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Caustic is an Elixir cryptocurrency library which containsalgorithms used in Bitcoin, Ethereum, and other blockchains.It also includes a rich cryptography, number theory,and general mathematics class library.You can use Caustic to quickly implement your own crypto walletor client. With the low-level math library, you can have fun withexploratory mathematics.

Warning: This library is developed for learning purposes. Please do notuse for production.

https://hexdocs.pm/caustic/

Add tomix.exs of your Elixir project:

defpdepsdo[{:caustic,"~> 0.1.22"}]end

And then run:

mix deps.get

You can generate Bitcoin private keys.

privkey=Caustic.Secp256k1.generate_private_key()# 55129182198667841522063226112743062531539377180872956850932941251085402073984privkey_base58check=Caustic.Utils.base58check_encode(<<privkey::size(256)>>,:private_key_wif,convert_from_hex:false)# 5Jjxv41cLxb3hBZRr5voBB7zj77MDo7QVVLf3XgK2tpdAoTNq9n

You can then digitally sign a message.

pubkey=Caustic.Secp256k1.public_key(privkey)# {6316467786437337873577388437635743649101330733943708346103893494005928771381, 36516277665018688612645564779200795235396005730419130160033716279021320193545}message="Hello, world!!!"hash=Caustic.Utils.hash256(message)signature=Caustic.Secp256k1.ecdsa_sign(hash,privkey)Caustic.Secp256k1.ecdsa_verify?(pubkey,hash,signature)# true

Caustic has many functions to deal with integers and their properties.For example you can do primality testing.

first_primes=1..20|>Enum.filter(&Caustic.Utils.prime?/1)# [2, 3, 5, 7, 11, 13, 17, 19]

So 7 is supposed to be a prime. Let's confirm by finding its divisors:

Caustic.Utils.divisors7# [1, 7]

This is in contrast to 6 for example, which has divisors other than 1and itself:

Caustic.Utils.divisors6# [1, 2, 3, 6]

The sum of 6's divisors other than itself (its proper divisors) equals to 6 again. Those kinds of numbersare called perfect numbers.

Caustic.Utils.proper_divisors6# [1, 2, 3]Caustic.Utils.proper_divisors_sum6# 6Caustic.Utils.perfect?6# true

We can easily find other perfect numbers.

1..10000|>Enum.filter(&Caustic.Utils.perfect?/1)# [6, 28, 496, 8128]

There aren't that many of them, it seems...

Now back to our list of first primes. You can find the primitive roots of those primes:

first_primes|>Enum.map(&{&1,Caustic.Utils.primitive_roots(&1)})# [#   {2, [1]},#   {3, [2]},#   {5, [2, 3]},#   {7, [3, 5]},#   {11, [2, 6, 7, 8]},#   {13, [2, 6, 7, 11]},#   {17, [3, 5, 6, 7, 10, 11, 12, 14]},#   {19, [2, 3, 10, 13, 14, 15]}# ]

We can see that 5 is a primitive root of 7. It means that repeatedexponentiation of 5 modulo 7 will generate all numbers relativelyprime to 7. Let's confirm it:

Caustic.Utils.order_multiplicative5,7# 61..6|>Enum.map(&Caustic.Utils.pow_mod(5,&1,7))# [5, 4, 6, 2, 3, 1]

First we check the order of 5 modulo 7. It is 6, which means that5^6 = 1 (mod 7), so further repeated multiplication (5^7 etc.) willjust repeat previous values.

Then we calculate 5^1 to 5^6 (mod 7), and as expected it cyclesthrough all numbers relatively prime to 7 because 5 is a primitiveroot of 7.

For more examples, please see the documentation ofCaustic.Utils.

Please send pull requests tohttps://github.com/agro1986/caustic

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