# Determine if an elliptic point is negative

How to determine if an elliptic point $$kG$$ is negative? Is $$k<0?$$

By example, for $$k=1234$$, the coordinate of the point $$1234G$$ using secp256k1 = $$X= 102884003323827292915668239759940053105992008087520207150474896054185180420338$$ $$Y= 49384988101491619794462775601349526588349137780292274540231125201115197157452$$

Note: $$1234 = 0x4D2 = 0b10011010010$$

The coordinate of a negative point is equal to (x, -y).

So, $$-1234G =$$

$$(102884003323827292915668239759940053105992008087520207150474896054185180420338,$$ $$\textbf{-49384988101491619794462775601349526588349137780292274540231125201115197157452})$$

We're in a field, so we apply modulo prime: $$-y \; \% \;prime = prime - y$$

So, $$-1234G =$$

$$(102884003323827292915668239759940053105992008087520207150474896054185180420338,$$ $$\textbf{prime} - 49384988101491619794462775601349526588349137780292274540231125201115197157452)$$

$$= (102884003323827292915668239759940053105992008087520207150474896054185180420338,$$ $$\textbf{66407101135824575629108209407338381264920846885348289499226458806793637514211})$$

Is there a trick to known which one correspond to -1234G or 1234G or more generally kG or -kG? Notes: We don't know k. I know that to uncompress a public key, we just know x, we use a bit to say take y less than p/2 or greater than p/2 but in this case it's always a positive k.

There is no usual and well-defined meaning to "negative" for a point $$P$$ of an elliptic curve on a finite field, as used in cryptography. If $$P=k\,G$$ for negative $$k$$, then $$P=k'\,G$$ for $$k'=k\bmod n$$, which is non-negative (it would be $$k'=k+n$$ if $$0<|k| ), and $$n$$ is the order of the group (or of $$G$$ )
For a given $$P$$, we could uniquely define $$k$$ as the integer with the smallest absolute value and $$P=k\,G$$, then consider the sign of $$k$$. But
• if the group is cryptographically strong, there is no efficient algorithm to determine the sign of that $$k$$ from an arbitrary $$P$$. Argument: if there was such method, it could be used as a subprogram to solve the Discrete Logarithm Problem (that is find $$k$$ for a given random $$P$$ in the group generate by $$G$$ ), by binary search.