integer-pairing

This library enables encodings of integer tuples as one integer. It implements two well-known types of encodings - Cantor and Szudzik. There is a great article on those two types. It also implements a slight generalization.

Usage

The base example is

from integer_pairing import cantor, szudzik

cantor.pair(11, 13) # 313
cantor.unpair(313) # (11, 13)

szudzik.pair(11, 13) # 180
szudzik.unpair(180) # (11, 13)

You can pair tuples of any size, but have to specify the size when unpairing

cantor.pair(11, 13, 17, 19, 23) # 1115111727200556569
cantor.unpair(1115111727200556569, dim=5) # (11, 13, 17, 19, 23)

It is also possible to include negative numbers, but you need to imply that when decoding

cantor.pair(11, 13, -1) # 726618
cantor.unpair(726618, dim=3, neg=True) # (11, 13, -1)

Naive implementations of the above algorithms fail to account for very large integers, as they use numeric calculation of the square root. Python allows for integers of any size to be stored, but converts them to float (64 bits) when doing numeric operations, so this approximation ruins the unpairing. Luckily this can be (efficiently) solved and is implemented here.

cantor.pair(655482261805334959278882253227, 730728447469919519177553911051)
# 960790065254702046274404114853633027146937669672812473623832
cantor.unpair(960790065254702046274404114853633027146937669672812473623832)
# (655482261805334959278882253227, 730728447469919519177553911051)

You can also pair (signed) integers in a way that encodes the tuple's dimension. This is called bundling and is done by encoding each number in a tuple in binary, then prepending those encodings by the number 2 or 22, depending on the number's sign. For space-efficiency, the string is then interpreted in a trinary base system.

from integer_pairing import bundle

bundle.pair(*range(-8,8))
# 1061264631713144962268472871675
bundle.unpair(1061264631713144962268472871675)
# (-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7)

The downside is that bundle.pair is not surjective, so not every number can be unpaired. Thus calling unpair on an invalid number will produce an exception

bundle.unpair(0)
              
Traceback (most recent call last):
  File "<pyshell#37>", line 1, in <module>
    bundle.unpair(0)
  File "...\integer_pairing\_interface.py", line 66, in unpair
    return self._unbundle(n)
  File "...\integer_pairing\_bundle.py", line 41, in _unbundle
    di = 1 if s[i+1] != '2' else 2
IndexError: string index out of range

Complexity

The pairing of n integers will result in an integer of the size of about their product.

Example usage from Cryptography

When encrypting messages deterministically, an attacker can always reproduce the encryption of any chosen messages. If the possibilities are few (e.g. true or false), those kinds of algorithms are pretty useless. This is solved by appending a random number, called salt, to the message. It can be useful to implement this appending via pairing.

from random import getrandbits

salt = getrandbits(128)
message = 0
encoded = szudzik.pair(message, salt)

Also, public-key cryptography can often only deal with integers, so messages have to be encoded accordingly. You can easily acomplish this with bundling.

txt2int = lambda m: bundle.pair(*map(ord, m))
int2txt = lambda n: ''.join(map(chr, bundle.unpair(n)))

message = 'hi there!'
message_enc = txt2int(message)
# 2050221782650890524283503336306989
message_dec = int2txt(message_enc)
# 'hi there!'

But there are better ways of doing this.