The Even/Odd Y-Coordinate

In order to reconstruct the full public key from a compressed version, we need a "Tie-Breaker" to tell us which of the two possible Y-coordinates to use. This is the role of the 02 and 03 prefixes.

1. Symmetry of the Curve

The Elliptic Curve is symmetric across the X-axis. This means that if $(x, y)$ is a point on the curve, then $(x, -y)$ is also a point on the curve.

• In "Modular Arithmetic" (which Bitcoin uses), there is no such thing as "Negative." Instead, $-y$ is represented as $(P - y)$.

• If $y$ is Even, then $(P - y)$ will be Odd (because $P$ is an odd prime number).

2. Choosing the Prefix

When a wallet compresses a public key, it looks at the Y-coordinate:

• 0x02: Used if the Y-coordinate is Even.

• 0x03: Used if the Y-coordinate is Odd.

3. Reconstructing Y

When a node receives a transaction, it performs the following steps:

1. Read the X-coordinate.

2. Calculate $y^2 = x^3 + 7 \pmod p$.

3. Calculate the Square Root of $y^2$. This gives two results ($y_1$ and $y_2$).

4. Check the prefix:

   • If prefix is 02, pick the Even result.
   • If prefix is 03, pick the Odd result.

4. Why it matters for Addresses

Because the Public Key Hash (PKH) is a hash of the serialized public key, hashing 04... (uncompressed) results in a completely different hash than hashing 02... (compressed).

• Key Insight: One private key has two public keys, which lead to two different addresses. Modern wallets stick strictly to compressed keys to ensure your funds are always in the same place.

Parity	Prefix	Math Logic
Even	0x02	$y \pmod 2 == 0$
Odd	0x03	$y \pmod 2 == 1$

In the final section, we will build a Python Public Key Auditor.