👋 Hey, thanks for the detailed response — you're absolutely right in the formal mathematical sense:

In elliptic curves over a finite field, there's no such thing as “distance,” “direction,” or “sign” in the usual linear sense.

The point at infinity is an abstract group identity element, and the coordinates of curve points are elements of
𝔽ₚ,
where the notion of "sign" or "closeness to zero" doesn't apply in the conventional way.

But that’s not what I meant. Let me clarify:

🔄 What I actually meant (and why it works)
Let’s assume we know the private key k lies within a limited range:

k ∈ [-1000, +1000] mod N
(This can happen in real-world scenarios — e.g., weak key generation or partial leakage.)

Now suppose we start from:


0⋅G
...and increment step-by-step with:

diff
Копировать
Редактировать
+1⋅G, +2⋅G, +3⋅G, ...
...comparing against a known public key P. If at some step i we reach -P instead of P, then:

mathematica

i⋅G = -P
⇒ i⋅G = (N - k)⋅G
⇒ i ≡ N - k  (mod N)
⇒ k ≈ N - i
Since i is small, k must be near N, which means it's “negative” in the limited range [-1000, +1000].

So, even though we don’t recover the exact value of k, we can infer its sign in that range.

That’s 1 bit of information recovered via curve symmetry and limited search space.

🧠 Why this matters
I'm not claiming you can infer point “direction” or “orientation” in the general case.

I'm saying: If you know that k is small (near 0), and you're doing incremental checks, you can determine whether it’s k or N - k that maps to the target point — i.e., you recover the sign of the scalar.

That’s useful:

In biased brute-force attacks

With weak RNGs

Or when targeting skewed key distributions

⚙️ Technically you’re right, but...
I wasn’t trying to redefine elliptic curve theory.

I was describing a practical case where a directional relationship between P and -P becomes observable because we’re restricting ourselves to a small enough scalar space and checking sequentially.

If that came across differently — fair enough, maybe I worded it poorly.

Still, in the dirty world of broken keygens and leaky bits, that one bit can be golden.
🏭🌀 We work in the rust and shadows — not everything is clean math.

❓Question:
Aside from using Y-parity (which obviously doesn’t help here since positive and negative values are evenly distributed),
do you know of any other properties or heuristics that might leak information about scalar direction or sign — in similarly constrained or biased scenarios?