ReconcileHashing

Privacy amplification, Toeplitz seeds and the authentication key bill

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Test	What it does	Result
test_amplify_toll	Privacy amplification costs 64.44 bits at eps = 1e-10 at every block length, so block cadence alone sets its key-rate share.	✅ pass
test_auth_epsilon	The 154 bits auth_cost bills carry a forgery probability of 1.79856e-10, under the 3e-10 of three messages at 1e-10, tag_length rounding each tag up to a whole bit.	✅ pass
test_auth_recycle	A one-time-padded tag reuses the 2k-bit hash seed, so a round costs k bits per message, not 3k.	✅ pass
test_engine_toll	mdi_length, pairing_length and the delta of sixstate_length and cv_finite rebuild bit for bit from hash_charge at each engine's own share, the Gaussian one two bits higher.	✅ pass
test_hash_lemma	The leftover hash lemma extracts $\lfloor H_{min}-2{\log }_2(1/(2ϵ))\rfloor$ bits, within $ϵ$ of uniform at that length.	✅ pass
test_lfsr_compare	DECLARED CORRECTION. At a seed twice the tag length Krawczyk's later bound sits a factor $2n/(n+k)$ BELOW his Theorem 9 one, not above as Fung, Ma & Chau state, and the costed tag uses the larger.	✅ pass
test_net_length	Fung, Ma & Chau's net key length subtracts every secret bit the round spent and clamps at zero.	✅ pass
test_poly_collision	The polynomial family is $\delta$-almost two-universal, $\delta =(r-1)/|F|$, so not two-universal, and growing with the message chunks.	✅ pass
test_tag_round	tag_length inverts Krawczyk's $n{2}^{-k+1}$: the tag meets the forgery probability and one bit shorter does not.	✅ pass
test_toeplitz_seed	A plain Toeplitz matrix needs n + l - 1 seed bits and the modified $(T\mid I)$ family n - 1.	✅ pass
test_verify_link	eps_cor is frame error rate times hash collision probability: Martinez-Mateo's 9.3e-5 at 2% over a 64-bit tag gives 5.0416e-24, four decades under the collision probability alone.	✅ pass