Speed up secp256k1 with endomorphism

In this article, we will look  secp256k1 at the endomorphism acceleration function that helps in optimizing the validation  ECDSA for the Bitcoin cryptocurrency, but first, a little history.

12 января 2009 года Satoshi Nakamoto sent Hal Finney   in the earliest bitcoin transactions  10 BTC.

That Satoshi Nakamoto chose Hal as the first recipient of Bitcoins is not surprising. Satoshi had great respect for Hal, who established himself as one of the world’s brightest programmers and cryptographers by developing the  PGP encryption system . Hal also laid an important foundation for the reusable proof-of-work that Satoshi would use in the development of Bitcoin.

Being one of the best cryptographers in the world, Hal realized that Bitcoin was a huge breakthrough immediately after he stumbled upon it.

Back in  it, 2008 году he called Bitcoin  «a very promising idea . »

This  tweet , posted by  11 января 2009 года, is proof enough that Hal  predicted the success of Bitcoin  before many even knew what it was.

Two years have passed and  2011 году Hal Finney as a developer and Bitcoin enthusiast wrote on the  Bitcointalk forum that  the secp256k1 endomorphism function can be used to speed up ECDSA signature verification

LAMBDA and BETA are the values ​​on the secp256k1 curve, where:

secp256k1 uses the following prime number for its x and y coordinates:

p = 0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc2f

and the order of the curve:

n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141

The first step is to calculate the values ​​of  LAMBDA and  BETA, such that for any point on the curve  Q = (x, y):

LAMBDA * Q = (BETA*х mod р, у)

This is the so-called effectively computable  endomorphism , and it means that you can multiply any point on the curve  secp256k1 by this special value  very quickly by LAMBDAdoing a single multiplication  mod p.

The meaning found and published by Hal Finney:

LAMBDA = 0x5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72
BETA = 0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee

Given that we can quickly multiply, the  LAMBDA, trick is to calculate  kQ. Let’s use the calculation first . Then   you need to break it into two parts   and  , each about half the width of  , so that: Q' = lambdaQkk1k2n

k = k1 + k2*LAMBDA  mod  n

then

k*Q = (k1 + k2*LAMBDA)*Q = k1*Q + k2*LAMBDA*Q = k1*Q + k2*Q'

This last expression can be computed efficiently using the double multiplication algorithm, and since  k1 and  k2 are half the length, we get a speedup.

The missing part splits  k into  k1 and  k2. The following  4 values ​​are used for this :

а1 = 0x3086d221a7d46bcde86c90e49284eb15
b1 = -0xe4437ed6010e88286f547fa90abfe4c3
а2 = 0x114ca50f7a8e2f3f657c1108d9d44cfd8
b2 = 0x3086d221a7d46bcde86c90e49284eb15

(it’s ok that a1 = b2)

We use them as follows to divide k:

c1 = RoundToNearestInteger(b2*k/n)
c2 = RoundToNearestInteger(-b1*k/n)
k1 = k - c1a1 - c2a2
k2 = -c1b1 - c2b2

We end up with roughly  20%-е a speedup due to the halving of the number of doublings.
This gives many algorithms that can be grouped on multiple points the performance they would have with twice the number of public keys.
This speedup at an equivalent level of optimization makes it  secp256k1 the fastest curve to test out of all the commonly used curves.

We learned about the existence of endomorphism in a more detailed study of the repository from the developer and researcher Jean Luc Pons

We previously published an article:  «Pollard’s Kangaroo find solutions to the discrete logarithm of secp256k1 PRIVATE KEY + NONCES in a known range»  ,  where we used the source code to build  Kangaroo  by  Jean Luc Pons .

Based on accelerated mechanism Jean Luc Pons pointed VanitySearch

Open  main.cpp

main.cpp

In lines  255  and  256  we see that Jean Luc Pons applied the elliptic curve acceleration function  secp256k1 using  endomorphism .

  lambda.SetBase16("5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72");
  lambda2.SetBase16("ac9c52b33fa3cf1f5ad9e3fd77ed9ba4a880b9fc8ec739c2e0cfc810b51283ce");

Let’s move on to the experimental part:

As we remember from the  article,  we analyzed transactions of Bitcoin Address  14NWDXkQwcGN1Pd9fboL8npVynD5SfyJAE
from the  Bitcoin Rich List  for a total of more than  10 million US dollars , take this Bitcoin Address as an example

Open Google Colab  [TerminalGoogleColab] in the terminal  and use the repository  «07EndomorphismSecp256k1»

git clone https://github.com/demining/CryptoDeepTools.git
cd CryptoDeepTools/07EndomorphismSecp256k1/
pip3 install base58

Open code  endomorphism.py  on line  145  we use all short values ​​to speed up  secp256k1 with  endomorphism

def split_scalar_endo(k):
    n = N
    a1 = 0x3086d221a7d46bcde86c90e49284eb15
    b1 = -0xe4437ed6010e88286f547fa90abfe4c3
    a2 = 0x114ca50f7a8e2f3f657c1108d9d44cfd8
    b2 = a1
    c1 = div_nearest(b2 * k, n)
    c2 = div_nearest(-b1 * k, n)
    k1 = mod(k - c1 * a1 - c2 * a2, n)
    k2 = mod(-c1 * b1 - c2 * b2, n)
    k1neg = k1 > POW_2_128
    k2neg = k2 > POW_2_128
    if k1neg:
        k1 = n - k1
    if k2neg:
        k2 = n - k2
    return (k1neg, k1, k2neg, k2)

Copy the private key in the  HEX-format that we published in the  article

HEX:  23d4a09295be678b21a5f1dceae1f634a69c1b41775f680ebf8165266471401b

Let’s run the Python script endomorphism.py specifying the private key:

python3 endomorphism.py 23d4a09295be678b21a5f1dceae1f634a69c1b41775f680ebf8165266471401b > pubkey.txt

The result of the public key will be saved to the file: pubkey.txt

Откроем файл: pubkey.txt и проверим:
cat pubkey.txt
04ca5606a1e820e7a2f6bb3ab090e8ade7b04a7e0b5909a68dda2744ae3b8ecbfa280a47639c811134d648e8ee8096c33b41611be509ebca837fbda10baaa1eb15

Next, get the Bitcoin Address by running the Python script pubtoaddr.py

python3 pubtoaddr.py
Откроем файл: BitcoinAddress.txt и проверим:
cat BitcoinAddress.txt
14NWDXkQwcGN1Pd9fboL8npVynD5SfyJAE

ADDR: 14NWDXkQwcGN1Pd9fboL8npVynD5SfyJAE
WIF:  5J64pq77XjeacCezwmAr2V1s7snvvJkuAz8sENxw7xCkikceV6e
HEX:  23d4a09295be678b21a5f1dceae1f634a69c1b41775f680ebf8165266471401b

Checking the private key on the bitaddress website

blockchain:

www.blockchain.com/btc/address/14NWDXkQwcGN1Pd9fboL8npVynD5SfyJAE

On this topic, you can read the literature:

• Gallant, Robert P., Robert J. Lambert, and Scott A. Wanston. «Faster point multiplication on elliptic curves with efficient endomorphisms» . Annual International Conference on Cryptology, pp. 190–200. Springer, Berlin, Heidelberg, (2001)
• Hankerson, Darrell, Alfred J. Menezes, and Scott Wanston. «A Guide to Elliptic Curve Cryptography» . Computer Reviews 46, no. 1 (2005)
• Hal Finney. bitcointalk —  «Acceleration of signature verification» . (2011)  https://bitcointalk.org/index.php?topic=3238.0
• Blahut, Richard E.  «Cryptography and Secure Communication» . Cambridge University Press, (2014)

This video was created for the  CRYPTO DEEP TECH portal  to ensure the financial security of data and cryptography on elliptic curves  secp256k1 against weak signatures  ECDSA in cryptocurrency BITCOIN

Source

Telegram :  https://t.me/cryptodeeptech

Video :  https://youtu.be/DH6FyNY-Gh0

Source : https://cryptodeeptech.ru/endomorphism

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