a5hash-128: an 18-byte every-seed family, and a 2^384 long-input family
1. 18 bytes
On the 17..32-byte path, c = lu32(m + len - 16)<<32 | lu32(m + len - 12) and d = the last 8 bytes enter one product umul128(c + Seed3, d + Seed4) with the public constants Seed3 = 0xA4093822299F31D0 and Seed4. Setting c = -Seed3 makes that product (0, 0) whatever d is. At 18 bytes, bytes 16..17 occur only in d, so any 16-byte prefix with c = -Seed3 followed by all 2^16 two-byte tails collides on the full 128-bit output for every seed (256 members at 17 bytes). The 64-bit a5hash has no such public product.
Example members (memory order), from mc_a5.cpp:
5e74ddc7f65b30ce60d68503aceb37f20000
5e74ddc7f65b30ce60d68503aceb37f20100| program | implementation | result | log |
|---|---|---|---|
mc_a5.cpp (x86-64) |
SMHasher3 a5hash.cpp v5.21 (89406B11, 14AD402C) |
18 B, 65,536 members: 65,536/65,536 seeds (full 128 bits); 17 B, 256 members: 65,536/65,536 | logs/mc_a5_xeon_s16.txt |
../museair/indep_museair_a5_murmur.cc (M2) |
upstream a5hash.h v5.25, unmodified; family regenerated
independently |
65,536 members, 4097/4097 seeds (full 128 bits) | logs/indep_museair_a5_murmur_m2.txt |
c++ -O2 -std=c++17 mc_a5.cpp -o mc_a5 && ./mc_a5 162. Long inputs: six zeroed products in the first 192 bytes
Only lane 1 carries the seed at the start of the bulk loop; lanes 2..4 start from public constants and stay public for one or two more 64-byte blocks. Zeroing words (block 0: m8 = -Seed3, m16 = -Seed5, m24 = -Seed7; block 1: m16 = -Seed3, m24 = -Seed5; block 2: m24 = -Seed3) make six public products (0, 0) and free their six partner words: 2^384 messages, anchored in the first 192 bytes, with any content and length after that (at least 96 bytes for block 0, 224 for all three).
| length | full 128-bit collisions | control (one zeroing word broken) |
|---|---|---|
| 96 B | 65,536/65,536 | 0/256 |
| 224 B, 256 B, 1 KiB, 64 KiB | 65,536/65,536 each | 0/256 |
| 1 MiB | 256/256 | 0/256 |
| 2^16 family at 1000 B | all members equal for 1024/1024 seeds |
a5_long.cpp:
c++ -O2 -std=c++17 a5_long.cpp -o a5_long && ./a5_long;
log logs/a5_long_xeon.txt.