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Theorem · T-SCH23-FINITE

Schiansky et al. 2023 · Supplementary Note 3C

P04 · Nat. Commun. 14, 3849 (published 2023-06-29; corrected 2023-08-21). Read after correction.

Status
VERIFIED FROM SOURCEField read in the primary source at the anchor given.VERIFIED_FROM_SOURCE; NOT_VERIFIED fields: chernoff_applicability
Game
D1
Anchor
Supplementary Note 3C, eqs.(19)-(23), SI PDF p.7-8; Supplementary Table 2 (SI p.10-11); main text Fig.4b
Source file
research/THEOREM_REGISTRY_SEED.yaml

Statement summary

Honest success p_h >= 1 - exp(-(delta_h^2/3) M(e_h,l_h) N) (20), subject to M(e_h,l_h)(1+delta_h) < (1/2)(1-p_m/2) (21). Dishonest success p_d = P[M(e_d,l_d) <= (1-delta_d)(1/2)(1-p_m/2)] <= exp(-(delta_d^2/2)(1/2)(1-p_m/2) N) (22), with (1/2)(1-p_m/2)(1-delta_d) >= M(e_h,l_h)(1+delta_h) (23). N = number of quantum states used to verify ONE bit of the cryptogram. Total token length lambda = N log2|m| = N log2 sqrt|C|.

Citations

Record

Every field of the source record, in file order.

Proof obligationproof_obligation
P-D1
Gamegame

D1

Source paper idsource_paper_id
P04
Source urlsource_url

https://doi.org/10.1038/s41467-023-39519-w

Anchoranchor

Supplementary Note 3C, eqs.(19)-(23), SI PDF p.7-8; Supplementary Table 2 (SI p.10-11); main text Fig.4b

Statusstatus

VERIFIED_FROM_SOURCE

Bound latexbound_latex
p_d\le\exp\!\left(-\frac{\delta_d^{2}}{2}\cdot\frac{1}{2}\left(1-\frac{p_m}{2}\right)N\right)
LaTeX source
Published valuespublished_values
evidence_kind: PUBLISHED_AGGREGATE
values:
  delta_h: 0.031
  delta_d: 0.01
  p_d: 5.911e-45
  p_d_unc: 8.8e-47
  N_fig4: 4.2e6
  N_SI_table2: 4482440
  p_m_SI: 0.0602
  p_m_main: 0.0676
Recomputationrecomputation

RECOMPUTED_BY_LANE_A: exp(-(0.01^2/2)*0.5*(1-0.0602/2)*N) gives 5.91e-45 only with N = 4.2e6 (exponent 101.84). With the SI Table 2 N = 4,482,440 it gives 6.3e-48. So the reported p_d uses N = 4.2e6 and p_m = 6.02%.

Caveatscaveats
  • Chernoff-Hoeffding (SI eqs 7-8) is for sums of independent {0,1} variables. M = (4+2 sqrt2) e + l is a weighted combination with weight > 1. Lane D must check whether the bound is applied within its hypotheses (open question, NOT a claim of error).
  • Independence across the N states is assumed in the Chernoff step.
  • MAC forgery p_t = 1/sqrt|C| (main text; SI Note 4) is a separate term. The overall cheating probability composition is not stated as a single theorem.
Verificationverification
statement_summary: VERIFIED_FROM_SOURCE
bound_latex: VERIFIED_FROM_SOURCE
published_values: VERIFIED_FROM_SOURCE
recomputation: RECOMPUTED_BY_LANE_A
chernoff_applicability: NOT_VERIFIED
Not verified reasonsnot_verified_reasons
chernoff_applicability: Needs a formal check by lane D. Lane A only flags it.

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