post-quantum cryptography · the geometry underneath

Proved, or assumed

Lattice security rests on how short a vector you can find. The searching is heuristic and always will be; the claims about what was found do not have to be. Solid here was decided in exact integers. Dashed was asserted.

progress— shortest ‖v‖— decided— lattice—

SVP Challenge, TU Darmstadt · determinants proved entry by entry · π bracketed by Machin · no floating point in any claim

view

run it

step

LLL on the real dimension-40 challenge basis, 20,154 steps and 10,219 swaps in 0.2 s. Every quantity is an integer: floating-point Gram–Schmidt does not survive a 121-digit modulus sitting beside unit entries, so there is none here. The lattice is checked unchanged at the end — det² = q², exactly.

marks

The geometric series assumption says a reduced profile is a straight line with root-Hermite factor 1.0219. It is an empirical claim about random lattices, asserted everywhere and proved nowhere, so it is drawn dashed. Against this basis it misses the measured profile by 0.15 decades and misses worst at the ends.

marks

926 published records. Each prints ‖v‖/GH to six figures as a float with no error bound, and none exceeds 1.04985 — so 1.05 is a hard wall and the table is pressed against it.

what was checked

37 records decided in exact rational arithmetic against 14 determinants, dimensions 40–210. All 37 published ratios are consistent; none is wrong.

The bar on a decided record is every ratio consistent with a true norm that prints as the integer printed. Comparing against the ratio at the rounded norm instead — the obvious thing to do — reports 32 of 37 as disagreeing, and that is an artefact of the rounding rather than a finding. Half a unit of norm is worth 1.7×10⁻⁴ here, which is the size of every apparent discrepancy.

The one open row. Dimension 119, seed 0, norm 2904: admissible if the norm is exactly 2904, refused if it is half a unit more. Its window straddles the wall, so it cannot be decided from what is published — it needs the vector, and the vectors are not published.

This folder audits. It does not propose a cryptosystem, a parameter set, or a variant of one, and it never will — that is written into its charter, not left to judgement.