Curve point divided by integer - is it possible?

[BlackHatCoiner]

So my output (the ones on the curve anyway) is correct? And not the test case output?

Yes, they're correct (the first five). Which one is the test case output? Every k value you've written except those that are outside the secp256k1 return correct results.

As for those last three; how did you manage to find their x and y coordinates since they don't belong there? Are they from another curve?

I noticed this from my testing, it looks like that k and n-k have the same X coordinate but inverted (curve order - y) Y coordinate.

Yep. Just learnt something new today.

115792089237316195423570985008687907852837564279074904382605163141518161494337 (highest value for k)
115792089237316195423570985008687907853269984665640564039457584007913129639936 (your k)

Actually, that was my mistake. I edited my last post. The highest value for k is this minus one. The k you quoted is n, but the highest k is n-1. So it's:

Code:

115792089237316195423570985008687907852837564279074904382605163141518161494336

[1714686028]

1714686028

[1714686028]

1714686028

[1714686028]

1714686028

[gmaxwell]

it is only in ECC that we only return numbers that are positive and between 0 and prime. ie. -1 ≡ 2 (mod 3).

Finite fields exist far beyond ECC.

[NotATether]

So my output (the ones on the curve anyway) is correct? And not the test case output?

Yes, they're correct (the first five). Which one is the test case output? Every k value you've written except those that are outside the secp256k1 return correct results.

As for those last three; how did you manage to find their x and y coordinates since they don't belong there? Are they from another curve?

All the k's are input and they output the X and Y values in the test.

The last three were in the StackExchange answer that had the test cases I used   I actually duplicated the very last test case three times by accident and skipped a bunch of k's that looked similar to that one, that's why their output looks all the same.

[pooya87]

By the way, I noticed that both 1 and n-1 give you the same x coordinate, but not the same y. What's the reasoning behind this?

When you negate a point on an elliptic curve (Q -> -Q) you change the y coordinate with -y and -y≡p-y (mod p).
If Q=k*G then -Q=-k*G and since -k≡n-k (mod n) => -P=(n-k)*G
Hence 1 and n-1 or x and n-x have the same x coordinate while having different y coordinates that is negative of one another.

P.S. Be careful that when negative a point the operations are mod p where p is the curve prime but when negating a private key (k) the operations are mod n where n is curve order.

[BlackHatCoiner]

Hence 1 and n-1 or x and n-x have the same x coordinate while having different y coordinates that is negative of one another.

Excuse me if that's a dumb question, but how do you recognize if a hexadecimal number is negative? For example, take a look in the y coordinates of those two points:

Code:

483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8 (for k = 1)
b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777 (for k = n-1)

When I convert them to decimal they both get huge, but with no sign.

Code:

32670510020758816978083085130507043184471273380659243275938904335757337482424
83121579216557378445487899878180864668798711284981320763518679672151497189239

I assume that this equation should always be true for k₁ + k₂ = n: y₁ + y₂ = 0

[pooya87]

Excuse me if that's a dumb question, but how do you recognize if a hexadecimal number is negative? For example, take a look in the y coordinates of those two points:

All numbers that come out of any Elliptic Curve related method (such as the point multiplication that you are probably using) are positive because by contract we only report positive numbers. Under the hood if any number is not positive the method should make it positive by computing its mod and then adding the prime so that is is always between 0 and the prime.

I assume that this equation should always be true for k₁ + k₂ = n: y₁ + y₂ = 0

That's right, when y₁=-y₂ the sum should be 0 as I explained above. The correct notation is this:
y₁ + y₂ ≡ 0 (mod p)

Here is an example:

Code:

k1 = 12233456
k2 = n -k1 = 115792089237316195423570985008687907852837564279074904382605163141518149260881

Q1.x = Q2.x = 114232938462135891686500124810364184602379329612856280684251352758526248132659
Q1.y = 17861562014024620061259601058584332743358430500525147265305253422039778750471
Q2.y = 97930527223291575362311383950103575109911554165115416774152330585869055921192

Q1.y + Q2.y = 115792089237316195423570985008687907853269984665640564039457584007908834671663 ≡ 0 (mod 115792089237316195423570985008687907853269984665640564039457584007908834671663)

[BlackHatCoiner]

Under the hood if any number is not positive the method should make it positive by computing its mod and then adding the prime so that is is always between 0 and the prime.

So, essentially, if the number is negative, we define it as following: number = (number mod n) + n. Correct? Wouldn't it be the same if we simply subtracted the number from n? (number = n - number)

The correct notation is this:
y1 + y2 ≡ 0 (mod p)

Why does y₁ + y₂ give us p and not n? (when y₁ = -y₂)

[NotATether]

The correct notation is this:
y1 + y2 ≡ 0 (mod p)

Why does y₁ + y₂ give us p and not n? (when y₁ = -y₂)

Y and X are point coordinates and not a multiplier and as such it has to be in the prime group (p) and not the curve order group (n).

[pooya87]

Wouldn't it be the same if we simply subtracted the number from n? (number = n - number)

It depends on where the negative number was produced, if you are sure that the equation is giving a negative number between -n and 0 exclusive then yes but it is not always like that. For example if you have -17 (mod 7) and add the prime (7) to it you'll still get a negative number (-10) and have to repeat it again (-3) and again (4) to finally get a positive number. But if you first compute the remainder then add the prime it will always be positive.
-17 ≡ -3 ≡ 4 (mod 7)

[BlackHatCoiner]

And when I was doing some calculations, I realized that something goes wrong.

I noticed this from my testing, it looks like that k and n-k have the same X coordinate but inverted (curve order - y) Y coordinate.

This is false or it's true only if k = 1.

Code:

Private key: 1
x: 79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
y: 483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

Private key: 115792089237316195423570985008687907852837564279074904382605163141518161494336
x: 79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
y: b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777

But, let's take another number. (k = 2)

Code:

Private key: 2
x: c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
y: 1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Private key: 115792089237316195423570985008687907852837564279074904382605163141518161494335
x: 2f01e5e15cca351daff3843fb70f3c2f0a1bdd05e5af888a67784ef3e10a2a01
y: a3b25758beac66b6d6c2f7d5ecd2ec4b3d1dec2945a489e84a25d3479342132b

Am I doing anything wrong or is the k and (n-k) rule incorrect for any k∈[2, n-2]?

[MixMAx123]

It is difficult to find another calculator to test my results against. Can someone verify if these are actually wrong (or correct, but the test vectors are wrong)?

Here is a tool of mine with which you can generate as many test vectors as you want.
- You can create and test signatures
- You can add points
- You can subtract points
- You can multiply points by a factor
- You can divide points by a factor
And you can carry out all the important operations for the Secp256k1 curve and test your code with it.


https://github.com/MrMaxweII/Secp256k1-Calculator

Download: https://github.com/MrMaxweII/Secp256k1-Calculator/releases/tag/V1.0.14



If you want to create test vectors, set "multiplication" and multiply "G" by whatever you want.

[NotATether]

Here is a tool of mine with which you can generate as many test vectors as you want.
You can create and test signatures.
And you can carry out all the important operations for the Secp256k1 curve and test your code with it.

~snip

If you want to create test vectors, set "multiplication" and multiply "G" by whatever you want.

It is nice, but, what do those other curve operations you've just shown in the screenshot do? (I'm talking about ':', 'sig', and 'ver' in the dropdown menu).

[MixMAx123]

Here is a tool of mine with which you can generate as many test vectors as you want.
You can create and test signatures.
And you can carry out all the important operations for the Secp256k1 curve and test your code with it.

~snip

If you want to create test vectors, set "multiplication" and multiply "G" by whatever you want.

It is nice, but, what do those other curve operations you've just shown in the screenshot do? (I'm talking about ':', 'sig', and 'ver' in the dropdown menu).

 +  You can add points                       https://i.ibb.co/5KfnfZ5/Unbenannt.png
 - You can subtract points                   https://i.ibb.co/bFWRCKX/Unbenannt.png
 * You can multiply points by a factor   https://i.ibb.co/RQn3SGz/Unbenannt.png
 : You can divide points by a factor.       https://i.ibb.co/m5dPQ5K/Unbenannt.png
sig: Sign https://i.ibb.co/w0hcCbC/Unbenannt.png
ver: Verify https://i.ibb.co/RzPKHTk/Unbenannt.png
verify:

[garlonicon]

Code:

Private key: 2
x: c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
y: 1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Private key: 115792089237316195423570985008687907852837564279074904382605163141518161494335
x: 2f01e5e15cca351daff3843fb70f3c2f0a1bdd05e5af888a67784ef3e10a2a01
y: a3b25758beac66b6d6c2f7d5ecd2ec4b3d1dec2945a489e84a25d3479342132b

How you calculated the second key? For example, after doubling "-1" key you can get this:

Code:

modulo=FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
c=3*x*x/2*y
c=3*(79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)^2/2*b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777
c=3*8550e7d238fcf3086ba9adcf0fb52a9de3652194d06cb5bb38d50229b854fc49/6f8a4b11b2b8773544b60807e3ddeeae05d0976eb2f557ccc7705edf09de52bf
c=8ff2b776aaf6d91942fd096d2f1f7fd9aa2f64be71462131aa7f067e28fef8ac/6f8a4b11b2b8773544b60807e3ddeeae05d0976eb2f557ccc7705edf09de52bf
c=8ff2b776aaf6d91942fd096d2f1f7fd9aa2f64be71462131aa7f067e28fef8ac*481ce5f9b128b2ceb2186fee3a0f5b953eaa9fdcac23c2cbf041511268985589
c=34ca4d7bd7efe5cfc1462edca66d539c0a77a83d09ce1196c92c514312232a7e
rx=c*c-2*x
rx=34ca4d7bd7efe5cfc1462edca66d539c0a77a83d09ce1196c92c514312232a7e^2-2*79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
rx=b9814c9235a6f4c5db86059a32ce92e661af8801e88b8e5a5f910c708a60d1e6-f37cccfdf3b97758ab40c52b9d0e160e0537f9b65b9c51b2b3e502b62df02f30
rx=c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
ry=c*(x-rx)-y
ry=34ca4d7bd7efe5cfc1462edca66d539c0a77a83d09ce1196c92c514312232a7e*(79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798-c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5)-b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777
ry=34ca4d7bd7efe5cfc1462edca66d539c0a77a83d09ce1196c92c514312232a7e*b3b9e6eab7ef3e3f255b222738c68e2ea6246e8fa0deec31ae4677a0ba8774e2-b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777
ry=9ce3bc8a331e7860fe957feaab820c680af219522713db054053fdc5b41f424d-b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777
ry=e51e970159c23cc65c3a7be6b99315110809cd9acd992f1edc9bce55af301705

So it should look like this:

Code:

Private key: 115792089237316195423570985008687907852837564279074904382605163141518161494335
x: C6047F9441ED7D6D3045406E95C07CD85C778E4B8CEF3CA7ABAC09B95C709EE5
y: E51E970159C23CC65C3A7BE6B99315110809CD9ACD992F1EDC9BCE55AF301705

[MixMAx123]

Here I have a program with TestVectors for you:

https://bitcointalk.org/index.php?topic=5363860.0

Here is an example:

Code:

{
 "88": {
  "p": "88d3ad1293d7ce5ed68dafebc0d14da58bee5e5670a16df7b489211854deab49",
  "x": "57c628c86c520d069b49efc06f91ccb303c89f73f4a5f914600ba81573f55af7",
  "y": "1281a83ac4de684ad5e5ba22d965c9424baa32705631bdd9bc37dfbee77fc988"
 },
 "89": {
  "p": "06d0db075b51c8314dce54230139ea1333dd38709fc7098fd7b35a7f0c0ef48f",
  "x": "78e2f762741b828fee2a63dbb62eabe80c141e84d7523137bc8de9c46b751779",
  "y": "2fc85c1b089afde6653e70c60bf84b8222fb1cb66492520bea65b51b4261f4d7"
 },
 "90": {
  "p": "5836a959cd2863cb9a73cb956955f3968dda107d7b321856ddd08e85351034b4",
  "x": "4fd730f63464a99b0e1cd8071c5df6e1bf27ce9a9829328db9f2f1c0eb8019e8",
  "y": "55b3b061d8d68c41e05ab2f0a48849ceb8edbb8291418b5b5eef2db5034abf81"
 },
 "91": {
  "p": "b07122c634bd0799a28bc8f41e36d58609a7413190f5dbc6d37c983be50e3b6b",
  "x": "c000a6555ceca19e2212e5792799d9fc3be8f6b434518927ec1f6fe11f995b9c",
  "y": "7ec6c1bbaa3aa31e9aa913b5b519fb2fddaac8fb2bf9a690bc024dcd3f63be83"
 },
 "92": {
  "p": "ba50f4040d50fd063d0c5f1231e559c76f1b114c67566500d5f9670673834622",
  "x": "a9233f427367a5001c3b3693c8424b64c4dd081b6d566e607b81af0b0cb8b803",
  "y": "bb898545d6483b0814b0ea3304bcf09e66fa3a60c2a10bfa8a40f6a4ee8fef6e"
 },
 "93": {
  "p": "b9b5c65b3fa260d6e1ebe6ef676ad4d0538f9f6e3c3cf6511a7ecaa6c9f74bee",
  "x": "37e0374418c3887b317b3ad0a4e471811c9e318c3bef113a1694ce002616331c",
  "y": "5fd2d3ffb7754e042381c9dfde694044c5c8e96c4ea8bc6c6c2b20d0bd291385"
 },
 "94": {
  "p": "f5e4a7e5ad0b3d2bb56712999f71c008f05f88cb43a1f6c156c0b1e5bba5eb64",
  "x": "f70b2b601ea34fff5ba9f0379ed80db5b31e7b7161a84e3160c652ff10f1da4c",
  "y": "b513958d11804caeee9bc9672371624f27bdd8eb02a48c7ca12fbe18f83774a3"
 },
 "95": {
  "p": "25ce7e13abae2e617ac0d3c9617b1681b2e04458ae942f0f13dc5a223da837b8",
  "x": "84bbcc9fc1917deacadde7e8c6ecbd36fca356925137bbd914431ddfa4543d3e",
  "y": "e891a7be911c7e84f2a7657c9b60ef98b9d224e1157fe4ead4d44128499b2a8d"
 },
 "96": {
  "p": "b21f24ad291a1a9e51b6fc21bfa83c5db4d96fbb96aa589320ba62f5c47cb674",
  "x": "3636400c28c7968528baba38bb7d8a2c6b142a87d63099cc333980c6169bf42c",
  "y": "d4e99af45629e46c90576ccb3ebb650f4d758fa9ef3f5bcd13951a3f5b988906"
 },
 "97": {
  "p": "791051af290b4954065dbe3b9289a4fb2e55f132f986941651acddfdc4517a96",
  "x": "59542726f9ea941fd055828314389d55eb8e9f4602a01b2d204b0ddf0da04631",
  "y": "5be58364ea5ddda5046800cd1e46aedecbfbb7c1bd3cbf961b555ce44a522a32"
 },
 "10": {
  "p": "92f27b3c63c7ae067d44c180fa3cc0902cef2f572310a63601d0bf6647bb6d4b",
  "x": "3f7bbe59fec46efb341296712c8d5cbc43cc46a5228200e64698ce3ed29938cf",
  "y": "e74d189369e0bd20c8fde9490f8b9114a57b133d40ef3388f743d6061afbc010"
 },
 "98": {
  "p": "c594c3c089918ccba583403606a137b5e074f6d055bfcf9a9c2f5ce327f1f39e",
  "x": "97f25d6149cc9f29e06a3073e8861b43a1e8f0646f611d8007e352accf5d5939",
  "y": "8733a77c651c36fd148eceb29c010e2d4646b0a4a78462e213f75cd9e6c7e948"
 },
 "11": {
  "p": "1aadb613eec0b1b2665fe70fdd5438a010b7b90bc99f7dca8d94462e5d0b9b1a",
  "x": "2bbc9bcef471f35ec47d94f1aed6b7ae38af86409263c480088c72ea08a41b3a",
  "y": "53b39c9d4db0c7aa50cfff4ac4b9867258febebcdc24f8bf9ed3a0181f7a4eab"
 },
 "99": {
  "p": "6a0bad07715f9b7130b6561d16644a3ce2eac7f63023bb1870d4c7e9dd6da428",
  "x": "dca01a18ef657c3a4b98327ce932a8f2e451bc4679cf5241f5fe2bfa1233f983",
  "y": "d3327fd6d82fed7821244a0e9fa61ee6f92595c1e03ca9e85e1000abef59fcd0"
 },
 "12": {
  "p": "0bcf7530baaa2028ff251bce43b4241e84cf72af63a588baa4c60397562e4241",
  "x": "7686c6ca99138b876c8f7cedc781b654e6ac9104aea681106bbaec85e1b0fa6d",
  "y": "0e09e016c1297c32a1f785acc7a3e00200517656c2e9425dbd64521739e5f9de"
 },
 "13": {
  "p": "a0dea83b2c23efbb79b3f2fce778a91d6a5236259f04e1a6bcf3b84f349fd0b1",
  "x": "a7457a11789c0f1233e0eacf80a9e1015cb72b8e305acf42e8d4fb675cb8bc78",
  "y": "119dc7c5e8d66d6203bede5cf11af18309d2dc84b19c0fbb736b31f6dd94c294"
 },
 "14": {
  "p": "db9fb6404ddd981a3ee720381c1539fd268e7d92ba1dd127f24e0867a27e36dc",
  "x": "e47ecfaebcf713fe7dd80c8953d5954f0fe0b2e7c0206d09cb765d587b612c5e",
  "y": "75fdd1ab65f38a001819be0fdb3490e329e668ce7aff5165d5645e1737de2634"
 },
 "15": {
  "p": "2be401c87ea73e169b88f76822800aa5117b6ec9cf68a4f750eab97e636d0bf1",
  "x": "928b8081b5811e70c60a9647a732ad92ad18b8eda4de28c7aa18333183a30ba3",
  "y": "3022c6953926aea83478bb93a60929310b9255a548b9b57c429faed4370b06db"
 },
 "16": {
  "p": "fe0e641366253e93bbe90de4f358c3c69ba9fe072fb927f8111deabe53d31c77",
  "x": "4ba526a6f12bdbdfc8dac0ee4c9d693998ea2fb8ffe729ff834cf5a87f253f28",
  "y": "cf6672bfd32fb194631eadf05c9efa0508655e80d196c2d174ae4b65328341ea"
 },
 "17": {
  "p": "618042111fbf2019f27c8f9796bc85c0f0215004854196e76176ce957e353baf",
  "x": "e520a425b922874388187ddcec82f364e4440fc7bed0425d0d0051ffa382c9f3",
  "y": "708a82a4f7e7041a9e36d1d7fed029fd77995383cc7af87ce2d130970584db65"
 },
 "18": {
  "p": "de176e801ca7b80214da1795dc9d9a3325c19e4a857b3d914520fb0b38ef228b",
  "x": "f287106a45562a841cd331fb1672f09c99f95d9250ee667aa839ada51ce826f7",
  "y": "60267179e640d8c41ad4589b0b397c2d22ff075f1e79f843036f206bb4bda7f9"
 },
 "19": {
  "p": "d500096548198775bdcd36d43bba81edbc24b8ff9bf05dc003ab131e4bd81f48",
  "x": "d86050de01e082e4239798ca4f5cf65fdb6985828e49cf04707d7e4dba64f4a2",
  "y": "66f0507d6bb0ea8ebfbe6675aa1ebc1e07725ad6944a527c96092c2250cb3630"
 },
 "0": {
  "p": "1741391819726679fd0f1c65ec26d4937c523467e6bbc88c3268a67257208cf9",
  "x": "c9fb7018b54a91062e118507025afc63d26f38da32a29958ce5b64e248afa750",
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  "x": "5f35cee558cce37bd146fd3b85a559e1e493e1f9df29ed999ccfc05454a4b49d",
  "y": "001735714d6e61317ef1777f26fba13a9b17da1599c425fe998cef58634b73d4"
 },
 "66": {
  "p": "bd5f6d6046f7a3c2ee61ae99bc8705bac4e4e4a0d7509c0a889c4176ec34589a",
  "x": "7765a6e84db0271a9760b2acbe01dd43735b81dc2fc4ac9533baf8d58608b9be",
  "y": "17338889d7cbaeac98734e6640c34ca22eafc6698b85aac2eb95f2e5df489e87"
 },
 "67": {
  "p": "1030eaae2fa9c36bb165fa99965dcddd7f208b9fe39f8ec470040cd2f80c1bd3",
  "x": "96f285a1abd6bde9e63859a616071d6fe4d8b7f22f59f96d4692c2d3f70a4506",
  "y": "d62adf21842b512132e26f5d6c816ce2e2e07490834001ccdbbe9f71810b96c0"
 },
 "68": {
  "p": "ce8a36223532ea9f5989f89376811dd0b4328cd9360acc8c98bc4e83cacf9919",
  "x": "c4e072e4a855ce1f62792ce6c6df7c90666b844a189a77735523431a574cf164",
  "y": "f0a351efe8d597311dcbbf2c4b0935f1fe9e20b5b88287f80014d85040c72d48"
 },
 "69": {
  "p": "ce1eae9eb6566073f49e0d64dfe616ecb07dc1a470b32ad17b5f94a1aed022d6",
  "x": "43703a427f36dccc6c75640e420596f6d9b5eed74516b0c3312efeddf1669b75",
  "y": "ece617a54eaf9bb2ef96f2777acac9a0d8d522dad718a3d1e7ff8ac51b3d6825"
 },
 "70": {
  "p": "3d721fd21934b7c749a85900e1e258caffac0f67a3fa11b62616f4b2be592aae",
  "x": "604c48b65967177f6f70ecce554f30512c02f736d30cd1641dc050cd47651274",
  "y": "50a99e816648359c7f2a7f2d30399023346b2a08b7c0b734bd7223541a674157"
 },
 "71": {
  "p": "16d54c630dd0e73895bd5d92f4f7d79ef98779af43bd79e0fc73af0a2a2fe190",
  "x": "4e23bb6da3fe9af5161d77fdb877ff99d6321ff348ddd86ca4ad1477cdea6d2f",
  "y": "05626b1b24131a869ac2a3686dc625ffde0e88e2b223d12c0f553c520279dce4"
 },
 "72": {
  "p": "dad98c1313307d7d87d067f879ba397e0c4920c9a293d5af8af337ce7290ab25",
  "x": "b70d6146058247f3a9c1e9f94a140f7543047192eb56d6ae0e790a3193ec7bfb",
  "y": "a974552cb63d4e7b4478899bc902c6fb49eede1d158987a0c5d604f3b6c502ac"
 },
 "73": {
  "p": "8a0cb463a70c3c3b66b383ff04574ab6de23bfa309b3cf718ad54df83cff2046",
  "x": "a7e1e3713b0a097845d5e4dc0d7e5ed6f7d1589e46d79af8df76d3ce9299dc4f",
  "y": "64ae2845707a667eb0529b1a8ceec0c2b0ba15639177d6a263e47566b1c74163"
 },
 "74": {
  "p": "b60cca1b4fc51f17882b5a4906bd5c8c898f5d8b24bd731c7ace58c4a1f45ec0",
  "x": "3796f964bd68eafcd441c9b87232bc06ac9cfe8140c0a009dfbef547380f2d68",
  "y": "c2903e6247e8017685fe7e5a9dca3cc79f71f1e5e48610ad38e4000097db366e"
 },
 "75": {
  "p": "1ff9ce604a7c46e317522ad1c8c060574aabc222b57f4c7af1f0682c3e74664f",
  "x": "a20fad464d83f5ffc866d34761e1185228689f92dcb2997cbc0c830a857a7fe8",
  "y": "d10c77672fd3bcb329af63a1ba0a1a7c16df60cee95d0edf3cac3e27a479b53b"
 },
 "76": {
  "p": "f655b5a7b0ff65ec97c997d7a0ed9cd21fc29b8fe533ae2647a0f7b4e98fd651",
  "x": "5fe600bce15be3dedf442560ca2b841e6b93c8854456049f5e52760b25b27d75",
  "y": "091b26426056e290542661dcff2683e93ea2f10f4df0ee7b207f59cfa54f3e0f"
 },
 "77": {
  "p": "a34e6bf30ff9bd0c5ab83e46a36da07140143da0445bb28f340ccb1a080f50ac",
  "x": "000ed83f4770584e11e4de34469fd39cf83e228db4e2f738aa98a1dadf8d9a79",
  "y": "d40f278e48c9d96c50435782979c265f17051623830650acb2c10ea247c2427a"
 },
 "78": {
  "p": "f146b54b021f1e697c11f1bdb0242cea310e6722ad3ec384b9c73254ab59441e",
  "x": "fbdbe968517624e7afb6984e206d5a81c64e1cd55bbc08a6f72a69baa5c71754",
  "y": "8a4f0aea37cb5e552706488d196d9587fa12ad66b93c0ae8cc45a1385c4a19bf"
 },
 "79": {
  "p": "7b3bc284b49e329de6951b31b054043377677f01b3f58d83f4da35ec0bf4d136",
  "x": "e93039d69e0b50225c9ab001bbf82b8e104b2ccbba70a168617e3ad97b7c94a3",
  "y": "3b8fd107fa22bfa9c24b79515cab5b9980889afd6965f786c669a3cd84cc4888"
 },
 "80": {
  "p": "ff1fa3f6ade93c4561631080a9f3ddcda0ecafa526eb5aa1a344a6943d4fbdb6",
  "x": "00d781ba41b795db8e6fb01a620e637c323ba7b744b54c43a459012f71b280f7",
  "y": "1e8cf4c3959f12bd7e46f788a1cfd65ce5b6ebfbf69ab4f925da92582e454f81"
 },
 "81": {
  "p": "d640a015ca6a02c8eeba76774768070347fe5f1557c777e2c0dc56733f975dab",
  "x": "bdb0c614ce737c2d98b79629f59289d738bbf1dbd82d7ea49f7176d376ffc27e",
  "y": "573ebc00966b8a14e3bb6a496aa980a243debeca51f6b22dd97116d70fe6e941"
 },
 "82": {
  "p": "4bc01d29a8b4715fded112c3367493a5540bcc74077cd13c72f4b8b2c5d419de",
  "x": "c5b702bb02a36019ba545d4c066c3579314bfa1ba244c6917b15d232c54c6042",
  "y": "d7da639ff679a2bfd0fcd18c5760dce825de7b2f7cc90091b08eb0a50be74959"
 },
 "83": {
  "p": "777bbfa764b890794aaa746f7ab5180075ab795a802465beb002f21e84a8db9e",
  "x": "b24c8767fb979ce04f34e178098a9ca84078f3fbbb40f23a2622d07c7df7a671",
  "y": "6499cd6c285efbfadb0d80e28b55d5560f37fd53eb97ae4ccf7a74d89c5c1e57"
 },
 "84": {
  "p": "6f414509b0e95cc934da1a024a5325d9e68ef39bbcefda4b3e7fd1d024f1e205",
  "x": "a8f813f910c25c3fa95d409c3ae52c72d251db97be13be842b08bd2b7028c5fe",
  "y": "3809c1e3f261c850ce0f23f31c6653b39d07b95476c4826aa4093b51c685c956"
 },
 "85": {
  "p": "4864e96d33d3fdd9a925eee68898661789277c8db29fca121061e0043c5e829d",
  "x": "7651c7a50ea4a27a974c72c46359c4085ddca4ff889db4ac075a71f11e9be4d4",
  "y": "4109fb32982de3ec42b84c207e4d5a66e588ab376f7665afc77ce7f34e5e8dc2"
 },
 "86": {
  "p": "cff2d926911366d4a7bd115d8101c102acaa62b2fe85120f77f4f7ae62d50988",
  "x": "0b4d018769711f4e140a17e4b3619e3fc8242f9fffc579c7a7d0fbe4914e9fa3",
  "y": "0672e26bb149eb7ebe0abf827a53d9bbcbc6c77b1570afd5519f23ea0ae149ce"
 },
 "87": {
  "p": "4b70fbe78d79338afa18a2e8e658d1bddb4a9111a6bff8b1f8e7cd875fdb6a1f",
  "x": "3a9df449a3e9214b74c9473d6545974bf9030a933b1fe0e8149c3035a19b2d8f",
  "y": "e5f31eb7c05d3b499f2efde3b0420b6aa5a9b8d02a9162081f29fad8cef0891b"
 }
}

[BlackHatCoiner]

How you calculated the second key?

I calculated it with MixMAx123's program called “Secp256k1 Calculator”. See it yourself: https://github.com/MrMaxweII/Secp256k1-Calculator

You're right, I just did a childish mistake. Instead of:

Code:

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD036413F

I wrote:

Code:

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364139

(40 -> 39 instead of 40 -> 3F)


So, sorry for my idiotic interruption and thanks for your time calculating this.