QRNG Min-Entropy Certification & Toeplitz Extraction

Raw Photon-Detection Source

Detector bias P(1) 0.53

Afterpulse correlation ρ 0.12

Raw block size n 256 bits

Security margin ε 10%

Entropy Certification

Empirical P(1)

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Min-entropy / bit

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Raw entropy budget

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Compression ratio

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Certified output bits m (Toeplitz-extracted)

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Certified throughput (auto-run)

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How it works

A real hardware QRNG never outputs perfectly unbiased, independent bits: detector inefficiency skews P(1) away from 0.5, and effects like APD afterpulsing/dead time correlate each bit with the previous one. This sim models the raw stream as an order-1 Markov source and certifies exactly how much true randomness survives.

Source:   P(1|prev=0)=p01,  P(1|prev=1)=p11   (fit from bias b, correlation ρ)
Predictor: worst case an adversary who knows "prev" guesses right with
           q = max(p01, 1-p01, p11, 1-p11)
Min-entropy/bit:  H_min = -log2(q)      (order-1 Markov estimate, cf. NIST SP 800-90B)

The raw n-bit block carries at most n·H_min bits of true randomness. A public, reusable Toeplitz matrix T (m×n, built from a random seed of length n+m-1, T[i][j] = seed[i+j]) then extracts a shorter output that is provably close to uniform, by the Leftover Hash Lemma:

output = T · raw   (mod 2, GF(2) matrix-vector product)
m  =  floor( n · H_min · (1 − ε) )       ε = security margin

Detector bias / correlation — worsen the source; watch H_min fall and m shrink even though n is unchanged.
Security margin ε — extra bits shaved off m to bound statistical distance from ideal uniform output (finite-block-size cushion).
Generate Block — draws a fresh raw block, computes stats, and runs the real Toeplitz extraction once.

This is the same two-stage design (raw quantum entropy source → certified randomness extractor) used by commercial QRNGs such as ID Quantique's Quantis, which must pass NIST SP 800-90B min-entropy certification before their extracted output is trusted for cryptographic keys.

Idle

Raw bit = 1 Raw bit = 0 Certified output bit

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