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Author Topic: How to get uncompressed public key from compressed one ?  (Read 2915 times)
Farghaly
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June 08, 2014, 11:48:14 PM
 #1

Uncompressed public key is:
0x04 + x-coordinate + y-coordinate

Compressed public key is:
0x02 + x-coordinate if y is even
0x03 + x-coordinate if y is odd

How to use this equation to derive the uncompressed public key

y^2 mod p = (x^3 + 7) mod p

 
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TimS
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June 09, 2014, 12:53:21 AM
 #2

From https://en.wikipedia.org/wiki/Quadratic_residue and http://mersennewiki.org/index.php/Modular_Square_Root, if

r^2 = a mod m where m = 3 mod 4 (as secp256k1's p does)
then
r = +-a^((m+1)/4) mod m

So:

y^2 mod p = (x^3 + 7) mod p
y mod p = +-(x^3 + 7)^((p+1)/4) mod p

So calculate (x^3 + 7)^((p+1)/4) mod p, and if the parity of the first answer you get is wrong, then take the negative of that answer (since we're working modulo an odd number, taking the negative will flip the even/odd parity).

Just for fun, here's some Python code that'll do the calculation in the blink of an eye, preset to work on the public key of (the random) private key 55255657523dd1c65a77d3cb53fcd050bf7fc2c11bb0bb6edabdbd41ea51f641

Code:
def pow_mod(x, y, z):
    "Calculate (x ** y) % z efficiently."
    number = 1
    while y:
        if y & 1:
            number = number * x % z
        y >>= 1
        x = x * x % z
    return number

p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
compressed_key = '0314fc03b8df87cd7b872996810db8458d61da8448e531569c8517b469a119d267'
y_parity = int(compressed_key[:2]) - 2
x = int(compressed_key[2:], 16)
a = (pow_mod(x, 3, p) + 7) % p
y = pow_mod(a, (p+1)//4, p)
if y % 2 != y_parity:
    y = -y % p
uncompressed_key = '04{:x}{:x}'.format(x, y)
print(uncompressed_key)
deepceleron
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June 09, 2014, 06:07:36 PM
 #3

Be mindful that there is no practical reason to "convert" between uncompressed and compressed public keys. Each has their own Bitcoin address - you cannot spend funds sent to the uncompressed public key's address with the compressed public key.
ftc-c
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May 09, 2015, 03:05:01 PM
 #4

Who can give a C++ implementation ?
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May 10, 2015, 01:01:24 AM
 #5

@ftc-c: Use libsecp256k1, AFAIK it has preprocessor stuff for C++ support, and it's easy enough to work through.

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May 10, 2015, 05:22:53 PM
 #6

The following link shows how to do it in C using the polarssl / mbedTLS library, you should be able to port this across a different crypto library so long as it has a proper 'bignum'/mpi implementation.

https://gist.github.com/flying-fury/6bc42c8bb60e5ea26631

The above example contains test data as well, from the comments '// Set y2 = X^3 + B' and '// Compute square root of y2' you can see where the important part is done.

I used the above to make a dll/so/dylib which does this useful conversion based on the above code.

What deepceleron said also applies, there are two distinct coin addresses which can be created from a single private key depending on whether the public key is 'compressed' or not.

The above conversion is still useful if you use ECC for things like verifying signatures.
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May 10, 2015, 05:28:55 PM
 #7

Welcome to use the CIYAM code which includes parts of the Bitcoin code as well (but might be easier to follow in terms of the C++ classes as it doesn't involve any Boost stuff).

https://github.com/ciyam/ciyam/blob/master/src/crypto_keys.cpp

With CIYAM anyone can create 100% generated C++ web applications in literally minutes.

GPG Public Key | 1ciyam3htJit1feGa26p2wQ4aw6KFTejU
Blockchain Mechanic
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January 09, 2017, 08:46:38 AM
 #8

Be mindful that there is no practical reason to "convert" between uncompressed and compressed public keys. Each has their own Bitcoin address - you cannot spend funds sent to the uncompressed public key's address with the compressed public key.

Hi

So I could use both addresses to receive funds so long as i have the one private key?


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January 09, 2017, 01:22:30 PM
 #9

Be mindful that there is no practical reason to "convert" between uncompressed and compressed public keys. Each has their own Bitcoin address - you cannot spend funds sent to the uncompressed public key's address with the compressed public key.

Hi

So I could use both addresses to receive funds so long as i have the one private key?


Yes. The compressed public key and the uncompressed ones have different public keys but they can be derived from the same private key. However, they may be different in the Wallet Import Format, private keys starting with 5 are compressed while the uncompressed ones starts from K or L.

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