HalftimeHash24 (advanced 24-byte API): which entry point, and one key condition for every leaf
halftime_hash::advanced::V1<3> ..
V4<3> return 24 bytes and use block widths b = 1, 2,
4, 8. The Encode3 encoder has distance 1: a low-half change in one
encoded word meets exactly one NH term, which vanishes when the high 32
bits of the matching core key word are zero (a 2^-32 key class). The
witness is 168b zero bytes versus the same with little-endian word 6b
equal to 1.
Scope: V1<3> versus V4<3>
For V4<3> (b = 8, the entry point the AVX-512 timing uses) every message shorter than 1344 bytes goes through the finalizer's tail path, where each word meets three NH terms. So the 168-byte pair that scores V1<3> does not collide on V4<3> even in the key class:
| entry point | pair | high32(core_key[6]) = 0 | outside the class | uniform keys |
|---|---|---|---|---|
| V1<3> (b = 1) | 168 B, word 6 | 1,048,576 / 1,048,576 | 0 / 1,048,576 | 0 / 1,048,576 |
| V4<3> (b = 8) | 1344 B, word 48 (byte 384 = 0x01) | 262,144 / 262,144 | 0 / 262,144 | 0 / 262,144 |
| V4<3> (b = 8) | 168 B, word 6 | 0 / 1,048,576 | 0 / 1,048,576 | 0 / 1,048,576 |
(logs/runs_xeon_avx512.log; the scalar build on arm64
repeats all three rows with 4,096 keys each,
logs/runs_m2_scalar.log.) With epsilon = 2^-32: V1<3>
caps at log2(21) + 32 = 36.39 bits (L = 21); V4<3> caps at
log2(168) + 32 = 39.39 bits (L = 168).
Long inputs: the condition frees every leaf
All leaves share the base-layer key, so the same single key condition frees the varied word in every 1344-byte leaf at once (V4<3>): a family varying 8 leaves of a 64 KB message (2^8 members) collides entirely for 64/64 class keys and 0/64 outside; varying 10 leaves of a 1 MB message (2^10 members), 16/16 class keys (AVX-512) and 128/128 (scalar arm64), 0 outside, 0 uniform. Since each of the 780 leaves of a 1 MB message offers 8 free 32-bit halves, the family has about 2^(32 * 8 * 780) ~ 2^200,000 members, all colliding on the same 2^-32 key class. Only the sub-families above were executed.
Key-free window (the raw core does not encode length): zero messages of lengths 0..63 (b = 8) or 0..7 (b = 1) collide for 2000/2000 keys, and the next length never joins them (0/2000).
Build and run
curl -LO https://raw.githubusercontent.com/jbapple/HalftimeHash/caf7924ceab4721f4e0cc33442b185558ba7f1c4/halftime-hash.hpp
c++ -O2 -march=native -std=c++17 -pthread -o halftime24_mc halftime24_mc.cpp
# arm64: the header's NEON dispatch does not compile (upstream issue #1); force the scalar path:
c++ -O2 -std=c++17 -pthread -U__ARM_NEON -U__ARM_NEON__ -o halftime24_mc halftime24_mc.cpp
./halftime24_mc class 1 168 6 6 1048576 101
./halftime24_mc class 8 1344 48 6 262144 102
./halftime24_mc class 8 168 6 6 1048576 103
./halftime24_mc long 1048576 10 16 802On x86-64 with AVX-512 the program first checks that the dispatched V2..V4 equal the header's scalar V2..V4 on 8,400 random inputs (0 mismatches in the logs).