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Description:

modnestexp 1.1.1

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
=================================================================