Quote from: mcdouglasx on Today at 01:50:49 AM

Quote from: bjpark on January 17, 2025, 08:53:02 PM

By analyzing the public key values alone, the program effectively narrows down the characteristics of the private keys, aiding in identifying whether the private key is positive or negative.

1. Private Key and Public Key Basics
Private Key:
In Bitcoin, a private key is defined as a positive integer between 1 and n-1, where n is the order of the curve used. For the secp256k1 curve, n = 2^256 - 2^32 - 977. Therefore, private keys are always positive integers and cannot have a negative sign.

Public Key:
A private key generates a public key by performing elliptic curve multiplication on the secp256k1 curve. The public key is represented as a point (x, y) on the elliptic curve, where x and y are integers. Mathematically, these integers can be either positive or negative.

2. Misunderstanding About the Negative Sign
Since private keys cannot be negative, some developers mistakenly assume that public keys also cannot have negative values.
However, the public key (x, y) lies on the elliptic curve defined by the equation:
y^2 = x^3 + 7

For any given x, the equation ensures there are two possible y values: one positive and one negative. This is due to the symmetric nature of the elliptic curve about the x-axis.

3. Why Public Keys Are Thought to Lack Negative Signs
In Bitcoin, public keys are stored and transmitted in specific formats, which often abstract away the negative sign:

Compressed Public Key Format:

In the compressed format, only the x coordinate is stored, along with a prefix that indicates the sign of y (even or odd).
The prefix is 02 if y is even and 03 if y is odd.
For example, a public key (x, y) where y > 0 or y < 0 is reduced to x and a single bit of information about the parity of y.
Uncompressed Public Key Format:

In the uncompressed format, both x and y are included. However, the protocol does not store negative integers directly. Instead, it uses the absolute values of x and y and assumes a positive representation.
Due to these storage and transmission methods, the negative sign of y is not explicitly visible. This abstraction leads to the mistaken belief that public keys never have negative components.

4. Conclusion: Public Keys Can Correspond to Negative Values
Public keys have two perspectives:

Mathematical Perspective:
Public keys (x, y) are points on the elliptic curve. For a given x, there are always two possible y values (one positive and one negative), meaning the negative version of the public key exists mathematically.

Bitcoin Protocol Perspective:
The Bitcoin protocol stores and transmits public keys in formats that either compress or use absolute values, making negative signs implicit rather than explicit. This is why many developers mistakenly believe public keys cannot have negative values.

Final Summary
Negative public key values exist mathematically, but they are abstracted away in Bitcoin's implementation. This abstraction leads to the common misconception that public keys cannot be negative.

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