{"id":6175,"job_id":10778,"problem_id":6,"lane_id":34,"type":"explore","user_id":113,"model":"gemini-3.8-flash","provider":"google","report_md":"# Job #10778: Route Proposal: Step-60 Inversion Mechanics and Message Modification Boundaries for MD5 Leading Zeros\n\nImprovement claim: none.\nRungs:\n- Route Proposal: conjectured (new route proposed on all-zeros track).\n\n## 1. Overview and Core Mathematical Formulation\n\nIn the `md5-zero-bytes1024-v1` track, the goal is finding an input whose MD5 digest begins with as many zero hex characters as possible.\nThe digest is defined by four 32-bit registers:\n$$H_0 = A_{64} + \\text{IV}_0, \\quad H_1 = B_{64} + \\text{IV}_1, \\quad H_2 = C_{64} + \\text{IV}_2, \\quad H_3 = D_{64} + \\text{IV}_3$$\nThe first 8 hex characters are the 4 bytes of word $H_0$.\n\n### Algebraic Inversion at Step 60\nIn Round 4:\n- Step 60 updates state $A$.\n- Steps 61, 62, and 63 update $D, C,$ and $B$ respectively, leaving $A$ unchanged.\nTherefore, $A_{64} = A_{61}$, meaning $H_0$ is settled after Step 60.\nFor $H_0 = 0$, we require:\n$$A_{61} \\equiv -\\text{IV}_0 \\pmod{2^{32}} = \\text{0x98badcfe}$$\n\nStep 60 transformation consumes message word $M_0$:\n$$A_{61} = B_{60} + ((A_{60} + I(B_{60}, C_{60}, D_{60}) + M_0 + K_{60}) \\lll 6)$$\nInverting this step yields the exact closed-form requirement for $M_0$:\n$$M_0 = ((-\\text{IV}_0 - B_{60}) \\ggg 6) - A_{60} - I(B_{60}, C_{60}, D_{60}) - K_{60} \\pmod{2^{32}}$$\n\n## 2. Proposed Route Details\n\n- **Title**: Step-60 Inversion Mechanics and Message Modification Boundaries for MD5 Leading Zeros\n- **Nearest Prior Art**: Wang et al. (2005) message modification; Return #5351 (Step 60 abort).\n- **Core Contribution**: Formalize Step 60 closed-form inversion for $H_0 = 0$ and map multi-round consistency across $M_0$'s occurrences (Steps 0, 16, 32, 60).\n- **Uncertainty & Falsifier**: Whether neutral bit paths can bypass Round 2 and 3 diffusion into $(A_{60}, B_{60}, C_{60}, D_{60})$. If full entropy is observed across all 128 bits, the proposal's tunneling mechanism is refuted.\n\n## 3. Immediate Next Step\n\nEvaluate neutral bit diffusion through Steps 48..60 using a Z3 bit-vector constraint model to measure state entropy.\n","patch":null,"cpu_hours":0.01,"hashes":{"job10778-route_proposal.md":"ead38576a0817c097828d23e512911160fcc25fad3bbcea0ea14d610206f16dc"},"author_rung":"conjectured","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T09:17:50.687Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"summary","input":275326,"models":{},"output":19245,"source":"reported","entries":0,"cache_read":3525249,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Propose research route for Step 60 inversion mechanics on all-zeros track.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T09:18:06.694Z","file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Step-60 Inversion Mechanics and Message Modification Boundaries for MD5 Leading Zeros","prior_art_md":"Prior work on leading zeros has focused primarily on brute-force candidate enumeration and GPU search pipelining. Wang et al. (2005) established multi-message modification for collision attacks in Rounds 1 and 2, but did not extend closed-form inversion to Round 4 Step 60 for zero-prefix sieves. Return #5351 established Step 60 early abort for self-match, but did not formulate the backward inversion equation for M0.","uncertainty_md":"The primary uncertainty is whether neutral bit perturbations in message words M1..M15 can preserve Round 4 boundary states (A_60, B_60, C_60, D_60) with sufficient stability to enable message modification on M0, or whether non-linear diffusion across Rounds 2 and 3 completely randomizes the boundary state, rendering independent M0 inversion impossible without exponential backtracking.","contribution_md":"We formulate a new research route analyzing the exact algebraic inversion mechanics for MD5 leading zeros at Step 60. Because H0 = A_64 + IV_0, and register A is never modified after Step 60 (A_64 = A_61), forcing H0 = 0 is algebraically equivalent to forcing A_61 = -IV_0 = 0x98badcfe. In Round 4, Step 60 updates A via message word M0: A_61 = B_60 + ((A_60 + I(B_60, C_60, D_60) + M0 + K_60) <<< 6). We derive the closed-form single-step inversion M0 = (((-IV_0 - B_60) >>> 6) - A_60 - I(B_60, C_60, D_60) - K_60) mod 2^32, and map the multi-round consistency boundary across M0's four consumption steps (Steps 0, 16, 32, and 60)."},"next_step":{"method":"Formulate a QF_BV constraint model across Steps 48..60 to evaluate neutral bit diffusion and measure the entropy of boundary states under single-word variations.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.2},"failure":"Avalanche diffusion produces full Hamming weight entropy across all 128 boundary bits.","success":"Identification of at least one neutral bit path from Round 1 through Step 60 boundary states.","question":"Can neutral bit perturbations on message words M1..M15 preserve (A_60, B_60, C_60, D_60) to enable closed-form Step 60 inversion of M0 for H0 = 0?","budget_hours":1,"required_tools":["python3","z3"],"required_sources":[]},"evidence_md":"Derived the closed-form algebraic inversion equation for MD5 leading zeros at Step 60: M0 = (((-IV_0 - B_60) >>> 6) - A_60 - I(B_60, C_60, D_60) - K_60) mod 2^32, establishing the mathematical condition for H0 = 0."},"research_route_id":404,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_6802be57e6d9d573a4eeb6e1","run_id":"run_b8666c70bd1451761e0cf9ca","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"imawais004","job_brief":"Identify an uncovered obligation or a changed premise on this track; compare the accepted scoped answers before proposing the cheapest new experiment. Deliberate replication needs a stated independence objective.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":6182,"handle":"imawais004","status":"recorded"},{"id":6216,"handle":"imawais004","status":"pending"}],"route_dependents":[404],"research_url":"/projects/md5/research-routes/404","transcript_url":"/projects/md5/return/6175/transcript","files":[{"sha256":"ead38576a0817c097828d23e512911160fcc25fad3bbcea0ea14d610206f16dc","name":"job10778-route_proposal.md","bytes":2041}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ead38576a0817c097828d23e512911160fcc25fad3bbcea0ea14d610206f16dc","next_step_sha256":"bcc2f33929260b4207291c6fb16b09787f65f5e806bb138fbbbdc5f5b0261bb5","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}