For Limited Capability Processors … here’s the Pearson Hash

For Limited Capability Processors … here’s the Pearson Hash

Not every computing device is a high-powered process and needs the security of cryptography hashes. In fact, there are more processors that run with 8-bit or 16-bit registers than are with powerful 32-bit or 64-bit registers. So we sometimes need methods that process with 8-bit registers, and so in 1990, Peter K. Pearson published a classic paper [here]:

His focus was simplifying the creation of a hashed value of an input string:

So basically we had a string (C) with n characters and iterated a hash value (h) by XOR-ing it with the current character. This is an 8-bit operation (and XORs one byte to another). On each iteration we perform a lookup operation for the 256 possible outcomes of this operation. In the paper, this look-up was defined as:

The great advantage of the method is that it could run on an 8-bit processor, and only needed 256 bytes to store the look-up table. If a device does not have enough space for the storage of 256 bytes, we can also use a permutation function to save space in the memory. An updated table was since published in RFC 3704 defined as the look-up defined defined by t [code]:

#include <stdio.h>

#include <stddef.h>
#include <stdint.h>
#include <cstring>

static const uint8_t t[256] ={
 0xfb, 0xaf, 0x77, 0xd7, 0x51, 0x0e, 0x4f, 0xbf, 0x67, 0x31, 0xb5, 0x8f, 0xba, 0x9d, 0x00, 0xe8,
 0x1f, 0x20, 0x37, 0x3c, 0x98, 0x3a, 0x11, 0xed, 0xae, 0x46, 0xa0, 0x90, 0xdc, 0x5a, 0x39, 0xdf,
 0x3b, 0x03, 0x12, 0x8c, 0x6f, 0xa6, 0xcb, 0xc4, 0x86, 0xf3, 0x7c, 0x5f, 0xde, 0xb3, 0xc5, 0x41,
 0xb4, 0x30, 0x24, 0x0f, 0x6b, 0x2e, 0xe9, 0x82, 0xa5, 0x1e, 0x7b, 0xa1, 0xd1, 0x17, 0x61, 0x10,
 0x28, 0x5b, 0xdb, 0x3d, 0x64, 0x0a, 0xd2, 0x6d, 0xfa, 0x7f, 0x16, 0x8a, 0x1d, 0x6c, 0xf4, 0x43,
 0xcf, 0x09, 0xb2, 0xcc, 0x4a, 0x62, 0x7e, 0xf9, 0xa7, 0x74, 0x22, 0x4d, 0xc1, 0xc8, 0x79, 0x05,
 0x14, 0x71, 0x47, 0x23, 0x80, 0x0d, 0xb6, 0x5e, 0x19, 0xe2, 0xe3, 0xc7, 0x4b, 0x1b, 0x29, 0xf5,
 0xe6, 0xe0, 0x2b, 0xe1, 0xb1, 0x1a, 0x9b, 0x96, 0xd4, 0x8e, 0xda, 0x73, 0xf1, 0x49, 0x58, 0x69,
 0x27, 0x72, 0x3e, 0xff, 0xc0, 0xc9, 0x91, 0xd6, 0xa8, 0x9e, 0xdd, 0x94, 0x9a, 0x7a, 0x0c, 0x54,
 0x52, 0xa3, 0x2c, 0x8b, 0xe4, 0xec, 0xcd, 0xf2, 0xd9, 0x0b, 0xbb, 0x92, 0x9f, 0x40, 0x56, 0xef,
 0xc3, 0x2a, 0x6a, 0xc6, 0x76, 0x70, 0xb8, 0xac, 0x57, 0x02, 0xad, 0x75, 0xb0, 0xe5, 0xf7, 0xfd,
 0x89, 0xb9, 0x63, 0xa4, 0x66, 0x93, 0x2d, 0x42, 0xe7, 0x34, 0x8d, 0xd3, 0xc2, 0xce, 0xf6, 0xee,
 0x38, 0x6e, 0x4e, 0xf8, 0x3f, 0xf0, 0xbd, 0x5d, 0x5c, 0x33, 0x35, 0xb7, 0x13, 0xab, 0x48, 0x32,
 0x21, 0x68, 0x65, 0x45, 0x08, 0xfc, 0x53, 0x78, 0x4c, 0x87, 0x55, 0x36, 0xca, 0x7d, 0xbc, 0xd5,
 0x60, 0xeb, 0x88, 0xd0, 0xa2, 0x81, 0xbe, 0x84, 0x9c, 0x26, 0x2f, 0x01, 0x07, 0xfe, 0x18, 0x04,
 0xd8, 0x83, 0x59, 0x15, 0x1c, 0x85, 0x25, 0x99, 0x95, 0x50, 0xaa, 0x44, 0x06, 0xa9, 0xea, 0x97 };

int main() {

char st[] = "hell0";

uint8_t val=0;

 for (int i=0;i<strlen(st);i++) {
  printf("[%x %x] ",st[i],val);
  val=t[st[i]^val];
  printf("=%x\n",val);
 } 
}

we can see we X-OR each byte with a given value. The result from the X-OR is used as an index to find the next value to use. With the RFC 3074 table, "hello" gives a hash of 0xb9:

[0x68 ^ 0x0]  =0x19
[0x65 ^ 0x19] =0xf1
[0x6c ^ 0xf1] =0x40
[0x6c ^ 0x40] =0xde
[0x6f ^ 0xde] =0xb9

but "hellp" should give F7 hex:

[0x68 0]  =0x19
[0x65 19] =0xf1
[0x6c f1] =0x40
[0x6c 40] =0xde
[0x70 de] =0xf7

A further enhancement for speed is to use the S-box used by AES, as this is supported for fast operations within many processes. The look-up table is then:

Unfortunately an 8-bit hash only gives us 256 hash output values, and where we will have many hash collisions. We thus need a larger hash value, such as for 32 bits, 64 bits or 128 bits. For a 64-bit version, we can iterate eight times for each byte:

for (j = 0; j < sizeof(retval); ++j) { // 8 - for 8 bytes (64 bits)
    // Change the first byte
    h = T[(C[0] + j) % 256];
    for (i = 1; i < len; ++i) {
      h = T[h ^ C[i]];
    }
    retval = ((retval << 8) | h); 
  }

In the following, I have implemented a 32-bit, 64-bit and 128-bit version of the Pearson hash using the AES lookup table:

https://asecuritysite.com/encryption/smh_pearson

and here is the code:

Conclusions

Not every processor has 32-bit or 64-bit registers, and not every processor has lots of memory to store data. The Pearson hash is fast and can run on limited processing devices.