Modular nested exponentiation

An algorithm that computes modular nested exponentiation efficiently.

🚩 Table of Contents

• Overview
• Prerequisites
• Installation
• Examples

🗺️ Overview

mod-nest-exp exports a Python function mod_nest_exp that takes as input an arbitrarily long sequence of positive integers a₁, a₂, ..., aₙ and a positive integer m and computes a₁^(a₂^(···^aₙ)) mod m efficiently (that is, without computing the value of the nested exponent).

To date, this problem was not solvable by computers in the general case.

🏳️ Prerequisites

mod-nest-exp requires Python v3.6+.

For best performance, install gmpy2 and sympy:

$ apt install libgmp-dev libmpfr-dev libmpc-dev # required for gmpy2
$ pip install gmpy2 sympy

The libraries offer more efficient alternatives to a number of functions used as subroutines in the core module.

🔧 Installation

The recommended installation method is from PyPI:

$ pip install mod-nest-exp

A development version can be installed from GitHub source using setuptools:

$ git clone https://github.com/avivbrook/modular-nested-exponentiation.git
$ cd modular-nested-exponentiation
$ python setup.py install

💡 Examples

Small inputs

>>> from mod_nest_exp import mod_nest_exp
>>> mod_nest_exp([6,5,4,3,2], 1948502738) # 6^(5^(4^(3^2))) mod 1948502738
951546056

Larger inputs

Here we demonstrate a computation that is not possible with simple modular exponentiation functions such as pow:

>>> from random import randint
>>> seq = [randint(1, 2**64) for _ in range(5)]
>>> seq
[6038140174510300905, 11769918488496772646, 2874847465098133786, 9008748983185995190, 13009674817390511365]
>>> m = randint(1, 2**64)
>>> m
6790053138492639247
>>> mod_nest_exp(seq, m)
3426314670852891859

Benchmark the main function

>>> from mod_nest_exp.core.benchmarks import benchmark_core
>>> benchmark_core(list_lengths=(10, 100, 1000), bit_lengths=(16, 128, 1024), mod_bit_lengths=(16, 32, 64))
Running mod_nest_exp on sequences of l pseudorandom b-bit positive integers over a B-bit modulus (1000 runs per table entry)
=================================================================
                            sequence length l
                  10               100               1000
          ----------------- ----------------- -----------------
  B     b     mean    stdev     mean    stdev     mean    stdev
-----------------------------------------------------------------
       16 |   0.08     0.04     0.08     0.03     0.10     0.03
 16   128 |   0.08     0.11     0.08     0.03     0.10     0.04
     1024 |   0.08     0.03     0.08     0.03     0.11     0.04
-----------------------------------------------------------------
       16 |   0.34     0.32     0.34     0.24     0.35     0.24
 32   128 |   0.33     0.23     0.34     0.23     0.36     0.23
     1024 |   0.33     0.22     0.33     0.24     0.37     0.24
-----------------------------------------------------------------
       16 |   8.82    34.83     6.20    21.27     7.18    30.35
 64   128 |   7.66    30.70     6.71    22.72     7.60    26.92
     1024 |   5.94    25.10     6.67    20.78     6.76    26.28
=================================================================