{"id":19915488,"url":"https://github.com/demining/polynonce-attack","last_synced_at":"2025-09-19T02:33:14.144Z","repository":{"id":188477957,"uuid":"678817615","full_name":"demining/Polynonce-Attack","owner":"demining","description":"POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private 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Polynonce Attack\n\n---\n\n\n* Tutorial: https://youtu.be/7nKs_KHtyn4\n* Tutorial: https://cryptodeeptech.ru/polynonce-attack\n\n\n---\n\n\n\u003cp\u003eIn this article, we will again touch on the topic:\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeeptech.ru/blockchain-attack-vectors\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e“Bitcoin’s Critical Vulnerability”\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;and use the brand new attack of 2023\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/polynonce-attack\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003e“POLYNONCE ATTACK”\u003c/strong\u003e\u003c/a\u003e\u0026nbsp;on all three examples .\u0026nbsp;The very first mention of this attack is described in an article from\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://research.kudelskisecurity.com/2023/03/06/polynonce-a-tale-of-a-novel-ecdsa-attack-and-bitcoin-tears/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e“Kudelski Security”\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;.\u003cstrong\u003e\u003ca href=\"https://cryptodeeptech.ru/blockchain-attack-vectors\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003c/a\u003e\u003c/strong\u003e\u003ca href=\"https://cryptodeeptech.ru/polynonce-attack\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003e\u003c/strong\u003e\u003c/a\u003e\u003cstrong\u003e\u003ca href=\"https://research.kudelskisecurity.com/2023/03/06/polynonce-a-tale-of-a-novel-ecdsa-attack-and-bitcoin-tears/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-140-1024x309.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2920\" width=\"167\" height=\"51\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-141-1024x372.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2921\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-142-1024x498.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2922\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-159-1024x403.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-3043\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ca href=\"https://research.kudelskisecurity.com/2023/03/06/polynonce-a-tale-of-a-novel-ecdsa-attack-and-bitcoin-tears/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003ehttps://research.kudelskisecurity.com/2023/03/06/polynonce-a-tale-of-a-novel-ecdsa-attack-and-bitcoin-tears/\u003c/code\u003e\u003c/a\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cp\u003eAs a practical basis, we will take materials from our earlier article\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/endomorphism\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e“\u0026nbsp;\u003cstrong\u003eSpeed ​​up secp256k1 with endomorphism”\u003c/strong\u003e\u003c/a\u003e\u0026nbsp;where the values ​​​​on the secp256k1 curve from\u0026nbsp;\u003ca href=\"https://en.wikipedia.org/wiki/Hal_Finney_(computer_scientist)\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eHal Finney\u0026nbsp;\u003c/a\u003e\u0026nbsp;\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eLAMBDA and BETA\u003c/a\u003e\u0026nbsp;hide the depth of uncertainty of Bitcoin elliptic curves.\u003c/p\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eWe can reveal a lot\u003ccode\u003eBinary number (4 digits):\u0026nbsp;\u003cstrong\u003e\"1111\"\u003c/strong\u003e\u0026nbsp;// Hex number\u003cem\u003e:\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\"F\"\u003c/strong\u003e\u003cem\u003e\u0026nbsp;//\u003c/em\u003e\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-143.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2923\" width=\"856\" height=\"757\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ca href=\"https://www.rapidtables.com/convert/number/hex-to-binary.html\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003ehttps://www.rapidtables.com/convert/number/hex-to-binary.html\u003c/code\u003e\u003c/a\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eWe also know perfectly well the order of the\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeep.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003esecp256k1\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;curve which consists of\u0026nbsp;\u003cstrong\u003e128 bits\u0026nbsp;\u003c/strong\u003e\u003cem\u003eBinary number (4 digits):\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e“1111”\u0026nbsp;\u003c/strong\u003e\u003cem\u003e// Hex number:\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e“F”\u0026nbsp;\u003c/strong\u003e\u003cem\u003e//\u003c/em\u003e\u003c/p\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003en = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141\n\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111010111010101011101101110011100110101011110100100010100000001110111011111111010010010111101000110011010000001101100100000101000001\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cp\u003eWe see that the polynomial as a unit in the binary code of an arbitrarily high degree of 128 bits modulo\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-147-1024x521.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2948\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cp\u003e\u003ca href=\"https://cryptodeeptech.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003eSpeed ​​up secp256k1 with endomorphism\u003c/code\u003e\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eGiven this fact, the initial bit of the Bitcoin private key will be\u0026nbsp;\u003cem\u003eBinary number (4 digits):\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e“1111”\u0026nbsp;\u003c/strong\u003e\u003cem\u003e// Hex number:\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e“F”\u0026nbsp;\u003c/strong\u003e\u003cem\u003e//\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003eFor a theoretical basis, we will take materials:\u003c/h2\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003e\u003ca href=\"https://attacksafe.ru/polynonce-attack-on-bitcoin/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e“Polynonce Attack on Bitcoin”\u003c/a\u003e\u003c/h2\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-146.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2940\" width=\"871\" height=\"1298\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ca href=\"https://attacksafe.ru/polynonce-attack-on-bitcoin\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003ehttps://attacksafe.ru/polynonce-attack-on-bitcoin\u003c/code\u003e\u003c/a\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eConsider an example with a Bitcoin Address:\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://btc1.trezor.io/address/1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003e1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-44.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2699\"\u003e\u003c/figure\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://btc1.trezor.io/tx/929d565c386a279cf7a0382ba48cab1f72d62e7cfb3ab97b4f211d5673bc4441\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003e929d565c386a279cf7a0382ba48cab1f72d62e7cfb3ab97b4f211d5673bc4441\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-45-1024x200.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2700\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eRawTX\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-85-1024x364.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2792\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s go to the official website:\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://colab.research.google.com/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ehttps://colab.research.google.com\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eSelect the option\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e“Upload notebook”\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-50.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2709\" width=\"861\" height=\"1478\"\u003e\u003c/figure\u003e\u003c/div\u003e\u003c/blockquote\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center\"\u003e\u003cem\u003eDownload the file:\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/POLYNONCE_ATTACK.ipynb\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ePOLYNONCE_ATTACK.ipynb\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-52.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2711\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center\"\u003eLoad\u0026nbsp;\u003ccode\u003eHEX\u003c/code\u003ethe -data through the utility\u0026nbsp;\u003ccode\u003eecho\u003c/code\u003e\u0026nbsp;and save it to a file:\u0026nbsp;\u003cstrong\u003eRawTX.txt\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!echo '02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000' \u0026gt; RawTX.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-53.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2716\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center\"\u003e\u003cstrong\u003eTo implement the attack, we will use the software\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cp class=\"has-text-align-center\"\u003e\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://attacksafe.ru/software/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e“ATTACKSAFE SOFTWARE”\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003ca href=\"https://attacksafe.ru/software/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-14.png\" alt=\"Implement Frey-Rück Attack to get the secret key \u0026quot;K\u0026quot; (NONCE)\" class=\"wp-image-705\"\u003e\u003c/a\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003cstrong\u003e\u003ccode\u003ewww.attacksafe.ru/software\u003c/code\u003e\u003c/strong\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eAccess rights:\u003c/h2\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!chmod +x attacksafe\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003els\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-54.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2717\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eApplication:\u003c/h2\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -help\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-57-1024x784.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2720\"\u003e\u003c/figure\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e  -version:  software version \n  -list:     list of bitcoin attacks\n  -tool:     indicate the attack\n  -gpu:      enable gpu\n  -time:     work timeout\n  -server:   server mode\n  -port:     server port\n  -open:     open file\n  -save:     save file\n  -search:   vulnerability search\n  -stop:     stop at mode\n  -max:      maximum quantity in mode\n  -min:      minimum quantity per mode\n  -speed:    boost speed for mode\n  -range:    specific range\n  -crack:    crack mode\n  -field:    starting field\n  -point:    starting point\n  -inject:   injection regimen\n  -decode:   decoding mode\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -version\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-60.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2724\"\u003e\u003c/figure\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003eVersion 5.3.3. [ATTACKSAFE SOFTWARE, © 2023]\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003ccode\u003e\"ATTACKSAFE SOFTWARE\"\u003c/code\u003e\u0026nbsp;includes all popular attacks on Bitcoin.\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLet’s run a list of all attacks:\u003c/h2\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -list\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-62-1024x758.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2728\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eLet’s choose\u003ccode\u003e\u0026nbsp;-tool: polynonce_attack\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo get a specific\u0026nbsp;\u003ccode\u003eHEX\u003c/code\u003evalue\u0026nbsp;\u003ccode\u003eR,S,Z\u003c/code\u003efor the signature\u0026nbsp;\u003ccode\u003eECDSA\u003c/code\u003e, we previously added data\u0026nbsp;\u0026nbsp;\u003ccode\u003eRawTX\u003c/code\u003e\u0026nbsp;through the utility\u0026nbsp;\u003ccode\u003eecho\u003c/code\u003eto a text document and saved it as a file\u0026nbsp;\u003ccode\u003eRawTX.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e02000000019e3de154f8b473a796b9e39dd279dff1d907a4d27a1d8b23a055f97b08ad4c6e310000006b483045022100b29bdfc27ddf6bebd0e77c84b31dc1bc64b5b2276c8d4147421e96ef85467e8d02204ddd8ff0ffa19658e3b417be5f64d9c425a4d9fcd76238b8538c1d605b229baf0121027b06fe78e39ced37586c42c9ac38d7b2d88ccdd4cd1bb38816c0933f9b8db695ffffffff0169020000000000001600145fc8e854994406f93ea5c7f3abccc5d319ae2a3100000000\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLaunch\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;using software\u0026nbsp;\u003ccode\u003e“ATTACKSAFE SOFTWARE”\u003c/code\u003e\u003c/h2\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-64-1024x182.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2734\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eWe launched this attack from\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;and the result was saved to a file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003eNow to see the successful result, open the file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-65-1024x179.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2736\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center\"\u003eIn order to calculate the private key to a Bitcoin Wallet from a file,\u0026nbsp;\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003ewe will install\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeeptech.ru/install-sagemath-in-google-colab/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eSageMath\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-27.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2188\" width=\"864\" height=\"245\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-28-1024x445.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2189\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp class=\"has-text-align-center\"\u003eEarlier we published\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/install-sagemath-in-google-colab/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ean article\u003c/a\u003e\u0026nbsp;, download\u0026nbsp;\u0026nbsp;\u003ccode\u003etar-file\u003c/code\u003e:\u0026nbsp;\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/sage-9.3-Ubuntu_20.04-x86_64.tar.bz2\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003esage-9.3-Ubuntu_20.04-x86_64.tar.bz2\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!wget https://cryptodeeptech.ru/sage-9.3-Ubuntu_20.04-x86_64.tar.bz2\n!tar -xf sage-9.3-Ubuntu_20.04-x86_64.tar.bz2\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-67-1024x319.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2744\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eLet’s go through the directory:\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ecd SageMath/\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003els\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-68-1024x723.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2746\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eRun\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003erelocate-once.py\u0026nbsp;\u003c/strong\u003e\u003cem\u003e\u0026nbsp;with the command:\u003c/em\u003e\u003ccode\u003ePython-script:\u003c/code\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003c/strong\u003e\u003cem\u003e\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!python3 relocate-once.py\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-69-1024x457.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2750\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eMove\u0026nbsp;\u003ccode\u003e\"AttackSafe\"\u003c/code\u003eto\u0026nbsp;\u003ccode\u003e\"SignatureRSZ.csv\"\u003c/code\u003efolder\u003ccode\u003e\"SageMath\"\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!mv '/content/attacksafe' '/content/SageMath/attacksafe'\n!mv '/content/SignatureRSZ.csv' '/content/SageMath/SignatureRSZ.csv'\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-70-1024x698.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2753\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003els\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-156-1024x344.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-3024\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eDownload the script\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ecrack_weak_ECDSA_nonces_with_LLL.py\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;from Dario Clavijo through the utility\u0026nbsp;\u003ccode\u003ewget\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!wget https://raw.githubusercontent.com/demining/CryptoDeepTools/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-72-1024x424.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2756\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eNow let’s run\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eSageMath\u003c/code\u003e\u003cem\u003e\u0026nbsp;the command:\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./sage -sh\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-74-1024x505.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2760\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo calculate the private key to the Bitcoin Wallet, run the script\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ecrack_weak_ECDSA_nonces_with_LLL.py\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;specifying the parameters\u003cstrong\u003e\u003ccode\u003e128 bits 4 sign\u003c/code\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003epython3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 \u0026gt; PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ecat PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-75-1024x442.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2766\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u003ccode\u003ePrivateKey.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003cem\u003eWe received the private key to the Bitcoin Wallet in\u0026nbsp;\u003ccode\u003eHEX\u003c/code\u003ethe format\u003c/em\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-78.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2769\" width=\"912\" height=\"207\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e\u003cstrong\u003ePrivKey = 0xf0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d\u003c/strong\u003e\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eCheck POLYNONCE for each ECDSA signature\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo do this, use the code from\u0026nbsp;\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example1/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003eGITHUB\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-150.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2972\" width=\"917\" height=\"1234\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example1/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003ehttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example1/POLYNONCE.py\u003c/code\u003e\u003c/a\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eResult:\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-149.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2963\"\u003e\u003c/figure\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eWe received four identical initial\u003ccode\u003e128 bits\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ePOLYNONCE \u0026gt;\u0026gt; 93e43392cb31d5d1f75175ee64ce16b7 efc86216627af576c29c9c52a0fd10fe\nPOLYNONCE \u0026gt;\u0026gt; 93e43392cb31d5d1f75175ee64ce16b7 f88ff4c8a9ea4b61b1e087d0c0988826\nPOLYNONCE \u0026gt;\u0026gt; 93e43392cb31d5d1f75175ee64ce16b7 6849e83cd03d103bcc37aca8323c8d2f\nPOLYNONCE \u0026gt;\u0026gt; 93e43392cb31d5d1f75175ee64ce16b7 efc86216627af576c29c9c52a0fd10fe\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eThanks to the value on the secp256k1 curve from\u0026nbsp;\u0026nbsp;\u003ca href=\"https://en.wikipedia.org/wiki/Hal_Finney_(computer_scientist)\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eHal Finney\u0026nbsp;\u003c/a\u003e\u0026nbsp;\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eLAMBDA and BETA\u003c/a\u003e\u0026nbsp;revealed to us the same initial bits\u0026nbsp;\u003ccode\u003e128 bits\u003c/code\u003e, since the initial bits of the private key to the Bitcoin Wallet begin with\u003cem\u003e\u003ccode\u003eBinary number (4 digits):\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"1111\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;// Hex number:\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"F\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;//\u003c/code\u003e\u003c/em\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLet’s check the HEX of the private key:\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eInstall the module\u003ccode\u003ebitcoin\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!pip3 install bitcoin\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-86-1024x219.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2799\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cstrong\u003eLet’s run the code:\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003efrom bitcoin import *\n\nwith open(\"PrivateKey.txt\",\"r\") as f:\n    content = f.readlines()\n\ncontent = [x.strip() for x in content]\nf.close()\n\n\noutfile = open(\"PrivateKeyAddr.txt\",\"w\")\nfor x in content:\n  outfile.write(x+\":\"+pubtoaddr(encode_pubkey(privtopub(x), \"bin_compressed\"))+\"\\n\")\n \noutfile.close()\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-87.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2800\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u003ccode\u003ePrivateKeyAddr.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-88.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2801\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ccode\u003ef0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d:1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\u003c/code\u003e\u003c/figcaption\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eLet’s open\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeep.ru/bitaddress.html\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ebitaddress\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;\u0026nbsp;and check:\u003c/p\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003eADDR: 1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\nWIF:  L5HV2GiosXifcmijGCpFWdYiMRuXh4x4JVK29urGjfAWyasBYoDX\nHEX:  f0a3e31646ce147bbd79bb6e45e6e9c8c4e51c535918c9b4cdca9528eb62172d\u003c/code\u003e\u003c/pre\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-bitaddress.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2779\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://www.blockchain.com/en/explorer/addresses/btc/1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\"\u003ehttps://www.blockchain.com/en/explorer/addresses/btc/1DxzwX4qC9PsWDSAzuWbJRzEwdGx3n9CJB\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-106.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2853\" width=\"873\" height=\"253\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-110.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2859\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-113-1024x217.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2864\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cp class=\"has-large-font-size\"\u003e\u003ccode\u003e\u003cstrong\u003eBALANCE: $ 3699.40\u003c/strong\u003e\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003eLet’s look at other examples:\u003c/h2\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center has-large-font-size\"\u003e\u003ccode\u003e\u003cstrong\u003e№2\u003c/strong\u003e\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eConsider example #2 with a Bitcoin Address:\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://btc1.trezor.io/address/137a6fqt13bhtAkGZWrgcGM98NLCotszR2\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e137a6fqt13bhtAkGZWrgcGM98NLCotszR2\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-82.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2788\"\u003e\u003c/figure\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://btc1.trezor.io/tx/c1da9d117e15883ba41539f558ac870f53865ea00f68a8ff8bc7e8a9ee67099b\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003ec1da9d117e15883ba41539f558ac870f53865ea00f68a8ff8bc7e8a9ee67099b\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-83-1024x193.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2789\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eRawTX\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-84-1024x363.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2790\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e010000000103ebc5c4b817124d45ad15e398ec32e9b9b7549c1fc10300ecbf36648c3cb5d42c0000006a47304402204e97dae0ab6e4eee9529f68687907c05db9037d9fbdba78dd01a3338a48d95b602207794cb7aa308243dfbdd5c20225777cd6e01bd7c4f76bf36948aa29290129c2b0121036360352efcff6a823eabb25578a29392eab4d302955fd54ece900578d2ab83b8ffffffff0162020000000000001976a914154813f71552c59487efa3b16d62bfb009dc5f1e88ac00000000\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s remove the files from the first example:\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!rm RawTX.txt\n!rm NoncesHEX.txt\n!rm PrivateKey.txt\n!rm SignatureRSZ.csv\n!rm PrivateKeyAddr.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-157.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-3027\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eLoad\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eHEX\u003c/code\u003e\u003cem\u003e the -data through the utility\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eecho\u003c/code\u003e\u003cem\u003e\u0026nbsp;and save it to a file:\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003eRawTX.txt\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!echo '010000000103ebc5c4b817124d45ad15e398ec32e9b9b7549c1fc10300ecbf36648c3cb5d42c0000006a47304402204e97dae0ab6e4eee9529f68687907c05db9037d9fbdba78dd01a3338a48d95b602207794cb7aa308243dfbdd5c20225777cd6e01bd7c4f76bf36948aa29290129c2b0121036360352efcff6a823eabb25578a29392eab4d302955fd54ece900578d2ab83b8ffffffff0162020000000000001976a914154813f71552c59487efa3b16d62bfb009dc5f1e88ac00000000' \u0026gt; RawTX.txt\n\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-94-1024x393.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2820\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003eLaunch\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;using software\u0026nbsp;\u003ccode\u003e“ATTACKSAFE SOFTWARE”\u003c/code\u003e\u003c/h2\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-95-1024x248.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2825\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eWe launched this attack from\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;and the result was saved to a file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003eNow to see the successful result, open the file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-96-1024x177.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2830\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eLet’s run\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eSageMath\u003c/code\u003e\u003cem\u003e\u0026nbsp;the command:\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./sage -sh\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-98-1024x494.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2837\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo calculate the private key to the Bitcoin Wallet, run the script\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ecrack_weak_ECDSA_nonces_with_LLL.py\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;\u0026nbsp;specifying the parameters\u0026nbsp;\u003cstrong\u003e\u003ccode\u003e128 bits 4 sign\u003c/code\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003epython3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 \u0026gt; PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ecat PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-99-1024x530.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2841\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u0026nbsp;\u003ccode\u003ePrivateKey.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003cem\u003eWe received the private key to the Bitcoin Wallet in\u0026nbsp;\u0026nbsp;\u003ccode\u003eHEX\u003c/code\u003e\u0026nbsp;the format\u003c/em\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-102.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2847\" width=\"907\" height=\"312\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e\u003cstrong\u003ePrivKey = 0xff0178fa717374f7e74d43f00150748967ea04b64241ec10a10f62debb70868c\u003c/strong\u003e\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eCheck POLYNONCE for each ECDSA signature\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo do this, use the code from\u0026nbsp;\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example2/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003eGITHUB\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-152.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2991\" width=\"916\" height=\"1250\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ccode\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example2/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ehttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example2/POLYNONCE.py\u003c/a\u003e\u003c/code\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eResult:\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-153.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2993\"\u003e\u003c/figure\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eWe received four identical initial\u003ccode\u003e128 bits\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ePOLYNONCE \u0026gt;\u0026gt; 5220dae0c281e1115b4dd69ea3500f70 c5f6da6334586ed2bdc88a05f37bcf95\nPOLYNONCE \u0026gt;\u0026gt; 5220dae0c281e1115b4dd69ea3500f70 6f82fbd847c138ab48e778135e908149\nPOLYNONCE \u0026gt;\u0026gt; 5220dae0c281e1115b4dd69ea3500f70 5541022f8aeac81e5ce62e018d1cd722\nPOLYNONCE \u0026gt;\u0026gt; 5220dae0c281e1115b4dd69ea3500f70 80e88efaff419ecd84d7ded17dc548a7\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eThanks to the value on the secp256k1 curve from\u0026nbsp;\u0026nbsp;\u003ca href=\"https://en.wikipedia.org/wiki/Hal_Finney_(computer_scientist)\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eHal Finney\u0026nbsp;\u003c/a\u003e\u0026nbsp;\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eLAMBDA and BETA\u003c/a\u003e\u0026nbsp;revealed to us the same initial bits\u0026nbsp;\u003ccode\u003e128 bits\u003c/code\u003e, since the initial bits of the private key to the Bitcoin Wallet begin with\u003cem\u003e\u003ccode\u003eBinary number (4 digits):\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"1111\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;// Hex number:\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"F\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;//\u003c/code\u003e\u003c/em\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLet’s check the HEX of the private key:\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cstrong\u003eLet’s run the code:\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003efrom bitcoin import *\n\nwith open(\"PrivateKey.txt\",\"r\") as f:\n    content = f.readlines()\n\ncontent = [x.strip() for x in content]\nf.close()\n\n\noutfile = open(\"PrivateKeyAddr.txt\",\"w\")\nfor x in content:\n  outfile.write(x+\":\"+pubtoaddr(encode_pubkey(privtopub(x), \"bin_compressed\"))+\"\\n\")\n \noutfile.close()\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-103-1024x451.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2848\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u0026nbsp;\u003ccode\u003ePrivateKeyAddr.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-104.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2850\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-large-font-size\"\u003eLet’s open\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeep.ru/bitaddress.html\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ebitaddress\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;\u0026nbsp;and check:\u003c/p\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003eADDR: 137a6fqt13bhtAkGZWrgcGM98NLCotszR2\nWIF:  L5mQfFuzR3rzLtneJ7Tcv64JrHjCpK64UN4JRdGDxCUTbQ8NfHxo\nHEX:  ff0178fa717374f7e74d43f00150748967ea04b64241ec10a10f62debb70868c\u003c/code\u003e\u003c/pre\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-bitaddress-2.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2877\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://www.blockchain.com/en/explorer/addresses/btc/137a6fqt13bhtAkGZWrgcGM98NLCotszR2\"\u003ehttps://www.blockchain.com/en/explorer/addresses/btc/137a6fqt13bhtAkGZWrgcGM98NLCotszR2\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-114.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2867\" width=\"838\" height=\"267\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-116.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2870\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-117-1024x199.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2871\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cp class=\"has-large-font-size\"\u003e\u003ccode\u003e\u003cstrong\u003eBALANCE: $ 1133.73\u003c/strong\u003e\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003eLet’s look at other examples:\u003c/h2\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-text-align-center has-large-font-size\"\u003e\u003ccode\u003e\u003cstrong\u003e№3\u003c/strong\u003e\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eConsider example #3 with a Bitcoin Address:\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://btc1.trezor.io/address/1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-119.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2883\"\u003e\u003c/figure\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://btc1.trezor.io/tx/fa80af660fc444d87853137506df02e5c75e8c2bf75dc44589b60356867a6d98\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003cstrong\u003efa80af660fc444d87853137506df02e5c75e8c2bf75dc44589b60356867a6d98\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-120-1024x193.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2884\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eRawTX\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-121-1024x353.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2885\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e01000000016eb80d35b08164302e49f88d8f86bf2827a91a5650149be38f4f73751ff41437060000006a473044022043d4c025a0f3be366a0d768c721b9b9191e0c3db6f2c6bfe34e8fb24af7f379102205a4fe2cc6944e00309c35619ff1242301b84d4728b863f97326f56dbd7a782220121027ccccf5f56ed78c2a761721ff3da0f76b792fbe4eae2ac73e7b4651bc3ef19cdffffffff01c057010000000000232103bec42e5d718b0e5b3853243c9bcf00dd671a335b0eb99fd8ca32f8d5784a9476ac00000000\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s remove the files from the second example:\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!rm RawTX.txt\n!rm NoncesHEX.txt\n!rm PrivateKey.txt\n!rm SignatureRSZ.csv\n!rm PrivateKeyAddr.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-158.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-3030\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eLoad\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eHEX\u003c/code\u003e\u003cem\u003e the -data through the utility\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eecho\u003c/code\u003e\u003cem\u003e\u0026nbsp;and save it to a file:\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003eRawTX.txt\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!echo '01000000016eb80d35b08164302e49f88d8f86bf2827a91a5650149be38f4f73751ff41437060000006a473044022043d4c025a0f3be366a0d768c721b9b9191e0c3db6f2c6bfe34e8fb24af7f379102205a4fe2cc6944e00309c35619ff1242301b84d4728b863f97326f56dbd7a782220121027ccccf5f56ed78c2a761721ff3da0f76b792fbe4eae2ac73e7b4651bc3ef19cdffffffff01c057010000000000232103bec42e5d718b0e5b3853243c9bcf00dd671a335b0eb99fd8ca32f8d5784a9476ac00000000' \u0026gt; RawTX.txt\n\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-124-1024x468.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2889\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading has-text-align-center\"\u003eLaunch\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;using software\u0026nbsp;\u003ccode\u003e“ATTACKSAFE SOFTWARE”\u003c/code\u003e\u003c/h2\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./attacksafe -tool polynonce_attack -open RawTX.txt -save SignatureRSZ.csv\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-125-1024x252.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2890\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eWe launched this attack from\u0026nbsp;\u0026nbsp;\u003ccode\u003e-tool polynonce_attack\u003c/code\u003e\u0026nbsp;and the result was saved to a file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003eNow to see the successful result, open the file\u0026nbsp;\u003ccode\u003eSignatureRSZ.csv\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-127-1024x184.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2892\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cem\u003eLet’s run\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003ccode\u003eSageMath\u003c/code\u003e\u003cem\u003e\u0026nbsp;the command:\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e!./sage -sh\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-98-1024x494.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2837\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo calculate the private key to the Bitcoin Wallet, run the script\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/crack_weak_ECDSA_nonces_with_LLL.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ecrack_weak_ECDSA_nonces_with_LLL.py\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;\u0026nbsp;specifying the parameters\u0026nbsp;\u003cstrong\u003e\u003ccode\u003e128 bits 4 sign\u003c/code\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003epython3 crack_weak_ECDSA_nonces_with_LLL.py SignatureRSZ.csv 128 4 \u0026gt; PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ecat PrivateKey.txt\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-131-1024x530.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2896\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u0026nbsp;\u003ccode\u003ePrivateKey.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cp\u003e\u003cem\u003eWe received the private key to the Bitcoin Wallet in\u0026nbsp;\u0026nbsp;\u003ccode\u003eHEX\u003c/code\u003e\u0026nbsp;the format\u003c/em\u003e\u003c/p\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-132.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2898\" width=\"922\" height=\"342\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003e\u003cstrong\u003ePrivKey = 0xfbc50a7158b3d9fd7fd58fe0874f20c10c650975dc118163debf442a44203fdf\u003c/strong\u003e\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eCheck POLYNONCE for each ECDSA signature\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eTo do this, use the code from\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example3/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eGITHUB\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-155.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2998\" width=\"918\" height=\"1219\"\u003e\u003cfigcaption class=\"wp-element-caption\"\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example3/POLYNONCE.py\" target=\"_blank\" rel=\"noreferrer noopener\"\u003e\u003ccode\u003ehttps://github.com/demining/CryptoDeepTools/blob/main/20PolynonceAttack/example3/POLYNONCE.py\u003c/code\u003e\u003c/a\u003e\u003c/figcaption\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eResult:\u003c/h2\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-154.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2997\"\u003e\u003c/figure\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eWe received four identical initial\u003ccode\u003e128 bits\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003ePOLYNONCE \u0026gt;\u0026gt; d7460c5b1a98f6d0443ae1cfe1f17814 fbc50a7158b3d9fd7fd58fe0874f20c1\nPOLYNONCE \u0026gt;\u0026gt; d7460c5b1a98f6d0443ae1cfe1f17814 d4de8d539655ecf0d50fd32187c3c467\nPOLYNONCE \u0026gt;\u0026gt; d7460c5b1a98f6d0443ae1cfe1f17814 6726aea1a6fd64d82dc657670352de72\nPOLYNONCE \u0026gt;\u0026gt; d7460c5b1a98f6d0443ae1cfe1f17814 89df16fd387156b39adca9a92464de18\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003eThanks to the value on the secp256k1 curve from\u0026nbsp;\u0026nbsp;\u003ca href=\"https://en.wikipedia.org/wiki/Hal_Finney_(computer_scientist)\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eHal Finney\u0026nbsp;\u003c/a\u003e\u0026nbsp;\u0026nbsp;\u003ca href=\"https://cryptodeeptech.ru/endomorphism/\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eLAMBDA and BETA\u003c/a\u003e\u0026nbsp;revealed to us the same initial bits\u0026nbsp;\u003ccode\u003e128 bits\u003c/code\u003e, since the initial bits of the private key to the Bitcoin Wallet begin with\u003cem\u003e\u003ccode\u003eBinary number (4 digits):\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"1111\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;// Hex number:\u0026nbsp;\u003c/code\u003e\u003c/em\u003e\u003cstrong\u003e\u003ccode\u003e\"F\"\u003c/code\u003e\u003c/strong\u003e\u003cem\u003e\u003ccode\u003e\u0026nbsp;//\u003c/code\u003e\u003c/em\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLet’s check the HEX of the private key:\u003c/h2\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003e\u003cstrong\u003eLet’s run the code:\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003efrom bitcoin import *\n\nwith open(\"PrivateKey.txt\",\"r\") as f:\n    content = f.readlines()\n\ncontent = [x.strip() for x in content]\nf.close()\n\n\noutfile = open(\"PrivateKeyAddr.txt\",\"w\")\nfor x in content:\n  outfile.write(x+\":\"+pubtoaddr(encode_pubkey(privtopub(x), \"bin_compressed\"))+\"\\n\")\n \noutfile.close()\u003c/code\u003e\u003c/pre\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-103-1024x451.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2848\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cblockquote class=\"wp-block-quote\"\u003e\n\u003cp\u003eLet’s open the file:\u0026nbsp;\u003ccode\u003ePrivateKeyAddr.txt\u003c/code\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\n\n\n\n\u003cfigure class=\"wp-block-image\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-133.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2900\"\u003e\u003c/figure\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp class=\"has-large-font-size\"\u003eLet’s open\u0026nbsp;\u0026nbsp;\u003cstrong\u003e\u003ca href=\"https://cryptodeep.ru/bitaddress.html\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ebitaddress\u003c/a\u003e\u003c/strong\u003e\u0026nbsp;\u0026nbsp;and check:\u003c/p\u003e\n\n\n\n\u003cpre class=\"wp-block-code\"\u003e\u003ccode\u003eADDR: 1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt\nWIF:  L5f7p5bReuXLm3d7rFkpPyGQ1GNpiGuj8QuQ6rNCKXC9bs3J9GEY\nHEX:  fbc50a7158b3d9fd7fd58fe0874f20c10c650975dc118163debf442a44203fdf\u003c/code\u003e\u003c/pre\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-bitaddress-3.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2906\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003e\u003ca href=\"https://www.blockchain.com/en/explorer/addresses/btc/1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt\"\u003ehttps://www.blockchain.com/en/explorer/addresses/btc/1HxrEeC2X8UEcSvsemPJtTqrnbAetGWYUt\u003c/a\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter is-resized\"\u003e\u003cimg decoding=\"async\" loading=\"lazy\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-137.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2909\" width=\"846\" height=\"258\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-138.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2910\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/image-139-1024x224.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2911\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003cp class=\"has-large-font-size\"\u003e\u003ccode\u003e\u003cstrong\u003eBALANCE: $ 459.24\u003c/strong\u003e\u003c/code\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003ch2 class=\"wp-block-heading\"\u003eLiterature:\u003c/h2\u003e\n\n\n\n\u003cul\u003e\n\u003cli\u003e\u003cem\u003eA Novel Related Nonce Attack for ECDSA, \u003ca href=\"https://www.researchgate.net/profile/Marco-Macchetti\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eMarco Macchetti\u003c/a\u003e \u003cstrong\u003e[Kudelski Security, Switzerland] (2023)\u003c/strong\u003e\u003c/em\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\n\n\n\u003cul\u003e\n\u003cli\u003e\u003cem\u003eGallant, Robert P., Robert J. Lambert, and Scott A. Wanston.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003cem\u003e“Faster point multiplication on elliptic curves with efficient endomorphisms”\u003c/em\u003e\u003c/strong\u003e\u003cem\u003e\u0026nbsp;.\u0026nbsp;Annual International Conference on Cryptology, pp. 190–200.\u0026nbsp;Springer, Berlin, Heidelberg, (2001)\u003c/em\u003e\u003c/li\u003e\n\n\n\n\u003cli\u003e\u003cem\u003eHankerson, Darrell, Alfred J. Menezes, and Scott Wanston.\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003cem\u003e“A Guide to Elliptic Curve Cryptography”\u003c/em\u003e\u003c/strong\u003e\u003cem\u003e\u0026nbsp;.\u0026nbsp;Computer Reviews 46, no.\u0026nbsp;1 (2005)\u003c/em\u003e\u003c/li\u003e\n\n\n\n\u003cli\u003e\u003cem\u003eHal Finney.\u0026nbsp;bitcointalk –\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003cem\u003e“Acceleration of signature verification”\u003c/em\u003e\u003c/strong\u003e\u003cem\u003e\u0026nbsp;.\u0026nbsp;(2011)\u0026nbsp;\u003c/em\u003e\u0026nbsp;\u003ca href=\"https://bitcointalk.org/index.php?topic=3238.0\" target=\"_blank\" rel=\"noreferrer noopener\"\u003ehttps://bitcointalk.org/index.php?topic=3238.0\u003c/a\u003e\u003c/li\u003e\n\n\n\n\u003cli\u003e\u003cem\u003eBlahut, Richard E.\u0026nbsp;\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e\u003cem\u003e“Cryptography and Secure Communication”\u003c/em\u003e\u003c/strong\u003e\u003cem\u003e\u0026nbsp;.\u0026nbsp;Cambridge University Press, (2014)\u003c/em\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://github.com/demining/CryptoDeepTools/tree/main/20PolynonceAttack\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eSource\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://attacksafe.ru/software\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eATTACKSAFE SOFTWARE\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://t.me/cryptodeeptech\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eTelegram: https://t.me/cryptodeeptech\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://youtu.be/7nKs_KHtyn4\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eVideo: https://youtu.be/7nKs_KHtyn4\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\"https://cryptodeeptech.ru/polynonce-attack\" target=\"_blank\" rel=\"noreferrer noopener\"\u003eSource: https://cryptodeeptech.ru/polynonce-attack\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\n\n\u003cdiv class=\"wp-block-image\"\u003e\n\u003cfigure class=\"aligncenter\"\u003e\u003cimg decoding=\"async\" src=\"./POLYNONCE ATTACK we use BITCOIN signatures as a Polynomial to an arbitrarily high power of 128 bits to get a Private Key - CRYPTO DEEP TECH_files/038-1024x576.png\" alt=\"POLYNONCE ATTACK use BITCOIN signatures as a polynomial to an arbitrarily high power of 128 bits to obtain a private key\" class=\"wp-image-2915\"\u003e\u003c/figure\u003e\u003c/div\u003e\n\n\n\u003chr class=\"wp-block-separator has-alpha-channel-opacity\"\u003e\n\t\u003c/div\u003e\u003c!-- .entry-content --\u003e\n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdemining%2Fpolynonce-attack","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fdemining%2Fpolynonce-attack","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdemining%2Fpolynonce-attack/lists"}