Is it true that Public keys with even y coordinate correspond to private key that are less than n/2 and vice versa? (Secp256k1)

Question

The question is somewhat complex and directed to clearing things out.

Suppose that $n$ is the order of the cyclic group. It $n - 1$ is the number of all private keys possible

n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141

We also know that every private and public key has its modular inverse. To get a modular inverse of a private key, we need to subtract the private key from $n$.

$$n - privKey$$

To get a modular inverse of a public key, we'll have to multiply its $y$ coordinate by $-1$ and modulo by the $p$ - order of the finite field. $$x,y = x, -y \bmod p$$

A modular inversed public key has the same $x$ coordinate as the original public key, but a different $y$ coordinate, and the $y$ coordinate is always different in its polarity. If the original $y$ was odd, in a modular inversed key it will be even, and vice versa.

The question is, if the $y$ coordinate of a public key is even, does it mean that the corresponding private key is less than $n/2$ by its value? If the $y$ is odd, the private key is more than $n/2$?

Is there any relationship between the evenness/oddness of the $y$ (or $x$) coordinate and the value of the corresponding private key?

Is there any way to know that the private key is more or less than $n/2$ while not knowing the private key itself?

Answer 1

A little test with Sagemath for the first 10 integers as the private key;

#secp256k1
p = Integer("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")

a = Integer("0x0000000000000000000000000000000000000000000000000000000000000000")
b = Integer("0x0000000000000000000000000000000000000000000000000000000000000007")

K = GF(p)
E = EllipticCurve(K,[a,b])

Gx = Integer(0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798)
Gy = Integer(0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)
G = E(Gx, Gy)

# Check the point

#prime curve
n = E.order()


for i in range(1,10):
    A = i* G
    print(A[0],A[1])

The below is the output $(x,y)$;

55066263022277343669578718895168534326250603453777594175500187360389116729240 32670510020758816978083085130507043184471273380659243275938904335757337482424
89565891926547004231252920425935692360644145829622209833684329913297188986597 12158399299693830322967808612713398636155367887041628176798871954788371653930
112711660439710606056748659173929673102114977341539408544630613555209775888121 25583027980570883691656905877401976406448868254816295069919888960541586679410
103388573995635080359749164254216598308788835304023601477803095234286494993683 37057141145242123013015316630864329550140216928701153669873286428255828810018
21505829891763648114329055987619236494102133314575206970830385799158076338148 98003708678762621233683240503080860129026887322874138805529884920309963580118
115780575977492633039504758427830329241728645270042306223540962614150928364886 78735063515800386211891312544505775871260717697865196436804966483607426560663
41948375291644419605210209193538855353224492619856392092318293986323063962044 48361766907851246668144012348516735800090617714386977531302791340517493990618
21262057306151627953595685090280431278183829487175876377991189246716355947009 41749993296225487051377864631615517161996906063147759678534462689479575333124
78173298682877769088723994436027545680738210601369041078747105985693655485630 92362876758821804597230797234617159328445543067760556585160674174871431781431

As we can see there are both even and odd cases for $X$ and $Y$ coordinates in this range. This concludes that there is no %100 relation.

Is it really indistinguishable, I think so, however, I can neither prove it nor calculate it!

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For a run for the first 1 million private keys we have;

Number of odd X'es 499822
Number of odd Y'es 499944

As we can see, quite close to the half.

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As a special note: the coordinate of the point should not be used for randomness since they are not uniform.