nx_u2048_millerrabin.nx

nx_u2048_millerrabin.nx -- F103e RUNG 5: probabilistic primality over u2048. The gate for RSA KEYGEN, which is the gate for signing, which is the gate for enrolling our own Secure Boot key (debt 1786237435: real EDK2 with MS keys refuses our unsigned .efi; the SAME secboot firmware in SETUP mode runs it, so the blocker is the enrolled key set). Unblocked by nx_rsa2048_mod_exp_big: Miller-Rabin needs a^d mod n with d ~1024 bits, and the estate's incumbent modexp takes the exponent as an i64 -- it structurally could not do this. ★NO RECALLED CONSTANTS. Every fixture's primality is established IN-TEST by independent i64 trial division; the u2048 Miller-Rabin path is then required to agree. Two methods, one answer. An RSA modulus recited from memory would be a fabricated constant, which is exactly what the estate's rules forbid -- so the key gets GENERATED (next rung), never quoted. ★THE ANTI-VACUITY TOOTH IS THE CARMICHAEL FAMILY. 561 = 3*11*17, 1105 = 5*13*17, 1729 = 7*13*19 -- built here AS PRODUCTS, so their compositeness is proven by construction rather than remembered. They satisfy the FERMAT test for every coprime base, so an implementation that computed a^(n-1) mod n instead of doing the square-root descent passes every other test in this file and fails only these. ★★★★★★A WEAKER ALGORITHM WEARING THE RIGHT NAME PASSES EVERY TEST EXCEPT THE ONE THAT WAS DESIGNED TO SEPARATE THEM. ★AND ONE TOOTH PROVES THE BASE LOOP MATTERS: 23*89 = 2047 is a strong pseudoprime to base 2, so MR with bases {2} must report PROBABLY-PRIME (wrong, and expected) while {2,3} reports COMPOSITE. Without this, a single-base implementation would look perfect. Usage: nx_u2048_millerrabin selftest Exit: 0 GREEN | 1 RED. Log -> knowledge/status/nishi_os.log, verdict= LAST. license_tier: ORIGINAL