elliptic

elliptic - elliptic curve digital signature algorithm (ECDSA) cryptography with OpenSSL made easy (incl. secp256k1 curve)

• home :: github.com/rubycoco/blockchain
• bugs :: github.com/rubycoco/blockchain/issues
• gem :: rubygems.org/gems/elliptic
• rdoc :: rubydoc.info/gems/elliptic

Usage

Intro

Did you know? All you need to open up a new account on a blockchain is an (unsigned) 256-bit / 32-byte integer number. Yes, that's it. No questions asked. The private key is the secret "magic" that unlocks your own bank.

Q: What's the maximum value for a 256-bit / 32-byte integer number (hint 2^256-1)?

Maximum value of 2^256-1 =

2**256-1
#=> 115792089237316195423570985008687907853269984665640564039457584007913129639935
(2**256-1).digits.size           # or to_s.length
#=> 78

Yes, that's 78 (!) decimal digits.

Let's (re)try the maximum value for a 256-bit (32-byte) integer number in hexadecimal (base 16) and binary (base 2) format?

(2**256-1).to_s(16)
#=> "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff"
(2**256-1).digits(16).size      # or to_s(16).length
#=> 64

(2**256-1).to_s(2)
#=> "1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111"
(2**256-1).digits(2).size       # or to_s(2).length
#=> 256

Surprise - a 256-bit number has 256 binary digits (0 and 1s).

BEWARE - Blockchain Bandits! If you use a low integer number e.g. 1, 2, etc. your account is guaranteed to get robbed by blockchain bandits in seconds.

(See A "Blockchain Bandit" Is Guessing Private Keys and Scoring Millions by Andy Greenberg, Wired Magazine, April 2019)

Private Key

An ECDSA (Elliptic Curve Digital Signature Algorithm) private key is a random number between 1 and the order of the elliptic curve group.

require 'elliptic'

# Auto-generate (random) private key
private_key = EC::PrivateKey.generate    # by default uses Secp256k1 curve (used in Bitcoin and Ethereum)

private_key.to_i
#=> 29170346885894798724849267297784761178669026868482995474159965944722616190552
private_key.to_s
#=> "407dd4ccde53d30f3a9cda74ceccb247f3997466964786b59e4d68e93e8f8658"

Derive / (Auto-)Calculate the Public Key - Enter Elliptic Curve (EC) Cryptography

The public key (K) are two numbers (that is, a point with the coordinates x and y) computed by multiplying the generator point (G) of the curve with the private key (k) e.g. K=k*G. This is equivalent to adding the generator to itself k times. Magic? Let's try:

# This private key is just an example. It should be much more secure!
private_key = EC::PrivateKey.new( 1234 )   # by default uses Secp256k1 curve (used in Bitcoin and Ethereum)

public_key =  private_key.public_key   ## the "magic" one-way K=k*G curve multiplication (K=public key,k=private key, G=generator point)
point = public_key.point

point.x
#=> 102884003323827292915668239759940053105992008087520207150474896054185180420338
point.y
#=> 49384988101491619794462775601349526588349137780292274540231125201115197157452

point.x.to_s(16)
#=> "e37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f2"
point.y.to_s(16)
#=> "6d2ee9a82d4158f164ae653e9c6fa7f982ed8c94347fc05c2d068ff1d38b304c"

Sign & Verify Transactions

Sign a transaction with an (elliptic curve) private key:

# Step 1 - Calculate the Transaction (tx) Hash
tx = 'from: Alice  to: Bob     cryptos: 43_000_000_000'
txhash = Digest::SHA256.digest( tx )

# Step 2 - Get the Signer's Private key
private_key = EC::PrivateKey.new( 1234 )     # This private key is just an example. It should be much more secure!

# Sign!
signature = private_key.sign( txhash )
# -or-
signature = EC.sign( txhash, private_key )

signature.r
#=> 80563021554295584320113598933963644829902821722081604563031030942154621916407
signature.s
#=> 58316177618967642068351252425530175807242657664855230973164972803783751708604

signature.r.to_s(16)
#=> "3306a2f81ad2b2f62ebe0faec129545bc772babe1ca5e70f6e56556b406464c0"
signature.s.to_s(16)
#=> "4fe202bb0835758f514cd4a0787986f8f6bf303df629dc98c5b1a438a426f49a"

Verify a signed transaction with an (elliptic curve) public key:

# Step 1 - Calculate the Transaction (tx) Hash
tx = 'from: Alice  to: Bob     cryptos: 43_000_000_000'
txhash = Digest::SHA256.digest( tx )

# Step 2 - Get the Signer's Public Key
public_key = EC::PublicKey.new(
   102884003323827292915668239759940053105992008087520207150474896054185180420338,
   49384988101491619794462775601349526588349137780292274540231125201115197157452
)

# Step 3 - Get the Transaction's Signature
signature = EC::Signature.new(
  80563021554295584320113598933963644829902821722081604563031030942154621916407,
  58316177618967642068351252425530175807242657664855230973164972803783751708604
)

# Don't Trust - Verify
public_key.verify?( txhash, signature )
# -or-
EC.verify?( txhash, signature, public_key )
#=> true


# or using hexadecimal numbers

public_key = EC::PublicKey.new(
  0xe37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f2,
  0x6d2ee9a82d4158f164ae653e9c6fa7f982ed8c94347fc05c2d068ff1d38b304c
)

signature = EC::Signature.new(
  0x3306a2f81ad2b2f62ebe0faec129545bc772babe1ca5e70f6e56556b406464c0,
  0x4fe202bb0835758f514cd4a0787986f8f6bf303df629dc98c5b1a438a426f49a
)

public_key.verify?( txhash, signature )
# -or-
EC.verify?( txhash, signature, public_key )
#=> true

To sum up:

• The (raw) private key is a 256-bit unsigned integer number
• The (raw) public key is a point (x,y), that is, two 256-bit unsigned integer numbers - derived (calculated) from the private key
• A (raw) signature is composed of (r,s), that is, two 256-bit unsigned integer numbers

That's all the magic.

Private / Public Key Formats

Intro

To get the all-in-one-string public key from a point with the coordinates x and y use the Standards for Efficient Cryptography (SEC) 1) uncompressed format or the 2) compressed format:

# 1) Uncompressed format (with prefix 04)
#   Convert to 64 hexstring characters (32 bytes) in length

prefix = '04'
pubkey = prefix + "%064x" % point.x + "%064x" % point.y
#=> "04e37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f26d2ee9a82d4158f164ae653e9c6fa7f982ed8c94347fc05c2d068ff1d38b304c"

# 2) Compressed format (with prefix - 02 = even / 03 = odd)
#   Instead of using both x and y coordinates,
#   just use the x-coordinate and whether y is even/odd

prefix = point.y % 2 == 0 ? '02' : '03'
pubkey = prefix + "%064x" % point.x
#=> "02e37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f2"

or use the builtin helpers:

# 1) Uncompressed format (with prefix 04)
#   Convert to 64 hexstring characters (32 bytes) in length

point.to_s   # or point.to_s( :uncompressed )
#=> "04e37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f26d2ee9a82d4158f164ae653e9c6fa7f982ed8c94347fc05c2d068ff1d38b304c"

# 2) Compressed format (with prefix - 02 = even / 03 = odd)
#   Instead of using both x and y coordinates,
#   just use the x-coordinate and whether y is even/odd

point.to_s( :compressed )
#=> "02e37648435c60dcd181b3d41d50857ba5b5abebe279429aa76558f6653f1658f2"

PEM, DER, BASE64(DER)

Export

To export a private or public key to the Privacy Enhanced Mail (PEM) format use to_pem:

private_key = EC::PrivateKey.generate
private_key.to_pem
#=> "-----BEGIN EC PRIVATE KEY-----
#    MHQCAQEEIDIWkCIC58Yo1E5noSiXbHdR/8zUqB+vvTK4nSk8tZ1RoAcGBSuBBAAK
#    oUQDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA13Z8Zif5Mh+qavIr/6HpI/Kq
#    Q0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA==
#    -----END EC PRIVATE KEY-----"

public_key = private_key.public_key
public_key.to_pem
#=> "-----BEGIN PUBLIC KEY-----
#    MFYwEAYHKoZIzj0CAQYFK4EEAAoDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA
#    13Z8Zif5Mh+qavIr/6HpI/KqQ0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA==
#    -----END PUBLIC KEY-----"

To export a private or public key to the (binary) Distinguished Encoding Rules (DER) in Abstract Syntax Notation One (ASN.1) format use to_der:

private_key.to_der
#=> "\xA1D\x03B\x00\x04\xA2Y|\xAD\x87\xAB|1
#     \xF8\xAB\xA9\xD3\xD6h\x93e\x9D\xF5\xE5\xAF\x01\x16\xC2\x80
#     \xD7v|f'\xF92\x1F\xAAj\xF2+\xFF\xA1\xE9#\xF2\xAACF\xE7\xB8
#     \xE6b\b?`\xA4B\x16\xA3\xB5F7\xCC\t\x81\xCB\xBAd"

public_key = private_key.public_key
public_key.to_der
#=> "0V0\x10\x06\a*\x86H\xCE=\x02\x01\x06\x05+\x81\x04\x00
#    \x03B\x00\x04\xA2Y|\xAD\x87\xAB|1\xF8\xAB\xA9
#    \xD3\xD6h\x93e\x9D\xF5\xE5\xAF\x01\x16\xC2\x80\xD7v|f'\xF92
#    \x1F\xAAj\xF2+\xFF\xA1\xE9#\xF2\xAACF\xE7\xB8\xE6b
#    \b?`\xA4B\x16\xA3\xB5F7\xCC\t\x81\xCB\xBAd"

To export a private or public key to the Base64-encoded Distinguished Encoding Rules (DER) in Abstract Syntax Notation One (ASN.1) format use to_base64:

private_key.to_base64
#=> "MHQCAQEEIDIWkCIC58Yo1E5noSiXbHdR/8zUqB+vvTK4nSk8tZ1RoAcGBSuBBAAK
#    oUQDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA13Z8Zif5Mh+qavIr/6HpI/Kq
#    Q0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA=="

public_key = private_key.public_key
public_key.to_base64
#=> "MFYwEAYHKoZIzj0CAQYFK4EEAAoDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA
#    13Z8Zif5Mh+qavIr/6HpI/KqQ0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA=="

Import

To import a private or public key in the PEM or DER format use the all-in-one convenience constructor:

private_key = EC::PrivateKey.new( "-----BEGIN EC PRIVATE KEY-----
    MHQCAQEEIDIWkCIC58Yo1E5noSiXbHdR/8zUqB+vvTK4nSk8tZ1RoAcGBSuBBAAK
    oUQDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA13Z8Zif5Mh+qavIr/6HpI/Kq
    Q0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA==
    -----END EC PRIVATE KEY-----" )

public_key = EC::PublicKey.new( "-----BEGIN PUBLIC KEY-----
    MFYwEAYHKoZIzj0CAQYFK4EEAAoDQgAEoll8rYerfDH4q6nT1miTZZ315a8BFsKA
    13Z8Zif5Mh+qavIr/6HpI/KqQ0bnuOZiCD9gpEIWo7VGN8wJgcu6ZA==
    -----END PUBLIC KEY-----" )

## or

private_key = EC::PrivateKey.new( "\xA1D\x03B\x00\x04\xA2Y|\xAD\x87\xAB|1
     \xF8\xAB\xA9\xD3\xD6h\x93e\x9D\xF5\xE5\xAF\x01\x16\xC2\x80
     \xD7v|f'\xF92\x1F\xAAj\xF2+\xFF\xA1\xE9#\xF2\xAACF\xE7\xB8
     \xE6b\b?`\xA4B\x16\xA3\xB5F7\xCC\t\x81\xCB\xBAd".b )

public_key = EC::PublicKey.new( "0V0\x10\x06\a*\x86H\xCE=\x02\x01\x06\x05+\x81\x04\x00
    \x03B\x00\x04\xA2Y|\xAD\x87\xAB|1\xF8\xAB\xA9
    \xD3\xD6h\x93e\x9D\xF5\xE5\xAF\x01\x16\xC2\x80\xD7v|f'\xF92
    \x1F\xAAj\xF2+\xFF\xA1\xE9#\xF2\xAACF\xE7\xB8\xE6b
    \b?`\xA4B\x16\xA3\xB5F7\xCC\t\x81\xCB\xBAd".b )

or use the decode/from helper:

private_key = EC::PrivateKey.decode_pem( ... )     # or from_pem( ... )
              EC::PrivateKey.decode_der( ... )     # or from_der( ... )
              EC::PrivateKey.decode_base64( ... )  # or from_base64( ... )

public_key  = EC::PublicKey.decode_pem( ... )      # or from_pem( ... )
              EC::PublicKey.decode_der( ... )      # or from_der( ... )
              EC::PublicKey.decode_base64( ... )   # or from_base64( ... )

That's it.

Aside - Elliptic What?

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

(Source: Elliptic-curve cryptography @ Wikipedia)

What's an Elliptic Curve?

This is a graph of secp256k1's elliptic curve y² = x³ + 7 over the real numbers. Note that because secp256k1 is actually defined over the field Zₚ, its graph will in reality look like random scattered points, not anything like this.

(Source: Secp256k1 @ Bitcoin Wiki)

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Install

Just install the gem:

$ gem install elliptic

License

The scripts are dedicated to the public domain. Use it as you please with no restrictions whatsoever.

Questions? Comments?

Send them along to the wwwmake forum. Thanks!