p = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
n = FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140
cryptography point multiplication private key 14base point 1x = 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
y = 483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
Note : point multiplication
private key is even only point doubling operation,
If private key is odd point addition with before doubled point or initial point
then double the point ( before doubled point )
when the privatekey is less then 1 operation ended
step 1 : private key is even (14) so point addition is none
and just double base point x,y , 1 * 2 = 2
x=c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
y=1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a
and private key divided by 2 get 7 , 14 / 2 = 7
step 2 : now private key is odd (7) so process addition function. step 1 addition value plus doubled point so the addition point is 0(none) + 2 = 2
x = c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
y = 1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a
Now double the doubled point to get point 2 * 2 = 4
x = e493dbf1c10d80f3581e4904930b1404cc6c13900ee0758474fa94abe8c4cd13
y = 51ed993ea0d455b75642e2098ea51448d967ae33bfbdfe40cfe97bdc47739922
and private key divided by 2 get 3 , 7 / 2 = 3
step 3 : now private key is odd (3) so process addition function. step 2 addition value plus doubled point so the addition point is 2 + 4 = 6
x = fff97bd5755eeea420453a14355235d382f6472f8568a18b2f057a1460297556
y = ae12777aacfbb620f3be96017f45c560de80f0f6518fe4a03c870c36b075f297
Now double the doubled point to get point 4 * 2 = 8
x = 2f01e5e15cca351daff3843fb70f3c2f0a1bdd05e5af888a67784ef3e10a2a01
y = 5c4da8a741539949293d082a132d13b4c2e213d6ba5b7617b5da2cb76cbde904
and privatekey divided by 2 get 3 , 3 / 2 = 1
step 4 : now private key is odd (1) so process addition function. step 3 addition value plus doubled point so the addition point is 6 + 8 = 14
x = 499fdf9e895e719cfd64e67f07d38e3226aa7b63678949e6e49b241a60e823e4
y = cac2f6c4b54e855190f044e4a7b3d464464279c27a3f95bcc65f40d403a13f5b
Now double the doubled point to get point 8 * 2 = 16
x = e60fce93b59e9ec53011aabc21c23e97b2a31369b87a5ae9c44ee89e2a6dec0a
y = f7e3507399e595929db99f34f57937101296891e44d23f0be1f32cce69616821
and privatekey divided by 2 get 1 , 1 / 2 = 0.5
operation ended
at last the output x , y
private key 14 public keys are
x = 499fdf9e895e719cfd64e67f07d38e3226aa7b63678949e6e49b241a60e823e4
y = cac2f6c4b54e855190f044e4a7b3d464464279c27a3f95bcc65f40d403a13f5b
==================================================================
Evensame point multiplication, public key to private key here some operation
points are 8 public key ( half of all public key are even )
x = 2f01e5e15cca351daff3843fb70f3c2f0a1bdd05e5af888a67784ef3e10a2a01
y = 5c4da8a741539949293d082a132d13b4c2e213d6ba5b7617b5da2cb76cbde904
step 1 : half of 8 public keys to get 4 public keys
x = e493dbf1c10d80f3581e4904930b1404cc6c13900ee0758474fa94abe8c4cd13
y = 51ed993ea0d455b75642e2098ea51448d967ae33bfbdfe40cfe97bdc47739922
and public keys is even ( 8 ) so just double, 1 * 2 = 2
step 2 : half of 4 public keys to get 2 public keys
x = c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
y = 1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a
and public keys is even (4) so just double, 2 * 2 = 4
step 3 : half of 2 public keys to get 1 public keys
x = 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
y = 483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
and public keys is even (2) so just double, 4 * 2 = 8
step 3 : public key is 1 so end the operation
at last the output 8
so the private key 8
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Oddpoints are 14 public key ( half of all public key even and odd public key entered ),
here we need to identify which public-key is odd and which is even
x = 499fdf9e895e719cfd64e67f07d38e3226aa7b63678949e6e49b241a60e823e4
y = cac2f6c4b54e855190f044e4a7b3d464464279c27a3f95bcc65f40d403a13f5b
step 1 : half of 14 public keys to get 7 public keys
x = 5cbdf0646e5db4eaa398f365f2ea7a0e3d419b7e0330e39ce92bddedcac4f9bc
y = 6aebca40ba255960a3178d6d861a54dba813d0b813fde7b5a5082628087264da
and public keys is even (14) so just double, 1 * 2 = 2
step 2 : half of 7 public key to get 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a4 public key
x = 592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031
y = 72dd2e1d26c233337760c49122a1df67d0aa792b453f97bd29765c83b47ba01d
you can double above public key to get 7 public key x,y But we expect public key of 3 then only we get private key back,
so identify which public key is odd and which is even
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here some analyse 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a4 + 1 public key are
x = 66954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859
y = f274e4ebbf160edc0c6379a70629b18201bf8e0f52578615bd114528b2c80733
double the above public key to get 9 public key
x = acd484e2f0c7f65309ad178a9f559abde09796974c57e714c35f110dfc27ccbe
y = cc338921b0a7d9fd64380971763b61e9add888a4375f8e0f05cc262ac64f9c37
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7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a4 - 1 public keys are
x = 5699b93fc6e1bd29e09a328d657a607b4155b61a6b5fcbedd7c12df7c67df8f5
y = 3eb81178ebcdab6894d1acb3196fd8b7d5a3cc237988cd5af7cc70f98cc978a7
double the above public key to get 5 public key
x = 2f8bde4d1a07209355b4a7250a5c5128e88b84bddc619ab7cba8d569b240efe4
y = d8ac222636e5e3d6d4dba9dda6c9c426f788271bab0d6840dca87d3aa6ac62d6
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Note: n = FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140
n / 2 = 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 5 == public key == half of 9 public key
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 4 == public key == half of 7 public key
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 3 == public key == half of 5 public key
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 2 == public key == half of 3 public key
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 1 == public key == - half of n public key
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Note:7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 --> <= this private key public keys are (+ half public key)
x = 3b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63
y = 3f3979bf72ae8202983dc989aec7f2ff2ed91bdd69ce02fc0700ca100e59ddf3
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a1 --> >= this private key public keys are (- half public key)
x = 3b78ce563f89a0ed9414f5aa28ad0d96d6795f9c63
y = c0c686408d517dfd67c2367651380d00d126e4229631fd03f8ff35eef1a61e3c
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my analyse half of even public key point goes down, but odd public key half goes higher >= - half
Example: 8 public key half is 4 public key, but 7 public key half is 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 4
even public key half examplex = 2f01e5e15cca351daff3843fb70f3c2f0a1bdd05e5af888a67784ef3e10a2a01 -----> 8 public key
y = 5c4da8a741539949293d082a132d13b4c2e213d6ba5b7617b5da2cb76cbde904
8 public key half below get 4 public keyx = e493dbf1c10d80f3581e4904930b1404cc6c13900ee0758474fa94abe8c4cd13
y = 51ed993ea0d455b75642e2098ea51448d967ae33bfbdfe40cfe97bdc47739922
odd public key half examplex = 5cbdf0646e5db4eaa398f365f2ea7a0e3d419b7e0330e39ce92bddedcac4f9bc -----> 7 public key
y = 6aebca40ba255960a3178d6d861a54dba813d0b813fde7b5a5082628087264da
7 public key half below get 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 4 public keyx = 592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031
y = 72dd2e1d26c233337760c49122a1df67d0aa792b453f97bd29765c83b47ba01d
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another analyse 7th half and 9th half public key make them further half to get different answer
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 5 == public key == half of 9 public key
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 4 == public key == half of 7 public key
now divide n into 4 half n/4 get 0x3fffffffffffffffffffffffffffffffaeabb739abd2280eeff497a3340d9050L
3fffffffffffffffffffffffffffffffaeabb739abd2280eeff497a3340d9050 --> below this, public keys are 1st half public keys
3fffffffffffffffffffffffffffffffaeabb739abd2280eeff497a3340d9050 --> above this, and 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 --> below this public keys are 2nd half public keys
7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 --> above this, and bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1 --> below this public keys are 3rds half public keys
bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f1 --> above this, public keys are 4th half public keys
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7 public key half below get 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 4 public key
x = 592152c398d6c719636a03a6dad64246a5a6814aa62c156b0ce5332f6759b031
y = 72dd2e1d26c233337760c49122a1df67d0aa792b453f97bd29765c83b47ba01d
half of above 7 publickey half to get 3fffffffffffffffffffffffffffffffaeabb739abd2280eeff497a3340d9052 public key it goes 2 nd half
x = 13780453c56d0cc911a106e74dbf49e743e46396dc09e42dba4aeaff970bacef
y = 8888029b003fb198c81ecabd9f7992df7a73d33734c0a71ca4ba0f4a7f607be6
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9 public key half below get 7fffffffffffffffffffffffffffffff5d576e7357a4501ddfe92f46681b20a0 + 5 public key
x = 66954eca0543426304036fc70fc0fe3381f5195e88433bc32c5a8a60341e2859
y = f274e4ebbf160edc0c6379a70629b18201bf8e0f52578615bd114528b2c80733
half of above 9 public key half to get bfffffffffffffffffffffffffffffff0c0325ad0376782ccfddc6e99c28b0f3 public key it goes 4 th half
x = a09125a449d92c03a1c677a0a276cf157cb3fc0ad313b2526aca7d6c9808363
y = 5bd9757b686012ee83b1546139bbb4f73b2ad4e0af198cdfdf7e649266b2a8c
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Below picture to understand clear
Even public keyhttp://virtacoin.do.am/images/1.pngOdd public key 7 http://virtacoin.do.am/images/2.pngOdd public key 9 http://virtacoin.do.am/images/3.png
