Bitcoin puzzle transaction ~32 BTC prize to who solves it

[Igor_cherkassy]

1PWo3JeB

40000000185CA03D74
4D5E8BDDD777CCC70C
5222222B15D38713BE
5A2910266666665984
60000000AE2BFA8AB8
617C59B99999909975
62473648CCCCCC0FCC
63888A96888FCCCF93
7811116F4661A55552
 
There are thousands of such Bitcoin keys

Anyone who stopped working with prefixes, please drop your keys for research.

[And24r]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

[brainless]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

[And24r]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

[brainless]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

Everyone have their own view to see angle
In short it's half, 1/2, 0.5

[kTimesG]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

No, it's "1/2" mod n = (n + 1) / 2 which has nothing to do with the group's elements or where you count from. It can be used to compute R from a known P such that 2*R = P, so R = [(n+1)/2]*P

[Grzegorz2022]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

Yes, that's the last point

[And24r]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

Yes, that's the last point

Have you considered deriving a formula such that adding the generator point to any given point results in a one‑point shift strictly towards y=0? This would allow you to determine whether the point lies on the upper or lower branch.
There is such a formula without the modulus p. However, when working with the modulus of a prime number, the presence of a square root causes complications, as it yields two roots.

[Grzegorz2022]

Who knows what this private key is? 7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0

Half of p mod
Everyone knows

Is this the upper branch of the elliptic curve, counting from the generator point?

Yes, that's the last point

Have you considered deriving a formula such that adding the generator point to any given point results in a one‑point shift strictly towards y=0? This would allow you to determine whether the point lies on the upper or lower branch.
There is such a formula without the modulus p. However, when working with the modulus of a prime number, the presence of a square root causes complications, as it yields two roots.

kTimesG is right algebraically — (n+1)/2 is 2⁻¹ mod n and is used for halving (2R = P → R = [(n+1)/2]·P).

But after my geometric research: The points (n/2)·G and (n/2+1)·G have the exact same X coordinate (the one with leading zeros: 00000000...3b78ce563f...) — they differ only in Y parity. This is a (k, n-k) pair.

More interestingly — if you unfold the keys around n/2, you'll see perfect mirror symmetry:
n/2 - 1  ↔  n/2 + 2   (same X, flipped Y)
n/2 - 2  ↔  n/2 + 3
n/2 - 3  ↔  n/2 + 4

The point n/2 is the symmetry axis of the cyclic group — a linear walk along the curve 'turns around' right here. This is not an 'upper branch' (such a concept doesn't exist in EC), but it is the reflection point in the group structure.

So (n+1)/2 is simultaneously: the multiplicative inverse of 2 mod n (halving), and the point where the k ↔ n-k symmetry 'meets.'

[ k -15 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747153
X:   6286bb9071f8887bc4fbb3f0a9ecb19efcdd2cf15e286da71bb60c9b5ff5e644
Pub: 026286bb9071f8887bc4fbb3f0a9ecb19efcdd2cf15e286da71bb60c9b5ff5e644  -> Adr: 15fS3Hy9oSSnoDHoj9wuVBThXjH5yTyQma
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747184
Pub: 036286bb9071f8887bc4fbb3f0a9ecb19efcdd2cf15e286da71bb60c9b5ff5e644  -> Adr: 1BE8jrkpvVpm1epztSXFRSJ7eYjfMT3TkX
----------------------------------------

[ k -14 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747154
X:   3905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b
Pub: 023905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b  -> Adr: 13xMdG58VuHFKESWs2wuhXbUXRP2jmGJBL
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747183
Pub: 033905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b  -> Adr: 1E77TAEg7VAdUcKLGUWMnwJuY5XqDtXy9Y
----------------------------------------

[ k -13 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747155
X:   561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e
Pub: 03561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e  -> Adr: 16mJRZXsiqw7TzVAZLAYeDxtHcDkAR8cbM
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747182
Pub: 02561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e  -> Adr: 135yPWdDgsB7PtMUdsEAf8jxkv9wesKker
----------------------------------------

[ k -12 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747156
X:   75bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5
Pub: 0375bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5  -> Adr: 1Jh1JmV3RxMPDjgBPEHZHJRzyTQ4ynULkC
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747181
Pub: 0275bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5  -> Adr: 1LZ35k6xrj62kaUxJpG45jFhEKu9Caspw6
----------------------------------------

[ k -11 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747157
X:   e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3
Pub: 02e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3  -> Adr: 1FewTQmntAiBarXqNCeC7YQfDd26kbsxRN
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747180
Pub: 03e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3  -> Adr: 17HemTH2qLRSdTPWw8B1fseBjoZWHGuuNE
----------------------------------------

[ k -10 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747158
X:   1fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e
Pub: 021fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e  -> Adr: 131BCaYih8kkF8Er5nzdWkuYSW3Ngmhyqe
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747179
Pub: 031fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e  -> Adr: 1BmdayqdTekMjor84epMyG9qjkhNnBMpTr
----------------------------------------

[ k -9 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747159
X:   1c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c
Pub: 031c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c  -> Adr: 17jrgGw1wCK5eexxXVitD2TKsxfCxC8fq7
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747178
Pub: 021c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c  -> Adr: 19KKCz4AcXTnqqs1i2DfSzBdbCeKaJNZ8q
----------------------------------------

[ k -8 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747160
X:   41f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8
Pub: 0241f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8  -> Adr: 16vVZ7R2q3DLLHX5PW1xA8uYLKV5Vp8zWc
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747177
Pub: 0341f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8  -> Adr: 1Krnz5cgFehW3jVMaFUVZgY85KvAS6FrNy
----------------------------------------

[ k -7 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747161
X:   e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60
Pub: 02e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60  -> Adr: 14owJG3qQCU4nXz8esjFiL85dEezjj5kxa
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747176
Pub: 03e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60  -> Adr: 1BF2E5wfufEGgiroV1YjtGttRuWYgJ36hR
----------------------------------------

[ k -6 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747162
X:   eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae
Pub: 02eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae  -> Adr: 17FNBNmR1FvJvc1jZ88uGsVeWnY8uiscV3
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747175
Pub: 03eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae  -> Adr: 19FqQtumUjgtwQK5wJvSRvjFvksNuHUE94
----------------------------------------

[ k -5 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747163
X:   5702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef
Pub: 035702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef  -> Adr: 1JCHSxKTt6orSkg9QHhennEZdybz2zSEbD
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747174
Pub: 025702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef  -> Adr: 1AXP4CRCPVA2KmfCJwCX2Taw6kmtZZbdkM
----------------------------------------

[ k -4 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747164
X:   66954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859
Pub: 0266954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859  -> Adr: 1Eepuc2uUoPfSRCUFHiuX5wtPt1hU8xdEd
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747173
Pub: 0366954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859  -> Adr: 18nHHZTwsNQwRFozvMjkUV7d8BwEPBpDft
----------------------------------------

[ k -3 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747165
X:   592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031
Pub: 02592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031  -> Adr: 1MiZBmHMMCmRSqmD5K1MwNgs2EEnm3Am1H
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747172
Pub: 03592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031  -> Adr: 18kSSgqXCr43XgesM2aFnDdebevBu4hXYn
----------------------------------------

[ k -2 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747166
X:   5699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5
Pub: 025699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5  -> Adr: 12NqA1nEZw2w2o2BrkmymQHDTMgcxBSH6w
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747171
Pub: 035699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5  -> Adr: 1LtmtRzYDJLEw6cjoD8AxqzAsL4NEBvRab
----------------------------------------

[ k -1 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747167
X:   c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413
Pub: 03c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413  -> Adr: 1AsPTrDFPaz6uYyuNgnmnrt8mEkfjYFmYq
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747170
Pub: 02c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413  -> Adr: 134yamsYAgAyWVr7z4KjH6h52UigkEnrL5
----------------------------------------

[ k +0 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747168
X:   00000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63
Pub: 0300000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63  -> Adr: 1LVAsnUyEtJgZ9HzLfbtiJZuZMzHLX1n6k
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747169
Pub: 0200000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63  -> Adr: 13see6qjfupx1YWgRefwEkccZeM8QGTAiJ
----------------------------------------

[ k +1 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747169
X:   00000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63
Pub: 0200000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63  -> Adr: 13see6qjfupx1YWgRefwEkccZeM8QGTAiJ
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747168
Pub: 0300000000000000000000003b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63  -> Adr: 1LVAsnUyEtJgZ9HzLfbtiJZuZMzHLX1n6k
----------------------------------------

[ k +2 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747170
X:   c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413
Pub: 02c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413  -> Adr: 134yamsYAgAyWVr7z4KjH6h52UigkEnrL5
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747167
Pub: 03c62c910e502cb615a27c58512b6cc2c94f5742f76cb3d12ec993400a3695d413  -> Adr: 1AsPTrDFPaz6uYyuNgnmnrt8mEkfjYFmYq
----------------------------------------

[ k +3 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747171
X:   5699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5
Pub: 035699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5  -> Adr: 1LtmtRzYDJLEw6cjoD8AxqzAsL4NEBvRab
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747166
Pub: 025699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5  -> Adr: 12NqA1nEZw2w2o2BrkmymQHDTMgcxBSH6w
----------------------------------------

[ k +4 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747172
X:   592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031
Pub: 03592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031  -> Adr: 18kSSgqXCr43XgesM2aFnDdebevBu4hXYn
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747165
Pub: 02592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031  -> Adr: 1MiZBmHMMCmRSqmD5K1MwNgs2EEnm3Am1H
----------------------------------------

[ k +5 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747173
X:   66954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859
Pub: 0366954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859  -> Adr: 18nHHZTwsNQwRFozvMjkUV7d8BwEPBpDft
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747164
Pub: 0266954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859  -> Adr: 1Eepuc2uUoPfSRCUFHiuX5wtPt1hU8xdEd
----------------------------------------

[ k +6 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747174
X:   5702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef
Pub: 025702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef  -> Adr: 1AXP4CRCPVA2KmfCJwCX2Taw6kmtZZbdkM
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747163
Pub: 035702fb8a2602f41f52699f688d4b005a128762e11dfd13fd22ea751ccbedb2ef  -> Adr: 1JCHSxKTt6orSkg9QHhennEZdybz2zSEbD
----------------------------------------

[ k +7 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747175
X:   eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae
Pub: 03eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae  -> Adr: 19FqQtumUjgtwQK5wJvSRvjFvksNuHUE94
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747162
Pub: 02eb3bc68c623b1f46ab905412c7f2d588fa25abb77a7bd782ba9bb3aac05a70ae  -> Adr: 17FNBNmR1FvJvc1jZ88uGsVeWnY8uiscV3
----------------------------------------

[ k +8 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747176
X:   e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60
Pub: 03e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60  -> Adr: 1BF2E5wfufEGgiroV1YjtGttRuWYgJ36hR
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747161
Pub: 02e5cbd62789c6a84325a2440789b88dbb1dc55afb9e8296e6bb8af7de57a50e60  -> Adr: 14owJG3qQCU4nXz8esjFiL85dEezjj5kxa
----------------------------------------

[ k +9 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747177
X:   41f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8
Pub: 0341f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8  -> Adr: 1Krnz5cgFehW3jVMaFUVZgY85KvAS6FrNy
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747160
Pub: 0241f7fa0a9a59513ae221e3b84b91995fc9d40eb5d120a6d8e663452ad92099c8  -> Adr: 16vVZ7R2q3DLLHX5PW1xA8uYLKV5Vp8zWc
----------------------------------------

[ k +10 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747178
X:   1c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c
Pub: 021c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c  -> Adr: 19KKCz4AcXTnqqs1i2DfSzBdbCeKaJNZ8q
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747159
Pub: 031c2bd878b94169da722a9de0c4e317cea8802aa96045830111a89d1d9de4270c  -> Adr: 17jrgGw1wCK5eexxXVitD2TKsxfCxC8fq7
----------------------------------------

[ k +11 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747179
X:   1fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e
Pub: 031fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e  -> Adr: 1BmdayqdTekMjor84epMyG9qjkhNnBMpTr
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747158
Pub: 021fb527e94e9c70e8657de7458b81ef9ee3c2b4e0128a675bf7e28980e18b201e  -> Adr: 131BCaYih8kkF8Er5nzdWkuYSW3Ngmhyqe
----------------------------------------

[ k +12 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747180
X:   e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3
Pub: 03e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3  -> Adr: 17HemTH2qLRSdTPWw8B1fseBjoZWHGuuNE
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747157
Pub: 02e881a840847aa2e22417cd3d3e798c561e6302905dff6bf7754d941998d401e3  -> Adr: 1FewTQmntAiBarXqNCeC7YQfDd26kbsxRN
----------------------------------------

[ k +13 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747181
X:   75bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5
Pub: 0275bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5  -> Adr: 1LZ35k6xrj62kaUxJpG45jFhEKu9Caspw6
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747156
Pub: 0375bdfa066a1a42a750f283e83ec91cc0a5b688296e6aa24a28a61e3365f378e5  -> Adr: 1Jh1JmV3RxMPDjgBPEHZHJRzyTQ4ynULkC
----------------------------------------

[ k +14 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747182
X:   561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e
Pub: 02561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e  -> Adr: 135yPWdDgsB7PtMUdsEAf8jxkv9wesKker
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747155
Pub: 03561a2ccdca12b67fdad28ee2c3cee78acf8117669e4a2543c81b1ca6eb4bd16e  -> Adr: 16mJRZXsiqw7TzVAZLAYeDxtHcDkAR8cbM
----------------------------------------

[ k +15 ]
k:   57896044618658097711785492504343953926418782139537452191302581570759080747183
X:   3905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b
Pub: 033905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b  -> Adr: 1E77TAEg7VAdUcKLGUWMnwJuY5XqDtXy9Y
n-k: 57896044618658097711785492504343953926418782139537452191302581570759080747154
Pub: 023905682b72282a782b8d8dba72cf147ade0025dca21521e1ea989040c248852b  -> Adr: 13xMdG58VuHFKESWs2wuhXbUXRP2jmGJBL
----------------------------------------

[And24r]

I understand that secp256k1 also has an analogue of the curve without the modulus p. Please pay attention to points 20, 30 and 40 in the list. Elliptic curve y^2=x^3+7
                                                                                        mod p = 59
                 X                                       Y  
                                                        
1=389.17997869568319>7677.60493098652       1=29 41  
2=97.2948906868066>959.702667793949            2=21 8
3=43.2416857982615>284.362761963552            3=38 3
4=24.3220588867655>119.979244620064            4=4 37
5=15.5630730229248>61.453397943244             5=45 6
6=10.8019990134261>35.6007250522935            6=2 29
7=7.92659885389531>22.473003580992             7=18 36
8=6.05390326510008>15.1285825296594            8=40 13
9=4.76140313990601>10.7212676235743            9=35 44
10=3.82586236563094>7.93725393285538           10=10 57
11=3.11992152882448>6.11302187457732           11=6 39
12=2.56626437079882>4.88883218767811           12=22 34
13=2.11545291568376>4.05794900766658           13=9 38
14=1.73445157514345>3.49539544037179           14=43 10
15=1.40036281759546>3.12187986816498           15=50 35
16=1.09687553663235>2.88438734809218           16=15 14
17=0.812210428639446>2.74514182899959          17=13 32
18=0.537936836476774>2.67500766772605          18=24 54
19=0.268320239516584>2.64939953725357          19=34 31
20=0.000000000000000>2.64575130934044          20=0 19
21=-0.268135371180904>2.6421055950599          21=19 50
22=-0.534978971777264>2.61665581832582         22=36 42
23=-0.797235910203483>2.54819322814125         23=47 7
24=-1.04954326142803>2.41741287261241          24=31 48
25=-1.28476029016542>2.20892804070141          25=26 58
26=-1.49448554407502>1.91365772659594          26=23 43
27=-1.66981280556291>1.53104637912539          27=53 26
28=-1.80227541928195>1.07044614297656          28=41 55
29=-1.88486133719283>0.551043426263376         29=51 12
30=-1.91293118194318>0.00000000000000          30=54 0
31=-1.88486133792247>-0.55104341920661         31=51 47
32=-1.80227542069929>-1.07044613652653         32=41 4
33=-1.66981280758989>-1.53104637359067         33=53 33
34=-1.4944855466083>-1.91365772216279          34=23 16
35=-1.28476029308939>-2.20892803742562         35=26 1
36=-1.04954326462774>-2.41741287042808         36=31 11
37=-0.797235913576117>-2.54819322688023        37=47 52
38=-0.534978975240392>-2.61665581775915        38=36 17
39=-0.26813537467757>-2.64210559491852         39=19 9
40=0.00000000000000>-2.64575130934233          40=0 40
41=0.268320236010652>-2.64939953711317         41=34 28
42=0.537936832937191>-2.67500766715381         42=24 5
43=0.812210425007265>-2.74514182769217         43=13 27
44=1.09687553281617>-2.88438734570725          44=15 45
45=1.40036281346522>-3.12187986427358          45=50 24
46=1.73445157051901>-3.49539543440092          46=43 49
47=2.11545291031484>-4.05794899878227          47=9 21
48=2.56626436433032>-4.88883217460587          48=22 25
49=3.11992152073606>-6.11302185525687          49=6 20
50=3.82586235512872>-7.93725390380444          50=10 2
51=4.76140312571999>-10.7212675785762          51=35 15
52=6.05390324508215>-15.1285824569148          52=40 46
53=7.92659882415904>-22.4730034562833          53=18 23
54=10.8019989663191>-35.6007248207001          54=2 30
55=15.5630729416093>-61.4533974625023          55=45 53
56=24.322058728008>-119.979243445921           56=4 22
57=43.2416857982615>-284.362761963552          57=38 56
58=97.2948906868066>-959.702667793949          58=21 51
59=389.17997869568319>-7677.60493098652        59=29 18

[Grzegorz2022]

I understand that secp256k1 also has an analogue of the curve without the modulus p. Please pay attention to points 20, 30 and 40 in the list. Elliptic curve y^2=x^3+7
                                                                                        mod p = 59
                 X                                       Y  
                                                        
1=389.17997869568319>7677.60493098652       1=29 41  
2=97.2948906868066>959.702667793949            2=21 8
3=43.2416857982615>284.362761963552            3=38 3
4=24.3220588867655>119.979244620064            4=4 37
5=15.5630730229248>61.453397943244             5=45 6
6=10.8019990134261>35.6007250522935            6=2 29
7=7.92659885389531>22.473003580992             7=18 36
8=6.05390326510008>15.1285825296594            8=40 13
9=4.76140313990601>10.7212676235743            9=35 44
10=3.82586236563094>7.93725393285538           10=10 57
11=3.11992152882448>6.11302187457732           11=6 39
12=2.56626437079882>4.88883218767811           12=22 34
13=2.11545291568376>4.05794900766658           13=9 38
14=1.73445157514345>3.49539544037179           14=43 10
15=1.40036281759546>3.12187986816498           15=50 35
16=1.09687553663235>2.88438734809218           16=15 14
17=0.812210428639446>2.74514182899959          17=13 32
18=0.537936836476774>2.67500766772605          18=24 54
19=0.268320239516584>2.64939953725357          19=34 31
20=0.000000000000000>2.64575130934044          20=0 19
21=-0.268135371180904>2.6421055950599          21=19 50
22=-0.534978971777264>2.61665581832582         22=36 42
23=-0.797235910203483>2.54819322814125         23=47 7
24=-1.04954326142803>2.41741287261241          24=31 48
25=-1.28476029016542>2.20892804070141          25=26 58
26=-1.49448554407502>1.91365772659594          26=23 43
27=-1.66981280556291>1.53104637912539          27=53 26
28=-1.80227541928195>1.07044614297656          28=41 55
29=-1.88486133719283>0.551043426263376         29=51 12
30=-1.91293118194318>0.00000000000000          30=54 0
31=-1.88486133792247>-0.55104341920661         31=51 47
32=-1.80227542069929>-1.07044613652653         32=41 4
33=-1.66981280758989>-1.53104637359067         33=53 33
34=-1.4944855466083>-1.91365772216279          34=23 16
35=-1.28476029308939>-2.20892803742562         35=26 1
36=-1.04954326462774>-2.41741287042808         36=31 11
37=-0.797235913576117>-2.54819322688023        37=47 52
38=-0.534978975240392>-2.61665581775915        38=36 17
39=-0.26813537467757>-2.64210559491852         39=19 9
40=0.00000000000000>-2.64575130934233          40=0 40
41=0.268320236010652>-2.64939953711317         41=34 28
42=0.537936832937191>-2.67500766715381         42=24 5
43=0.812210425007265>-2.74514182769217         43=13 27
44=1.09687553281617>-2.88438734570725          44=15 45
45=1.40036281346522>-3.12187986427358          45=50 24
46=1.73445157051901>-3.49539543440092          46=43 49
47=2.11545291031484>-4.05794899878227          47=9 21
48=2.56626436433032>-4.88883217460587          48=22 25
49=3.11992152073606>-6.11302185525687          49=6 20
50=3.82586235512872>-7.93725390380444          50=10 2
51=4.76140312571999>-10.7212675785762          51=35 15
52=6.05390324508215>-15.1285824569148          52=40 46
53=7.92659882415904>-22.4730034562833          53=18 23
54=10.8019989663191>-35.6007248207001          54=2 30
55=15.5630729416093>-61.4533974625023          55=45 53
56=24.322058728008>-119.979243445921           56=4 22
57=43.2416857982615>-284.362761963552          57=38 56
58=97.2948906868066>-959.702667793949          58=21 51
59=389.17997869568319>-7677.60493098652        59=29 18

You're right that on the mod 59 curve there is an upper and lower half with a clear transition point. Your table shows this beautifully point k=30 (n/2) has y=0, points k=20 and k=40 have x=0. These special points divide the group into segments.
I studied this thoroughly on secp256k1. The result:
Point y=0 requires x³+7 ≡ 0 (mod p), meaning -7 must be a cubic residue mod p. It is not. Point y=0 does not exist on secp256k1.
Point x=0 requires y²=7 (mod p), meaning 7 must be a quadratic residue mod p. It is not. Point x=0 also does not exist on secp256k1.
On your mod 59 curve, both exist because 59 is small and has different residue properties. secp256k1 uses b=7 with a p that deliberately eliminates both of these reference points. The only special point is O (infinity) abstract, with no x,y coordinates.
The symmetry k ↔ n-k still works (same X, flipped Y) and can be used for double coverage. But without the y=0 and x=0 points, you have no axis of division you can't determine 'upper or lower half' with just the pubkey. This is a deliberate design choice in Bitcoin. As I mentioned to you earlier, the key issue is that these properties don't carry over from small curves to secp256k1 the calculations change depending on the properties of p and the residues. But the division axis exists in a different form. On secp256k1, the pair of points around n/2 has an identical X coordinate (with leading zeros) and flipped Y:
k = (n-1)/2 → X = 0000...3b78ce563f... parity 03
k = (n+1)/2 → X = 0000...3b78ce563f... parity 02
The symmetry around this pair is perfect: (n/2)-1 ↔ (n/2)+2, (n/2)-2 ↔ (n/2)+3, etc. same X, flipped Y. This is the proper counterpart of k=30 from the table, except instead of a single y=0 point, we have a pair of points directly next to each other.
The problem remains the same: this axis is only visible when you know k. Given just the pubkey (X, parity), you don't know which side of n/2 you're on

[And24r]

I understand that secp256k1 also has an analogue of the curve without the modulus p. Please pay attention to points 20, 30 and 40 in the list. Elliptic curve y^2=x^3+7
                                                                                        mod p = 59
                 X                                       Y  
                                                        
1=389.17997869568319>7677.60493098652       1=29 41  
2=97.2948906868066>959.702667793949            2=21 8
3=43.2416857982615>284.362761963552            3=38 3
4=24.3220588867655>119.979244620064            4=4 37
5=15.5630730229248>61.453397943244             5=45 6
6=10.8019990134261>35.6007250522935            6=2 29
7=7.92659885389531>22.473003580992             7=18 36
8=6.05390326510008>15.1285825296594            8=40 13
9=4.76140313990601>10.7212676235743            9=35 44
10=3.82586236563094>7.93725393285538           10=10 57
11=3.11992152882448>6.11302187457732           11=6 39
12=2.56626437079882>4.88883218767811           12=22 34
13=2.11545291568376>4.05794900766658           13=9 38
14=1.73445157514345>3.49539544037179           14=43 10
15=1.40036281759546>3.12187986816498           15=50 35
16=1.09687553663235>2.88438734809218           16=15 14
17=0.812210428639446>2.74514182899959          17=13 32
18=0.537936836476774>2.67500766772605          18=24 54
19=0.268320239516584>2.64939953725357          19=34 31
20=0.000000000000000>2.64575130934044          20=0 19
21=-0.268135371180904>2.6421055950599          21=19 50
22=-0.534978971777264>2.61665581832582         22=36 42
23=-0.797235910203483>2.54819322814125         23=47 7
24=-1.04954326142803>2.41741287261241          24=31 48
25=-1.28476029016542>2.20892804070141          25=26 58
26=-1.49448554407502>1.91365772659594          26=23 43
27=-1.66981280556291>1.53104637912539          27=53 26
28=-1.80227541928195>1.07044614297656          28=41 55
29=-1.88486133719283>0.551043426263376         29=51 12
30=-1.91293118194318>0.00000000000000          30=54 0
31=-1.88486133792247>-0.55104341920661         31=51 47
32=-1.80227542069929>-1.07044613652653         32=41 4
33=-1.66981280758989>-1.53104637359067         33=53 33
34=-1.4944855466083>-1.91365772216279          34=23 16
35=-1.28476029308939>-2.20892803742562         35=26 1
36=-1.04954326462774>-2.41741287042808         36=31 11
37=-0.797235913576117>-2.54819322688023        37=47 52
38=-0.534978975240392>-2.61665581775915        38=36 17
39=-0.26813537467757>-2.64210559491852         39=19 9
40=0.00000000000000>-2.64575130934233          40=0 40
41=0.268320236010652>-2.64939953711317         41=34 28
42=0.537936832937191>-2.67500766715381         42=24 5
43=0.812210425007265>-2.74514182769217         43=13 27
44=1.09687553281617>-2.88438734570725          44=15 45
45=1.40036281346522>-3.12187986427358          45=50 24
46=1.73445157051901>-3.49539543440092          46=43 49
47=2.11545291031484>-4.05794899878227          47=9 21
48=2.56626436433032>-4.88883217460587          48=22 25
49=3.11992152073606>-6.11302185525687          49=6 20
50=3.82586235512872>-7.93725390380444          50=10 2
51=4.76140312571999>-10.7212675785762          51=35 15
52=6.05390324508215>-15.1285824569148          52=40 46
53=7.92659882415904>-22.4730034562833          53=18 23
54=10.8019989663191>-35.6007248207001          54=2 30
55=15.5630729416093>-61.4533974625023          55=45 53
56=24.322058728008>-119.979243445921           56=4 22
57=43.2416857982615>-284.362761963552          57=38 56
58=97.2948906868066>-959.702667793949          58=21 51
59=389.17997869568319>-7677.60493098652        59=29 18

You're right that on the mod 59 curve there is an upper and lower half with a clear transition point. Your table shows this beautifully point k=30 (n/2) has y=0, points k=20 and k=40 have x=0. These special points divide the group into segments.
I studied this thoroughly on secp256k1. The result:
Point y=0 requires x³+7 ≡ 0 (mod p), meaning -7 must be a cubic residue mod p. It is not. Point y=0 does not exist on secp256k1.
Point x=0 requires y²=7 (mod p), meaning 7 must be a quadratic residue mod p. It is not. Point x=0 also does not exist on secp256k1.
On your mod 59 curve, both exist because 59 is small and has different residue properties. secp256k1 uses b=7 with a p that deliberately eliminates both of these reference points. The only special point is O (infinity) abstract, with no x,y coordinates.
The symmetry k ↔ n-k still works (same X, flipped Y) and can be used for double coverage. But without the y=0 and x=0 points, you have no axis of division you can't determine 'upper or lower half' with just the pubkey. This is a deliberate design choice in Bitcoin. As I mentioned to you earlier, the key issue is that these properties don't carry over from small curves to secp256k1 the calculations change depending on the properties of p and the residues. But the division axis exists in a different form. On secp256k1, the pair of points around n/2 has an identical X coordinate (with leading zeros) and flipped Y:
k = (n-1)/2 → X = 0000...3b78ce563f... parity 03
k = (n+1)/2 → X = 0000...3b78ce563f... parity 02
The symmetry around this pair is perfect: (n/2)-1 ↔ (n/2)+2, (n/2)-2 ↔ (n/2)+3, etc. same X, flipped Y. This is the proper counterpart of k=30 from the table, except instead of a single y=0 point, we have a pair of points directly next to each other.
The problem remains the same: this axis is only visible when you know k. Given just the pubkey (X, parity), you don't know which side of n/2 you're on

I know that the secp256k1 curve has no points where y=0 and x=0. However, this does not change anything. I have other examples where these points are also absent, but the symmetry of the curve modulo is preserved, and the number of points remains the same.

[Grzegorz2022]

I understand that secp256k1 also has an analogue of the curve without the modulus p. Please pay attention to points 20, 30 and 40 in the list. Elliptic curve y^2=x^3+7
                                                                                        mod p = 59
                 X                                       Y  
                                                        
1=389.17997869568319>7677.60493098652       1=29 41  
2=97.2948906868066>959.702667793949            2=21 8
3=43.2416857982615>284.362761963552            3=38 3
4=24.3220588867655>119.979244620064            4=4 37
5=15.5630730229248>61.453397943244             5=45 6
6=10.8019990134261>35.6007250522935            6=2 29
7=7.92659885389531>22.473003580992             7=18 36
8=6.05390326510008>15.1285825296594            8=40 13
9=4.76140313990601>10.7212676235743            9=35 44
10=3.82586236563094>7.93725393285538           10=10 57
11=3.11992152882448>6.11302187457732           11=6 39
12=2.56626437079882>4.88883218767811           12=22 34
13=2.11545291568376>4.05794900766658           13=9 38
14=1.73445157514345>3.49539544037179           14=43 10
15=1.40036281759546>3.12187986816498           15=50 35
16=1.09687553663235>2.88438734809218           16=15 14
17=0.812210428639446>2.74514182899959          17=13 32
18=0.537936836476774>2.67500766772605          18=24 54
19=0.268320239516584>2.64939953725357          19=34 31
20=0.000000000000000>2.64575130934044          20=0 19
21=-0.268135371180904>2.6421055950599          21=19 50
22=-0.534978971777264>2.61665581832582         22=36 42
23=-0.797235910203483>2.54819322814125         23=47 7
24=-1.04954326142803>2.41741287261241          24=31 48
25=-1.28476029016542>2.20892804070141          25=26 58
26=-1.49448554407502>1.91365772659594          26=23 43
27=-1.66981280556291>1.53104637912539          27=53 26
28=-1.80227541928195>1.07044614297656          28=41 55
29=-1.88486133719283>0.551043426263376         29=51 12
30=-1.91293118194318>0.00000000000000          30=54 0
31=-1.88486133792247>-0.55104341920661         31=51 47
32=-1.80227542069929>-1.07044613652653         32=41 4
33=-1.66981280758989>-1.53104637359067         33=53 33
34=-1.4944855466083>-1.91365772216279          34=23 16
35=-1.28476029308939>-2.20892803742562         35=26 1
36=-1.04954326462774>-2.41741287042808         36=31 11
37=-0.797235913576117>-2.54819322688023        37=47 52
38=-0.534978975240392>-2.61665581775915        38=36 17
39=-0.26813537467757>-2.64210559491852         39=19 9
40=0.00000000000000>-2.64575130934233          40=0 40
41=0.268320236010652>-2.64939953711317         41=34 28
42=0.537936832937191>-2.67500766715381         42=24 5
43=0.812210425007265>-2.74514182769217         43=13 27
44=1.09687553281617>-2.88438734570725          44=15 45
45=1.40036281346522>-3.12187986427358          45=50 24
46=1.73445157051901>-3.49539543440092          46=43 49
47=2.11545291031484>-4.05794899878227          47=9 21
48=2.56626436433032>-4.88883217460587          48=22 25
49=3.11992152073606>-6.11302185525687          49=6 20
50=3.82586235512872>-7.93725390380444          50=10 2
51=4.76140312571999>-10.7212675785762          51=35 15
52=6.05390324508215>-15.1285824569148          52=40 46
53=7.92659882415904>-22.4730034562833          53=18 23
54=10.8019989663191>-35.6007248207001          54=2 30
55=15.5630729416093>-61.4533974625023          55=45 53
56=24.322058728008>-119.979243445921           56=4 22
57=43.2416857982615>-284.362761963552          57=38 56
58=97.2948906868066>-959.702667793949          58=21 51
59=389.17997869568319>-7677.60493098652        59=29 18

You're right that on the mod 59 curve there is an upper and lower half with a clear transition point. Your table shows this beautifully point k=30 (n/2) has y=0, points k=20 and k=40 have x=0. These special points divide the group into segments.
I studied this thoroughly on secp256k1. The result:
Point y=0 requires x³+7 ≡ 0 (mod p), meaning -7 must be a cubic residue mod p. It is not. Point y=0 does not exist on secp256k1.
Point x=0 requires y²=7 (mod p), meaning 7 must be a quadratic residue mod p. It is not. Point x=0 also does not exist on secp256k1.
On your mod 59 curve, both exist because 59 is small and has different residue properties. secp256k1 uses b=7 with a p that deliberately eliminates both of these reference points. The only special point is O (infinity) abstract, with no x,y coordinates.
The symmetry k ↔ n-k still works (same X, flipped Y) and can be used for double coverage. But without the y=0 and x=0 points, you have no axis of division you can't determine 'upper or lower half' with just the pubkey. This is a deliberate design choice in Bitcoin. As I mentioned to you earlier, the key issue is that these properties don't carry over from small curves to secp256k1 the calculations change depending on the properties of p and the residues. But the division axis exists in a different form. On secp256k1, the pair of points around n/2 has an identical X coordinate (with leading zeros) and flipped Y:
k = (n-1)/2 → X = 0000...3b78ce563f... parity 03
k = (n+1)/2 → X = 0000...3b78ce563f... parity 02
The symmetry around this pair is perfect: (n/2)-1 ↔ (n/2)+2, (n/2)-2 ↔ (n/2)+3, etc. same X, flipped Y. This is the proper counterpart of k=30 from the table, except instead of a single y=0 point, we have a pair of points directly next to each other.
The problem remains the same: this axis is only visible when you know k. Given just the pubkey (X, parity), you don't know which side of n/2 you're on

I know that the secp256k1 curve has no points where y=0 and x=0. However, this does not change anything. I have other examples where these points are also absent, but the symmetry of the curve modulo is preserved, and the number of points remains the same.

Correct the symmetry k ↔ n-k is a property of every cyclic group of an elliptic curve, regardless of whether the y=0 and x=0 points exist. The question is not 'does the symmetry exist' (it always does), but 'can you extract information about the key's position from it, given only the pubkey?' My answer: NO!

[And24r]

I understand that secp256k1 also has an analogue of the curve without the modulus p. Please pay attention to points 20, 30 and 40 in the list. Elliptic curve y^2=x^3+7
                                                                                        mod p = 59
                 X                                       Y  
                                                        
1=389.17997869568319>7677.60493098652       1=29 41  
2=97.2948906868066>959.702667793949            2=21 8
3=43.2416857982615>284.362761963552            3=38 3
4=24.3220588867655>119.979244620064            4=4 37
5=15.5630730229248>61.453397943244             5=45 6
6=10.8019990134261>35.6007250522935            6=2 29
7=7.92659885389531>22.473003580992             7=18 36
8=6.05390326510008>15.1285825296594            8=40 13
9=4.76140313990601>10.7212676235743            9=35 44
10=3.82586236563094>7.93725393285538           10=10 57
11=3.11992152882448>6.11302187457732           11=6 39
12=2.56626437079882>4.88883218767811           12=22 34
13=2.11545291568376>4.05794900766658           13=9 38
14=1.73445157514345>3.49539544037179           14=43 10
15=1.40036281759546>3.12187986816498           15=50 35
16=1.09687553663235>2.88438734809218           16=15 14
17=0.812210428639446>2.74514182899959          17=13 32
18=0.537936836476774>2.67500766772605          18=24 54
19=0.268320239516584>2.64939953725357          19=34 31
20=0.000000000000000>2.64575130934044          20=0 19
21=-0.268135371180904>2.6421055950599          21=19 50
22=-0.534978971777264>2.61665581832582         22=36 42
23=-0.797235910203483>2.54819322814125         23=47 7
24=-1.04954326142803>2.41741287261241          24=31 48
25=-1.28476029016542>2.20892804070141          25=26 58
26=-1.49448554407502>1.91365772659594          26=23 43
27=-1.66981280556291>1.53104637912539          27=53 26
28=-1.80227541928195>1.07044614297656          28=41 55
29=-1.88486133719283>0.551043426263376         29=51 12
30=-1.91293118194318>0.00000000000000          30=54 0
31=-1.88486133792247>-0.55104341920661         31=51 47
32=-1.80227542069929>-1.07044613652653         32=41 4
33=-1.66981280758989>-1.53104637359067         33=53 33
34=-1.4944855466083>-1.91365772216279          34=23 16
35=-1.28476029308939>-2.20892803742562         35=26 1
36=-1.04954326462774>-2.41741287042808         36=31 11
37=-0.797235913576117>-2.54819322688023        37=47 52
38=-0.534978975240392>-2.61665581775915        38=36 17
39=-0.26813537467757>-2.64210559491852         39=19 9
40=0.00000000000000>-2.64575130934233          40=0 40
41=0.268320236010652>-2.64939953711317         41=34 28
42=0.537936832937191>-2.67500766715381         42=24 5
43=0.812210425007265>-2.74514182769217         43=13 27
44=1.09687553281617>-2.88438734570725          44=15 45
45=1.40036281346522>-3.12187986427358          45=50 24
46=1.73445157051901>-3.49539543440092          46=43 49
47=2.11545291031484>-4.05794899878227          47=9 21
48=2.56626436433032>-4.88883217460587          48=22 25
49=3.11992152073606>-6.11302185525687          49=6 20
50=3.82586235512872>-7.93725390380444          50=10 2
51=4.76140312571999>-10.7212675785762          51=35 15
52=6.05390324508215>-15.1285824569148          52=40 46
53=7.92659882415904>-22.4730034562833          53=18 23
54=10.8019989663191>-35.6007248207001          54=2 30
55=15.5630729416093>-61.4533974625023          55=45 53
56=24.322058728008>-119.979243445921           56=4 22
57=43.2416857982615>-284.362761963552          57=38 56
58=97.2948906868066>-959.702667793949          58=21 51
59=389.17997869568319>-7677.60493098652        59=29 18

You're right that on the mod 59 curve there is an upper and lower half with a clear transition point. Your table shows this beautifully point k=30 (n/2) has y=0, points k=20 and k=40 have x=0. These special points divide the group into segments.
I studied this thoroughly on secp256k1. The result:
Point y=0 requires x³+7 ≡ 0 (mod p), meaning -7 must be a cubic residue mod p. It is not. Point y=0 does not exist on secp256k1.
Point x=0 requires y²=7 (mod p), meaning 7 must be a quadratic residue mod p. It is not. Point x=0 also does not exist on secp256k1.
On your mod 59 curve, both exist because 59 is small and has different residue properties. secp256k1 uses b=7 with a p that deliberately eliminates both of these reference points. The only special point is O (infinity) abstract, with no x,y coordinates.
The symmetry k ↔ n-k still works (same X, flipped Y) and can be used for double coverage. But without the y=0 and x=0 points, you have no axis of division you can't determine 'upper or lower half' with just the pubkey. This is a deliberate design choice in Bitcoin. As I mentioned to you earlier, the key issue is that these properties don't carry over from small curves to secp256k1 the calculations change depending on the properties of p and the residues. But the division axis exists in a different form. On secp256k1, the pair of points around n/2 has an identical X coordinate (with leading zeros) and flipped Y:
k = (n-1)/2 → X = 0000...3b78ce563f... parity 03
k = (n+1)/2 → X = 0000...3b78ce563f... parity 02
The symmetry around this pair is perfect: (n/2)-1 ↔ (n/2)+2, (n/2)-2 ↔ (n/2)+3, etc. same X, flipped Y. This is the proper counterpart of k=30 from the table, except instead of a single y=0 point, we have a pair of points directly next to each other.
The problem remains the same: this axis is only visible when you know k. Given just the pubkey (X, parity), you don't know which side of n/2 you're on

I know that the secp256k1 curve has no points where y=0 and x=0. However, this does not change anything. I have other examples where these points are also absent, but the symmetry of the curve modulo is preserved, and the number of points remains the same.

Correct the symmetry k ↔ n-k is a property of every cyclic group of an elliptic curve, regardless of whether the y=0 and x=0 points exist. The question is not 'does the symmetry exist' (it always does), but 'can you extract information about the key's position from it, given only the pubkey?' My answer: NO!

My answer is yes. We simply don’t know the mathematical formulas, but they do exist. And my statement is correct. Because in mathematics it’s never the case that all the initial data are known, but the answer is unknown. It’s not like a hash function, where information is partially lost in the process. I have mathematical formulas that would greatly surprise you, but I can’t show them. And +7 is not magic, as I’ve already said — it’s just mathematics.

[speed_user_113]

Andr24 wtf? You start to sound like other user....what are you talking about there? If you have formulas why are you stil here with loosers? You are like a little child that say he is the best but he don't want to show....wtf? Grow up!!!
If you come with afirmations, come also with proofs otherwise you just have another theory of conspiration...

The spart guys who have something they stay in shadow, not talk like a child who discover the ice cream.....

You need to know how to talk technical stuff and share if you have something, and others will share to you....so you take info from others, but you keep for you....why anybody else will share to you or confirm anything if you are selfish?

[And24r]

Andr24 wtf? You start to sound like other user....what are you talking about there? If you have formulas why are you stil here with loosers? You are like a little child that say he is the best but he don't want to show....wtf? Grow up!!!
If you come with afirmations, come also with proofs otherwise you just have another theory of conspiration...

The spart guys who have something they stay in shadow, not talk like a child who discover the ice cream.....

You need to know how to talk technical stuff and share if you have something, and others will share to you....so you take info from others, but you keep for you....why anybody else will share to you or confirm anything if you are selfish?

I didn’t say I have formulas for obtaining a private key. But there are interesting formulas and prospects for this. Since you’re so smart, look at what I’m telling you: divide the X‑coordinate of the base point by the X‑coordinate of the public key and extract the square root. You’ll get the private key with a remainder. There’s also a formula for removing the remainder, but it needs to be refined. And once I refine it, we can move on to calculations with the modulus of a prime number and obtain the private key.

[damiankopacz87]

There’s also a formula for removing the remainder, but it needs to be refined.

If You love mathematics and if You have skills, You can shorten Your suffering and make mathematical proof that it is imposible to separate residue. This is the beauty of eliptic curve multiplication.

BR
Damian

[kTimesG]

But after my geometric research: The points (n/2)·G and (n/2+1)·G have the exact same X coordinate (the one with leading zeros: 00000000...3b78ce563f...) — they differ only in Y parity. This is a (k, n-k) pair.

More interestingly — if you unfold the keys around n/2, you'll see perfect mirror symmetry:
n/2 - 1  ↔  n/2 + 2   (same X, flipped Y)
n/2 - 2  ↔  n/2 + 3
n/2 - 3  ↔  n/2 + 4

There is no (n/2)G point because n = 0 mod n. It is the point at infinity. You probably meant (n-1)/2 and (n+1)/2, e.g. -1/2 and 1/2.

So, after fixing your typo, yes, both 0 and "n/2" are symmetry "checkpoints". Think of them like opposite "inexistent inter-points" on a circle, where the circle points are consecutive points on the curve. It's not so interesting because we already know that 0 == n/2 so the mirror folding is known, as -1/2 + 1 = 1/2. But it is useful for some speed optimizations

The rest of the talk in this page is BS non-sense.

[And24r]

There’s also a formula for removing the remainder, but it needs to be refined.

If You love mathematics and if You have skills, You can shorten Your suffering and make mathematical proof that it is imposible to separate residue. This is the beauty of eliptic curve multiplication.

BR
Damian

Yes, previously I was told that it’s impossible to obtain even an approximate value of the private key. But as you can see, it turned out to be not the case.

[GinnyBanzz]

Andr24 wtf? You start to sound like other user....what are you talking about there? If you have formulas why are you stil here with loosers? You are like a little child that say he is the best but he don't want to show....wtf? Grow up!!!
If you come with afirmations, come also with proofs otherwise you just have another theory of conspiration...

The spart guys who have something they stay in shadow, not talk like a child who discover the ice cream.....

You need to know how to talk technical stuff and share if you have something, and others will share to you....so you take info from others, but you keep for you....why anybody else will share to you or confirm anything if you are selfish?

I didn’t say I have formulas for obtaining a private key. But there are interesting formulas and prospects for this. Since you’re so smart, look at what I’m telling you: divide the X‑coordinate of the base point by the X‑coordinate of the public key and extract the square root. You’ll get the private key with a remainder. There’s also a formula for removing the remainder, but it needs to be refined. And once I refine it, we can move on to calculations with the modulus of a prime number and obtain the private key.

And after you've done that you'll still have no advantage to anybody else in finding the private key.