# MIT License
#
# Copyright (c) 2019-2023 Knoxen
#
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# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
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# The above copyright notice and this permission notice shall be included in all
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#
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# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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# SOFTWARE.
defmodule Puid.Entropy do
@moduledoc """
[Entropy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) related calculations
The implementation is based on mathematical approximations to the solution of what is often
referred to as the [Birthday
Problem](https://en.wikipedia.org/wiki/Birthday_problem#Calculating_the_probability).
"""
@doc """
Entropy bits necessary to generate `total` number of `puid`s with `risk` risk of repeat.
The total number of possible `puid`s is 2<sup>bits</sup>.
Risk is expressed as a 1 in `risk` chance, so the probability of a repeat is `1/risk`.
## Example
iex> Puid.Entropy.bits(10.0e6, 1.0e12)
85.37013046707142
"""
@spec bits(non_neg_integer(), non_neg_integer()) :: float()
def bits(0, _), do: 0
def bits(1, _), do: 0
def bits(_, 0), do: 0
def bits(_, 1), do: 0
def bits(total, risk) do
n =
cond do
total < 1000 ->
:math.log2(total) + :math.log2(total - 1)
true ->
2 * :math.log2(total)
end
n + :math.log2(risk) - 1
end
@doc """
Approximate total number of `puid`s which can be generated using `bits` bits entropy at a `risk` risk of repeat.
The total number of possible `puid`s is 2<sup>bits</sup>.
Risk is expressed as a 1 in `risk` chance, so the probability of a repeat is `1/risk`.
Due to approximations used in the entropy calculation this value is also approximate; the approximation is
conservative, however, so the calculated total will not exceed the specified `risk`.
## Example
iex> bits = 64
iex> risk = 1.0e9
iex> Puid.Entropy.total(bits, risk)
192077
"""
@spec total(bits :: float(), risk :: float()) :: integer()
def total(0, _), do: 0
def total(1, _), do: 1
def total(_, 0), do: 1
def total(_, 1), do: 1
def total(bits, risk) do
round(2 ** ((bits + 1) / 2) * :math.sqrt(:math.log(risk / (risk - 1))))
end
@doc """
Risk of repeat in `total` number of events with `bits` bits entropy.
The total number of possible `puid`s is 2<sup>bits</sup>.
Risk is expressed as a 1 in `risk` chance, so the probability of a repeat is `1/risk`.
Due to approximations used in the entropy calculation this value is also approximate; the approximation is
conservative, however, so the calculated risk will not be exceed for the specified `total`.
## Example
iex> bits = 96
iex> total = 1.0e7
iex> Puid.Entropy.risk(bits, total)
1501199875790165
iex> 1.0 / 1501199875790165
6.661338147750941e-16
"""
@spec risk(bits :: float(), total :: float()) :: integer()
def risk(0, _), do: 0
def risk(1, _), do: 1
def risk(_, 0), do: 1
def risk(_, 1), do: 1
def risk(bits, total) do
events = 2 ** bits
exponent = -1.0 * ((total - 1) * total) / (2 * events)
round(1 / (1 - :math.exp(exponent)))
end
@doc """
Entropy bits per `chars` character.
`chars` must be valid as per `Chars.charlist/1`.
## Example
iex> Puid.Entropy.bits_per_char(:alphanum)
{:ok, 5.954196310386875}
iex> Puid.Entropy.bits_per_char("dingosky")
{:ok, 3.0}
"""
@spec bits_per_char(Puid.Chars.puid_chars()) :: {:ok, float()} | Puid.Error.t()
def bits_per_char(chars) do
with {:ok, charlist} <- chars |> Puid.Chars.charlist() do
{:ok, charlist |> length() |> :math.log2()}
else
error ->
error
end
end
@doc """
Same as `bits_per_char/1` but either returns __bits__ or raises a `Puid.Error`
## Example
iex> Puid.Entropy.bits_per_char!(:alphanum)
5.954196310386875
Puid.Entropy.bits_per_char!("dingosky")
3.0
"""
@spec bits_per_char!(Puid.Chars.puid_chars()) :: float()
def bits_per_char!(chars) do
with {:ok, ebpc} <- bits_per_char(chars) do
ebpc
else
{:error, reason} ->
raise(Puid.Error, reason)
end
end
@doc """
Entropy bits for a binary of length `len` comprised of `chars` characters.
`chars` must be valid as per `Chars.charlist/1`.
## Example
iex> Puid.Entropy.bits_for_len(:alphanum, 14)
{:ok, 83}
iex> Puid.Entropy.bits_for_len('dingosky', 14)
{:ok, 42}
"""
@spec bits_for_len(Puid.Chars.puid_chars(), non_neg_integer()) ::
{:ok, non_neg_integer()} | Puid.Error.t()
def bits_for_len(chars, len) do
with {:ok, ebpc} <- bits_per_char(chars) do
{:ok, (len * ebpc) |> trunc()}
else
error ->
error
end
end
@doc """
Same as `Puid.Entropy.bits_for_len/2` but either returns __bits__ or raises a
`Puid.Error`
## Example
iex> Puid.Entropy.bits_for_len!(:alphanum, 14)
83
iex> Puid.Entropy.bits_for_len!("dingosky", 14)
42
"""
@spec bits_for_len!(Puid.Chars.puid_chars(), non_neg_integer()) :: non_neg_integer()
def bits_for_len!(chars, len) do
with {:ok, ebpc} <- bits_for_len(chars, len) do
ebpc
else
{:error, reason} ->
raise(Puid.Error, reason)
end
end
@doc """
Length needed for a string generated from `chars` to have entropy `bits`.
`chars` must be valid as per `Chars.charlist/1`.
## Example
iex> Puid.Entropy.len_for_bits(:alphanum, 128)
{:ok, 22}
iex> Puid.Entropy.len_for_bits("dingosky", 128)
{:ok, 43}
"""
@spec len_for_bits(Puid.Chars.puid_chars(), non_neg_integer()) ::
{:ok, non_neg_integer()} | Puid.Error.t()
def len_for_bits(chars, bits) do
with {:ok, ebpc} <- bits_per_char(chars) do
{:ok, (bits / ebpc) |> :math.ceil() |> round()}
else
error ->
error
end
end
@doc """
Same as `Puid.Entropy.len_for_bits/2` but either returns __len__ or raises a
`Puid.Error`
## Example
iex> Puid.Entropy.len_for_bits!(:alphanum, 128)
22
iex> Puid.Entropy.len_for_bits!('dingosky', 128)
43
"""
@spec len_for_bits!(Puid.Chars.puid_chars(), non_neg_integer()) ::
non_neg_integer() | Puid.Error.t()
def len_for_bits!(chars, bits) do
with {:ok, len} <- len_for_bits(chars, bits) do
len
else
{:error, reason} ->
raise(Puid.Error, reason)
end
end
end
