS ✓ |
| L02 | 3.9448 | 98.6% | 19.750 | −10.83 | PASS ✓ |
| L03 | 3.9525 | 98.8% | 16.625 | −13.95 | PASS ✓ |
| L04 | 3.9337 | 98.3% | 22.375 | −8.20 | PASS ✓ |
| L05 | 3.9351 | 98.4% | 24.000 | −6.58 | PASS ✓ |
| L06 | 3.9191 | 98.0% | 26.125 | −4.45 | PASS ✓ |
| L07 | 3.9483 | 98.7% | 18.500 | −12.08 | PASS ✓ |
| L08 | 3.9304 | 98.3% | 24.625 | −5.95 | PASS ✓ |
| L09 | 3.9174 | 97.9% | 30.000 | −0.58 | PASS ✓ |
| L10 | 3.9046 | 97.6% | 31.625 | +1.05 | MARGINAL |
| L11 | 3.8806 | 97.0% | 44.625 | +14.05 | MARGINAL |
| L12 | 3.8993 | 97.5% | 36.000 | +5.42 | MARGINAL |
| L13 | 3.9728 | 99.3% | 9.625 | −20.95 | PASS ✓ |
| L14 | 3.9331 | 98.3% | 24.375 | −6.20 | PASS ✓ |
| L15 | 3.9820 | 99.6% | 6.250 | −24.33 | PASS ✓ |
| L16 | 3.9263 | 98.2% | 25.500 | −5.08 | PASS ✓ |
| Metric | Value |
|---|---|
| Average H (16 layers) | 3.9330 bits (98.3% of max 4.0) |
| Average χ² | 23.648 |
| Layers passing χ² test | 13 / 16 (81%) |
| Marginal layers (30.578–45) | 3 / 16 |
| Rejected layers (>45) | 0 / 16 |
| Lowest layer H | 3.8806 bits (L11) |
| Highest layer χ² | 44.625 (L11) |
Walk delta δ = 524,288 represents approximately 1/16 of domain q = 8,380,417. Each step moves a coefficient by 6.25% of the full range — the Dilithium-2 rejection sampling bound chosen to preserve uniform distribution after rejection.
Average Shannon entropy across all 16 layers: 3.9330 bits (98.3% of the 4.0-bit maximum). The walk maintains near-maximum entropy throughout — confirming the ±δ step size does not introduce systematic band concentration or measurable entropy decay.
Note: Chi-square is measured independently per layer and averaged. Flattening all layers before testing would artificially inflate χ² due to the correlated directional march of the γ&sub1; walk vectors across layers.
Recovering identity seed ζ from the observed cube requires:
Inversion complexity is bounded below by SHA-256 preimage resistance: 2¹²&sup8; operations at NIST Security Level 3.
| Metric | Value |
|---|---|
| Total values | 4,096 |
| Shannon entropy | 3.9758 bits (99.4% of max 4.0) |
| Avg χ² (16 layers) | 23.648 PASS ✓ |
| Min value | 1,391 |
| Max value | 8,377,175 |
| Mean | 4,171,094 (expected: ~4,190,209) |