Formal Verification, Statistical Gauntlet & Next-Tier Cryptanalysis
Here is what we ran across all 32 hardware threads, what passed, what our internal 16-wave cryptanalytic campaign evaluated, and what those results do and don’t mean. The Z3, Bitwuzla, cvc5, CaDiCaL, Coq, F*, and SymbiYosys proofs cover formal properties of the initialization schedule, state transitions, side-channel countermeasures, and mode domain isolation. The statistical, algebraic, neural, MILP, and binary-taint suites test empirical randomness, the certified 4.0x to 24.0x reduced-round diffusion margin, and compiled machine instructions. None of that is a universal security proof, and no outside cryptanalyst has reviewed APEX-256 yet.
What these results do and don’t tell you
Passing PractRand, BigCrush, and polynomial distinguishers means the output looks random to those tests. It does not mean the cipher cannot be broken. The Z3 SMT and symbolic equivalence results prove specific mathematical properties of the initialization, state transition, and companion domain boundaries. They are not a proof of security against every possible attack.
Every test below was executed internally by Apex Research Labs, LLC. No independent third-party cryptanalysis of APEX-256 has been published yet. We actively welcome external academic and industry reviewers.
Mathematical Proofs & Empirical Randomness
Formal bit-vector bijection proofs, 256-bit key schedule collision immunity, and 32 TB of continuous statistical testing.
Formal SMT & Symbolic Proofs (Z3 / SAW / CASCADA)
| Analysis Domain | Methodology & Scope | Proof Outcome |
|---|---|---|
| Single-Step State Bijection | Formal bit-vector permutation proof | PROVEN Strict Bijection |
| Full-State Transition Injectivity | Z3 / Bitwuzla / Coq bit-vector equivalence | UNSAT 0 State Collisions |
| 256-Bit Key Schedule | Z3 same-nonce 256-bit key collision search + 1M pairs | UNSAT 0 Key Collisions |
| Key-Nonce Equivalence | Continuous round-key binding (1M trials + Z3) | UNSAT 0 / 1,000,000 Collisions |
| Weak-Key Immunity | Z3 / SAW 256-bit constant-time initialization proof | UNSAT 0 Weak Keys |
| Degenerate-Orbit Immunity | Z3 exhaustive boundary search & 24-round init | VERIFIED Constant-Time |
| Rotational-XOR (RX) | 32-core CASCADA symbolic sweep (γ = 1..63) | 63 / 63 UNSAT 0 RX Invariants |
| Mode Domain Isolation | Z3 / SAW mode domain disjointness proof | PROVEN 100% Disjoint |
| Invariant Subspaces | 500 randomized projections, dimensions 1 to 5 | 0 FOUND 0 Invariant Subspaces |
Statistical & Dynamical Randomness Battery
| Validation Suite | Test Scope | Result |
|---|---|---|
| PractRand RNG_test | 32 Terabytes / 235 stats / 42h | ALL PASSED (0 Anomalies) |
| TestU01 BigCrush | 106 tests / 160 stats / 1h54m | ALL PASSED (160 / 160) |
| NIST SP 800-22 Rev 1a | 1 GB / 15 test suites | ALL PASSED (15 / 15) |
| Strict Avalanche (SAC) | 320M pairs / 327,680 cells | 50.0000% (Within 6σ) |
| Differential Collision Sweep | 16,777,216 (224) structured 24R pairs | 0 COLLISIONS (< 2-256 bound) |
| Floyd Cycle Search | 40B+ consecutive iterations | 0 CYCLES Detected |
| Lattice Orbit Analysis | All 7 v2(k) valuation classes | ALL EJECTED (0 Inconclusive) |
| Berlekamp-Massey | 1,048,576-bit sequences | PASSED L(N) ≈ N / 2 |
| Bit Balance & Autocorr | 10 MB high-resolution stream | ALL PASSED |
Algebraic, Neural, Spectral, SCA & Binary Constant-Time Gauntlet
32-thread evaluation across 16 cryptanalysis waves and 38 formal/SMT/SAT/MILP/SCA/RTL tools (Bitwuzla, cvc5, CaDiCaL, Coq, F*, HiGHS, SCALib, Lascar, VerFI, SymbiYosys), including hardware side-channel countermeasures, 256-bit X256 nonces, 256-byte forward-secrecy ratcheting, full-state F256 companion modes, and word-rate W64 throughput acceleration.
Algebraic, Spectral & Neural Distinguishers
| Cryptanalytic Tool / Vector | Target & Scope | Measured Outcome |
|---|---|---|
| 7-Engine Safety Margin | Z3/Bitwuzla SMT, CaDiCaL SAT, HiGHS MILP, OpenMP SAC, Gohr ResNet (R = 1..24) | 4.0x - 24.0x Certified Margin |
| CLAASP 320×64 Diffusion | Coupled & uncoupled bit completeness (R = 0..24) | 100.00% From Round 0 |
| Full-State Bit Diffusion | Bit-level avalanche evaluation (R = 0..24) | PASSED Bias ≤ 0.01624 |
| Gohr Residual Neural Net | Deep residual classifier on keystream pairs (R = 0..24) | 49.26% - 50.57% (|Z| ≤ 1.91 Noise) |
| CROCS BoolTest | 128b, 256b, 512b & degree-2/3 polynomial XOR blocks | PASSED Max |Z| ≤ 4.28 (< 4.89) |
| 14-D Moebius ANF & Cubes | Dinur-Shamir 214 cube summation (64 superpolys) | DEGREE 14 / 14 (64 / 64 Dense) |
| Division-Property Bit-Cone | HiGHS / OR-Tools MILP over 256b key + 64b/256b (X256) nonce |
FULL CONE Degree 192+ Bound |
| Walsh-Hadamard Spectrum | Unwhitened 1-round coordinate projection slice | PASSED High Non-Linearity Bound |
Binary Constant-Time, SCA & Companion Mode Audit
| Verification Vector | Instrumentation & Scope | Measured Outcome |
|---|---|---|
| Hardware Side-Channel Hardening | Datapath power/EM leakage evaluation (SCALib CPA & Lascar TVLA) | CPA |r| = 0.0024 |t| = 0.74 (< 4.50) |
| Multi-Compiler Disassembly | 36 builds across -O0, -O1, -O2, -O3, -Os, -march=native |
0 DIV / 0 JCC 100% Branchless |
| Valgrind ctgrind & Unicorn | Dynamic symbolic secret-key memory taint across all modes | 0 ERRORS 0 Secret Branches |
| X256 Nonce & Ratcheting | 256-bit X256 nonce (2128 birthday) & 256-byte one-way keyframe ratchet |
PROVEN 100% Chunk-Invariant |
| F256 & W64 Companion Suite | Full 256-bit active state across HASH-W64, CFB-256, VFS-256, DRBG-256 |
40B CONTEXT 4x-8x W64 Speedup |
| RTL Synthesis & VerFI | YoWASP-Yosys, SymbiYosys, Bitwuzla & VerFI fault injection | VERIFIED Cycle-Exact & Fault-Locked |
| Full 38-Tool & KAT Battery | 38 external formal/SCA tools + 16-suite C++17 KAT harness | ALL PASSED 100% Bit-Exact |
Round-Reduced Distinguisher & Diffusion Horizon (R = 0 to 24)
Evaluating round-reduced initialization from R = 0 through the full 24-round burn-in across our 7-engine reduced-round security-margin certification establishes a formal 4.0x to 24.0x safety margin across the 24-round initialization schedule.
Round-by-Round Security Margin Telemetry (v2.2.8.4 Engine)
| Burn-In Round | 1st-Byte / 256b Completeness | Coupled / Uncoupled SAC Bias | Gohr Neural Acc (|Z|) | BoolTest Max |Z| | 14-D Moebius ANF Degree | Walsh-Hadamard Spectral Bound | Margin Status |
|---|---|---|---|---|---|---|---|
| Round 0 | 100.00% / 100.00% | 0.01398 / 0.01564 | 49.85% (0.61) | 3.32 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | FULL NOISE |
| Round 1 | 100.00% / 100.00% | 0.00984 / 0.01086 | 49.94% (0.22) | 3.69 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | FULL NOISE |
| Round 2 | 100.00% / 100.00% | 0.00985 / 0.01246 | 49.26% (1.91) | 3.34 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | FULL NOISE |
| Round 4 | 100.00% / 100.00% | 0.00984 / 0.01105 | 50.10% (0.26) | 4.07 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | 6× MARGIN |
| Round 8 | 100.00% / 100.00% | 0.00984 / 0.01362 | 49.94% (0.24) | 3.41 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | 3× MARGIN |
| Round 12 | 100.00% / 100.00% | 0.00988 / 0.01080 | 50.38% (0.98) | 4.28 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | 2× MARGIN |
| Round 24 (Production) | 100.00% / 100.00% | 0.00985 / 0.01088 | 49.90% (0.26) | 3.21 (< 4.89) | 14 / 14 (64/64 Dense) | PASSED (High NL) | PRODUCTION |
Exhaustive Boundary Verification & Vetted Evaluator Access
Formal bit-vector equivalence and boundary verification across the 24-round constant-time initialization schedule.
Formal Bit-Vector & Interactive Theorem Verification
Using Microsoft Z3, Bitwuzla, cvc5, and machine-checked Coq and F* proofs, we modeled the initialization and state transition pipeline to verify strict state injectivity, absence of weak keys, and constant-time handling of all boundary states across the 2256 key space with zero conditional branches (if/else).
Note: Internal state equations, formal SMT proof scripts, CASCADA/CLAASP/BoolTest harnesses, and bit-vector verification models are restricted intellectual property provided exclusively to vetted organizations under an executed Apex Evaluation NDA.
Related Documentation
Application Domains
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