Updated Vernier/SG: a hand cipher kernel that replaces the one-time pad.

The Vernier cipher uses an OTP at its core. OTPs are information-theoretically secure but operationally expensive — pad pages must be generated, distributed, stored, used once, destroyed. Possession is incriminating. The pad is the liability.

Vernier/SG replaces the pad with a digit stream both parties derive independently from a single shared number: a Sophie Germain prime.

A Sophie Germain prime p is one where 2p+1 is also prime. The second prime — q, the "safe prime" — is bonded to p; knowing p determines q.

The stream comes from a multiplicative walk modulo q:

s(n) = g * s(n-1) mod q

where g is the smallest primitive root of q with g >= 5. Each pad digit is the last digit of the digit sum of s(n). The period is exactly 2p, guaranteed by Fermat-Euler.

Worked example, p=83, q=167, g=5:

s: 1, 5, 25, 125, 124, 119, 94, 136, 12, 60, 133, ...
pad: 1, 5, 7, 8, 7, 1, 3, 0, 3, 6, 7, ...

Two parties starting from the same p generate the identical stream. Adding it to a Vernier-encoded message mod 10 is the encryption.

When a kernel's stream exhausts at 2p digits, both parties take the last full value and find the nearest SG prime strictly greater. That's the next kernel. The system self-extends indefinitely from the initial seed alone.

Kerckhoffs's principle holds completely. The algorithm is public, including the lookup table of SG primes — that's mathematics, not key material. The only secret is one number.

Key distribution gets interesting. Verbal: "eighty-three." Contextual: a year both parties remember, walked forward to the nearest SG prime. Dice: roll a handful of d10s in front of the other party, concatenate the digits, walk forward. The dice carry no kernel until you do the walk; the walk is mechanical and reproducible from the rolled digits alone.

Three rejected designs taught the math:

Direct mod-10 extraction produces long runs of repeated digits.
Additive walks produce linear arithmetic progressions with predictable staircases.
Fibonacci recurrences have unpredictable Pisano periods. p=1103 has period 64, shorter than one block.

The multiplicative walk with digit-sum extraction passes chi-squared and autocorrelation across all 318 SG primes tested.

For broadcast use, SKP-1 wraps it: HKDF-SHA512 stream, 10-digit kernel fingerprint, 20-digit HMAC, date-derived counter. Header is 24 digits; the rest is ciphertext plus authentication.

Threat model: informed human adversaries with intercepted traffic and patience. Not nation-state SIGINT — the binary mode with ChaCha20-Poly1305 is for that. What Vernier/SG provides: nothing physical to find, full algorithmic transparency, mathematical deniability, recovery via rotation if a kernel burns.

The pad is gone. The number remains.

https://thetrailingdot.com/posts/vernier-sg/

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