A Walk Through Ethereum Classic Digital Signature Code

3/15/2018 Christian Seberino

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Ethereum Classic (ETC) digital signatures secure transactions. These involve elliptic curve cryptography and the Elliptic Curve Digital Signature Algorithm (ECDSA). I will describe ETC digital signatures without these topics using only small Python functions.

Basics


![](./1*yw1934-mAqp5DM4FWgbNqQ.jpeg)

Signing and verifying will be implemented using the following four constants and three functions:

N  = 115792089237316195423570985008687907852837564279074904382605163141518161494337
P  = 115792089237316195423570985008687907853269984665640564039457584007908834671663
Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424

def invert(number, modulus):
        """
        Finds the inverses of natural numbers.
        """

        result = 1
        power  = number
        for e in bin(modulus - 2)[2:][::-1]:
                if int(e):
                        result = (result * power) % modulus
                power = (power ** 2) % modulus

        return result

def add(pair_1, pair_2):
        """
        Finds the sums of two pairs of natural numbers.
        """

        if   pair_1 == [0, 0]:
                result = pair_2
        elif pair_2 == [0, 0]:
                result = pair_1
        else:
                if pair_1 == pair_2:
                        temp = 3 * pair_1[0] ** 2
                        temp = (temp * invert(2 * pair_1[1], P)) % P
                else:
                        temp = pair_2[1] - pair_1[1]
                        temp = (temp * invert(pair_2[0] - pair_1[0], P)) % P
                result = (temp ** 2 - pair_1[0]  - pair_2[0]) % P
                result = [result, (temp * (pair_1[0] - result) - pair_1[1]) % P]

        return result

def multiply(number, pair):
        """
        Finds the products of natural numbers and pairs of natural numbers.
        """

        result = [0, 0]
        power  = pair[:]
        for e in bin(number)[2:][::-1]:
                if int(e):
                        result = add(result, power)
                power = add(power, power)

        return result

The invert function defines an operation on numbers in terms of other numbers referred to as moduli. The add function defines an operation on *pairs* of numbers. The multiply function defines an operation on a number and a pair of numbers. Here are examples of their usage: ``` >>> invert(82856, 7164661) 3032150

add([84672, 5768], [15684, 471346]) [98868508778765247164450388534339365517943901419260061027507991295919394382071, 110531019976596004792591549651085191890711482591841040377832420464376026143223]

multiply(82716, [31616, 837454]) [82708077205483544970470074583740846828577431856187364454411787387343982212318, 30836796656275663256542662990890163662171092281704208118107591167423888588304]

### Private & Public Keys
<br/>![](./1*0Y8TNbhhEQytGNYJH5uPTg.jpeg)

Private keys are any nonzero numbers less than the constant N. Public keys are the products of these private keys and the pair (Gx, Gy ). For example:

>>> private_key = 296921718

>>> multiply(private_key, (Gx, Gy))
[29493341745186804828936410559976490896704930101972775917156948978213464516647, 14120583959514503052816414068611328686827638581568335296615875235402122319824]

Notice that public keys are pairs of numbers.

Signing

1 na0d3BXnFL nSj5mNOsE2g Signing transactions involves an operation on the Keccak 256 hashes of the transactions and private keys. The following function implements this operation:

import random

def sign(hash, priv_key):
        """
        Signs the hashes of transactions.
        """

        result = [0, 0]
        while 0 in result or result[1] > N / 2:
                temp      = random.randint(1, N - 1)
                result[0] = multiply(temp, (Gx, Gy))[0] % N
                result[1] = invert(temp, N) * (hash + priv_key * result[0])
                result[1] = result[1] % N

        return result

For example: ``` >>> hash = 0xf62d00f14db9521c03a39c20e94aa10a82ff5f5a614772b25e36757a95a71048

private_key = 296921718

sign(hash, private_key) [12676003675279000995677412431399004760576311052126257887715931882164427686866, 17853929027942611176839390215748157599052991088042356790746129338653342477382]

sign(hash, private_key) [18783324464633387734826042295911802941026009108876130700727156896210203356179, 41959562951157235894396660120771158332032804144867595196194581439345450008533]

<br/>Notice that digital signatures are pairs of numbers. Notice also that the sign function can give different results for the *same* inputs!

Verifying

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Verifying digital signatures involves confirming certain properties with regards to the Keccak 256 hashes and public keys. The following function implements these checks:

def verify(sig, hash, pub_key):
        """
        Verifies the signatures of the hashes of transactions.
        """

        temp_1 = multiply((invert(sig[1], N) * hash)   % N, (Gx, Gy))
        temp_2 = multiply((invert(sig[1], N) * sig[0]) % N, pub_key)
        sum    = add(temp_1, temp_2)
        test_1 = (0 < sig[0] < N) and (0 < sig[1] < N)
        test_2 = sum != (0, 0)
        test_3 = sig[0] == sum[0] % N

        return test_1 and test_2 and test_3

For example: ``` >>> hash = 0xf62d00f14db9521c03a39c20e94aa10a82ff5f5a614772b25e36757a95a71048

private_key = 296921718

public_key = multiply(private_key, (Gx, Gy))

public_key [29493341745186804828936410559976490896704930101972775917156948978213464516647, 14120583959514503052816414068611328686827638581568335296615875235402122319824]

signature = sign(hash, private_key)

signature [54728868372105873293629977757277092827353030346967592768173610703187933361202, 18974025727476367931183775600389145833964496722266015570370178285290252701715]

verify(signature, hash, public_key) True

<br/>To verify that public keys correspond to specific ETC account addresses, confirm that the rightmost 20 bytes of the public key Keccak 256 hashes equal those addresses.

Recovery Identifiers


![](./6RuyzCB.png)
Strictly speaking, ETC digital signatures include additional small numbers referred to as *recovery identifiers*. These allow public keys to be determined solely from the signed transactions.

Conclusion


![](./1*c2zDUxyTF1IidCj15Xa4yg.jpeg)

I have explained ETC digital signatures using code rather than mathematics. Hopefully seeing how signing and verifying can be implemented with these tiny functions has been useful.

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Acknowledgements

I would like to thank IOHK (Input Output Hong Kong) for funding this effort.

License

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This work is licensed under the Creative Commons Attribution ShareAlike 4.0 International License.


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