Colin Faife

3 years ago

The brand-new USB Rubber Ducky is much riskier than before.

The brand-new USB Rubber Ducky is much riskier than before.

With its own programming language, the well-liked hacking tool may now pwn you.

With a vengeance, the USB Rubber Ducky is back.

This year's Def Con hacking conference saw the release of a new version of the well-liked hacking tool, and its author, Darren Kitchen, was on hand to explain it. We put a few of the new features to the test and discovered that the most recent version is riskier than ever.

WHAT IS IT?

The USB Rubber Ducky seems to the untrained eye to be an ordinary USB flash drive. However, when you connect it to a computer, the computer recognizes it as a USB keyboard and will accept keystroke commands from the device exactly like a person would type them in.

Kitchen explained to me, "It takes use of the trust model built in, where computers have been taught to trust a human, in that anything it types is trusted to the same degree as the user is trusted. And a computer is aware that clicks and keystrokes are how people generally connect with it.

The USB Rubber Ducky, a brainchild of Darren Kitchen Corin

Over ten years ago, the first Rubber Ducky was published, quickly becoming a hacker favorite (it was even featured in a Mr. Robot scene). Since then, there have been a number of small upgrades, but the most recent Rubber Ducky takes a giant step ahead with a number of new features that significantly increase its flexibility and capability.

WHERE IS ITS USE?

The options are nearly unlimited with the proper strategy.

The Rubber Ducky has already been used to launch attacks including making a phony Windows pop-up window to collect a user's login information or tricking Chrome into sending all saved passwords to an attacker's web server. However, these attacks lacked the adaptability to operate across platforms and had to be specifically designed for particular operating systems and software versions.

The nuances of DuckyScript 3.0 are described in a new manual.

The most recent Rubber Ducky seeks to get around these restrictions. The DuckyScript programming language, which is used to construct the commands that the Rubber Ducky will enter into a target machine, receives a significant improvement with it. DuckyScript 3.0 is a feature-rich language that allows users to write functions, store variables, and apply logic flow controls, in contrast to earlier versions that were primarily limited to scripting keystroke sequences (i.e., if this... then that).

This implies that, for instance, the new Ducky can check to see if it is hooked into a Windows or Mac computer and then conditionally run code specific to each one, or it can disable itself if it has been attached to the incorrect target. In order to provide a more human effect, it can also generate pseudorandom numbers and utilize them to add a configurable delay between keystrokes.

The ability to steal data from a target computer by encoding it in binary code and transferring it through the signals intended to instruct a keyboard when the CapsLock or NumLock LEDs should light up is perhaps its most astounding feature. By using this technique, a hacker may plug it in for a brief period of time, excuse themselves by saying, "Sorry, I think that USB drive is faulty," and then take it away with all the credentials stored on it.

HOW SERIOUS IS THE RISK?

In other words, it may be a significant one, but because physical device access is required, the majority of people aren't at risk of being a target.

The 500 or so new Rubber Duckies that Hak5 brought to Def Con, according to Kitchen, were his company's most popular item at the convention, and they were all gone on the first day. It's safe to suppose that hundreds of hackers already possess one, and demand is likely to persist for some time.

Additionally, it has an online development toolkit that can be used to create attack payloads, compile them, and then load them onto the target device. A "payload hub" part of the website makes it simple for hackers to share what they've generated, and the Hak5 Discord is also busy with conversation and helpful advice. This makes it simple for users of the product to connect with a larger community.

It's too expensive for most individuals to distribute in volume, so unless your favorite cafe is renowned for being a hangout among vulnerable targets, it's doubtful that someone will leave a few of them there. To that end, if you intend to plug in a USB device that you discovered outside in a public area, pause to consider your decision.

WOULD IT WORK FOR ME?

Although the device is quite straightforward to use, there are a few things that could cause you trouble if you have no prior expertise writing or debugging code. For a while, during testing on a Mac, I was unable to get the Ducky to press the F4 key to activate the launchpad, but after forcing it to identify itself using an alternative Apple keyboard device ID, the problem was resolved.

From there, I was able to create a script that, when the Ducky was plugged in, would instantly run Chrome, open a new browser tab, and then immediately close it once more without requiring any action from the laptop user. Not bad for only a few hours of testing, and something that could be readily changed to perform duties other than reading technology news.

More on Technology

CyberPunkMetalHead

2 years ago

I made ChatGPT copy itself.

OpenAI's ChatGPT is an amazing AI-powered chatbot. It can write poetry, stories, and marketing campaigns, and it's also adept at coding. ChatGPT's coding is so amazing that it replicated itself. What happened:

Me: Hi

Hello! Can I help? I'll answer any inquiries you have.

Me: Build a python and flask AI chatbot that responds to user messages on the frontend.

CGPT:

from flask import Flask, request, jsonify

import openai
openai.api_key = "YOUR_API_KEY"

@app.route('/chatbot', methods=['POST'])
def chatbot():
    user_message = request.get_json()['message']
    response = chatbot_response(user_message)
    return jsonify({'response': response})


def chatbot_response(user_message):
    response = openai.Completion.create(
        engine="text-davinci-002",
        prompt=user_message,
        max_tokens=1024,
        n = 1,
        temperature=0.5
    )
    return response['choices'][0]['text']

if __name__ == '__main__':
    app.run()

To integrate a chatbot with a frontend, use Flask to serve an HTML page with a form for user messages and the chatbot's answer. AJAX can submit the form to the /chatbot route, and JavaScript can show the chatbot's response.

I imported code snippets at the top of the file.

Let's add a frontend interface, I said.

CGPT:

<!DOCTYPE html>
<html>
<head>
    <title>AI Chatbot</title>
</head>
<body>
    <h1>AI Chatbot</h1>
    <form id="chatbot-form">
        <label for="message">Message:</label><br>
        <input type="text" id="message" name="message"><br>
        <input type="submit" value="Submit">
    </form> 
    <div id="chatbot-response"></div>

    <script src="https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js"></script>
    <script>
        $(document).ready(function() {
            $("#chatbot-form").submit(function(e) {
                e.preventDefault();
                var userMessage = $("#message").val();
                $.ajax({
                    type: "POST",
                    url: "/chatbot",
                    data: JSON.stringify({message: userMessage}),
                    contentType: "application/json; charset=utf-8",
                    dataType: "json",
                    success: function(response) {
                        $("#chatbot-response").text(response.response);
                    }
                });
            });
        });
    </script>
</body>
</html>

To serve this HTML page using Flask, build a new route that returns the page as a GET response.

So far, so good.

Looked quite basic so I got chatGPT to add some tailwind to the project.

Me: Tailwind the chatbot.html page, please. Give it a futuristic style with purple and neon

After a few more prompts, Talk GPT aligned the form to the center, changed some colors, and added padding.

CGPT:

<!DOCTYPE html>
<html>
<head>
    <title>AI Chatbot</title>
    <link href="https://unpkg.com/tailwindcss@^1.0/dist/tailwind.min.css" rel="stylesheet">
</head>
<body class="bg-black flex flex-col justify-center items-center">
    <h1 class="sm:text-center text-purple-500">AI Chatbot</h1>
    <form id="chatbot-form" class="border-purple-500 bg-purple-800 rounded">
        <label for="message">Message:</label><br>
        <input type="text" id="message" name="message" class="min-w-30 rounded"><br>
        <input type="submit" value="Submit" class="bg-purple-500 border-red-500 transform hover:scale-110 rounded">
    </form> 
    <div class="flex justify-center">
        <div id="chatbot-response" class="text-white rounded"></div>
    </div>

    <script src="https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js"></script>
    <script>
        var colors = ['red', 'orange', 'yellow', 'green', 'blue', 'indigo', 'violet'];
        var colorIndex = 0;
        
        $(document).ready(function() {
            setInterval(function() {
                $("#chatbot-form").css('border-color', colors[colorIndex]);
                colorIndex = (colorIndex + 1) % colors.length;
            }, 1000);

            $("#chatbot-form").submit(function(e) {
                e.preventDefault();
                var userMessage = $("#message").val();
                $.ajax({
                    type: "POST",
                    url: "/chatbot",
                    data: JSON.stringify({message: userMessage}),
                    contentType: "application/json; charset=utf-8",
                    dataType: "json",
                    success: function(response) {
                        $("#chatbot-response").text(response.response);
                    }
                });
            });
        });
    </script>
</body>
</html>

Improved:

A few more prompts like “Add more vertical and horizontal padding to the submit button and vertical margin, and vertically align this file's body.”

The result:

Is it effective? Now, I'll chat with ChatGPT's AI Chatbot. I intervened minimally throughout construction. Chat GPT created numerous files, connected and interacted with its own API, and presented the user request response on the front end by itself.

Me to AI Chatbot: Hey, can you give me 6 name suggestions for a black horse?

Me to AI Chatbot: Write a poem in the style of Jim Morrison. Make sure to mention cacti and black holes

ChatGPTs Chatbot:

In the desert night, I am confronted by cacti,
Their arms reaching out, As if to grab me and take me away.
The stars in the sky, Beckon me to come,
To a world of black holes, Where I can be free.
I am in a trance, My body and mind,
In a state of limbo, As I wander the night.
I hear a voice, Calling me to the unknown,
I cannot resist, The lure of the black hole.
The cacti watch, As I take my leave,
To a place of mystery, Where I can find peace.

Of course, the next question I gave it was to build a chatbot!

Me: Build a python and flask AI chatbot that responds to user messages on the frontend.

ChatbotGPTs:

import os from flask 
import Flask, request, jsonify 
from chatterbot 
import ChatBot 
from chatterbot.trainers 
import ListTrainer 

# Create a new chatbot 
bot = ChatBot( 
  "My Bot", 
  storage_adapter="chatterbot.storage.SQLStorageAdapter", 
  logic_adapters=[ 
    "chatterbot.logic.BestMatch", 
    { "import_path": "chatterbot.logic.BestMatch", 
      "statement_comparison_function": "chatterbot.comparisons.levenshtein_distance", 
       "response_selection_method": "chatterbot.response_selection.get_first_response" } 
    ], 
    database_uri="sqlite:///database.sqlite3" ) 

# Train the chatbot with a few sample conversations 
conversation = [ "Hello", "Hi there!", "How are you doing?", "I'm doing great.", "That is good to hear", "Thank you.", "You're welcome." ] 
trainer = ListTrainer(bot) trainer.train(conversation) 

# Create a Flask app 
app = Flask(__name__) 

# Define a route for the default URL 
@app.route("/") def index(): return "<h1>Chatbot</h1>" 

# Define a route for the /get_response URL 
@app.route("/get_response", methods=["POST"]) 

def get_response(): 
# Get the user's message 
user_message = request.form["user_message"] 
# Get the bot's response 
bot_response = bot.get_response(user_message) 
# Return the bot's response 
return str(bot_response) 
# Run the Flask 
app if __name__ == "__main__": app.run()

No matter how many times I tried, the second generation always used chatterbot instead of the ChatGPT API. Even when I told it to use the ChatGPT API, it didn't.

ChatGTP's ability to reproduce or construct other machine learning algorithms is interesting and possibly terrifying. Nothing prevents ChatGPT from replicating itself ad infinitum throughout the Internet other than a lack of desire. This may be the first time a machine repeats itself, so I've preserved the project as a reference. Adding a requirements.txt file and python env for easier deployment is the only change to the code.

I hope you enjoyed this.

Ossiana Tepfenhart

3 years ago

Has anyone noticed what an absolute shitshow LinkedIn is?

After viewing its insanity, I had to leave this platform.

I joined LinkedIn recently. That's how I aim to increase my readership and gain recognition. LinkedIn's premise appealed to me: a Facebook-like platform for professional networking.

I don't use Facebook since it's full of propaganda. It seems like a professional, apolitical space, right?

I expected people to:

be more formal and respectful than on Facebook.
Talk about the inclusiveness of the workplace. Studies consistently demonstrate that inclusive, progressive workplaces outperform those that adhere to established practices.
Talk about business in their industry. Yep. I wanted to read articles with advice on how to write better and reach a wider audience.

Oh, sh*t. I hadn't anticipated that.

After posting and reading about inclusivity and pro-choice, I was startled by how many professionals acted unprofessionally. I've seen:

Men have approached me in the DMs in a really aggressive manner. Yikes. huge yikes Not at all professional.
I've heard pro-choice women referred to as infant killers by many people. If I were the CEO of a company and I witnessed one of my employees acting that poorly, I would immediately fire them.
Many posts are anti-LGBTQIA+, as I've noticed. a lot, like, a lot. Some are subtly stating that the world doesn't need to know, while others are openly making fun of transgender persons like myself.
Several medical professionals were posting explicitly racist comments. Even if you are as white as a sheet like me, you should be alarmed by this. Who's to guarantee a patient who is black won't unintentionally die?
I won't even get into how many men in STEM I observed pushing for the exclusion of women from their fields. I shouldn't be surprised considering the majority of those men I've encountered have a passionate dislike for women, but goddamn, dude.

Many people appear entirely too at ease displaying their bigotry on their professional profiles.

As a white female, I'm always shocked by people's open hostility. Professional environments are very important.

I don't know if this is still true (people seem too politicized to care), but if I heard many of these statements in person, I'd suppose they feel ashamed. Really.

Are you not ashamed of being so mean? Are you so weak that competing with others terrifies you? Isn't this embarrassing?

LinkedIn isn't great at censoring offensive comments. These people aren't getting warnings. So they were safe while others were unsafe.

The CEO in me would want to know if I had placed a bigot on my staff.

I always wondered if people's employers knew about their online behavior. If they know how horrible they appear, they don't care.

As a manager, I was picky about hiring. Obviously. In most industries, it costs $1,000 or more to hire a full-time employee, so be sure it pays off.

Companies that embrace diversity and tolerance (and are intolerant of intolerance) are more profitable, likely to recruit top personnel, and successful.

People avoid businesses that alienate them. That's why I don't eat at Chic-Fil-A and why folks avoid MyPillow. Being inclusive is good business.

CEOs are harmed by online bigots. Image is an issue. If you're a business owner, you can fire staff who don't help you.

On the one hand, I'm delighted it makes it simpler to identify those with whom not to do business.

Don’t get me wrong. I'm glad I know who to avoid when hiring, getting references, or searching for a job. When people are bad, it saves me time.

What's up with professionalism?

Really. I need to know. I've crossed the boundary between acceptable and unacceptable behavior, but never on a professional platform. I got in trouble for not wearing bras even though it's not part of my gender expression.

If I behaved like that at my last two office jobs, my supervisors would have fired me immediately. Some of the behavior I've seen is so outrageous, I can't believe these people have employment. Some are even leaders.

Like…how? Is hatred now normalized?

Please pay attention whether you're seeking for a job or even simply a side gig.

Do not add to the tragedy that LinkedIn comments can be, or at least don't make uninformed comments. Even if you weren't banned, the site may still bite you.

Recruiters can and do look at your activity. Your writing goes on your résumé. The wrong comment might lose you a job.

Recruiters and CEOs might reject candidates whose principles contradict with their corporate culture. Bigotry will get you banned from many companies, especially if others report you.

If you want a high-paying job, avoid being a LinkedIn asshole. People care even if you think no one does. Before speaking, ponder. Is this how you want to be perceived?

Better advice:

If your politics might turn off an employer, stop posting about them online and ask yourself why you hold such objectionable ideas.

Liz Martin

3 years ago

A Search Engine From Apple?

Apple's search engine has long been rumored. Recent Google developments may confirm the rumor. Is Apple about to become Google's biggest rival?

Here's a video:

People noted Apple's changes in 2020. AppleBot, a web crawler that downloads and caches Internet content, was more active than in the last five years.

Apple hired search engine developers, including ex-Googlers, such as John Giannandrea, Google's former search chief.

Apple also changed the way iPhones search. With iOS 14, Apple's search results arrived before Google's.

These facts fueled rumors that Apple was developing a search engine.

Apple and Google Have a Contract

Many skeptics said Apple couldn't compete with Google. This didn't affect the company's competitiveness.

Apple is the only business with the resources and scale to be a Google rival, with 1.8 billion active devices and a $2 trillion market cap.

Still, people doubted that due to a license deal. Google pays Apple $8 to $12 billion annually to be the default iPhone and iPad search engine.

Apple can't build an independent search product under this arrangement.

Why would Apple enter search if it's being paid to stay out?

Ironically, this partnership has many people believing Apple is getting into search.

A New Default Search Engine May Be Needed

Google was sued for antitrust in 2020. It is accused of anticompetitive and exclusionary behavior. Justice wants to end Google's monopoly.

Authorities could restrict Apple and Google's licensing deal due to its likely effect on market competitiveness. Hence Apple needs a new default search engine.

Apple Already Has a Search Engine

The company already has a search engine, Spotlight.

Since 2004, Spotlight has aired. It was developed to help users find photos, documents, apps, music, and system preferences.

Apple's search engine could do more than organize files, texts, and apps.

Spotlight Search was updated in 2014 with iOS 8. Web, App Store, and iTunes searches became available. You could find nearby places, movie showtimes, and news.

This search engine has subsequently been updated and improved. Spotlight added rich search results last year.

If you search for a TV show, movie, or song, photos and carousels will appear at the top of the page.

This resembles Google's rich search results.

When Will the Apple Search Engine Be Available?

When will Apple's search launch? Robert Scoble says it's near.

Scoble tweeted a number of hints before this year's Worldwide Developer Conference.

Scoble bases his prediction on insider information and deductive reasoning. January 2023 is expected.

Will you use Apple's search engine?

Hasan AboulHasan

2 years ago

High attachment products can help you earn money automatically.

Affiliate marketing is a popular online moneymaker. You promote others' products and get commissions. Affiliate marketing requires constant product promotion.

Affiliate marketing can be profitable even without much promotion. Yes, this is Autopilot Money.

Screenshot of my profits following this strategy (Just From One Product)

How to Pick an Affiliate Program to Generate Income Autonomously

Autopilot moneymaking requires a recurring affiliate marketing program.

Finding the best product and testing it takes a lot of time and effort.

Here are three ways to choose the best service or product to promote:

Find a good attachment-rate product or service.

When choosing a product, ask if you can easily switch to another service. Attachment rate is how much people like a product.

Higher attachment rates mean better Autopilot products.

Consider promoting GetResponse. It's a 33% recurring commission email marketing tool. This means you get 33% of the customer's plan as long as he pays.

GetResponse has a high attachment rate because it's hard to leave and start over with another tool.

2. Pick a good or service with a lot of affiliate assets.

Check if a program has affiliate assets or creatives before joining.

Images and banners to promote the product in your business.

They save time; I look for promotional creatives. Creatives or affiliate assets are website banners or images. This reduces design time.

3. Select a service or item that consumers already adore.

New products are hard to sell. Choosing a trusted company's popular product or service is helpful.

As a beginner, let people buy a product they already love.

Online entrepreneurs and digital marketers love Systeme.io. It offers tools for creating pages, email marketing, funnels, and more. This product guarantees a high ROI.

Make the product known!

Affiliate marketers struggle to get traffic. Using affiliate marketing to make money is easier than you think if you have a solid marketing strategy.

Your plan should include:

1- Publish affiliate-related blog posts and SEO-optimize them

2- Sending new visitors product-related emails

3- Create a product resource page.

4-Review products

5-Make YouTube videos with links in the description.

6- Answering FAQs about your products and services on your blog and Quora.

7- Create an eCourse on how to use this product.

8- Adding Affiliate Banners to Your Website.

With these tips, you can promote your products and make money on autopilot.

Nir Zicherman

3 years ago

The Great Organizational Conundrum

Only two of the following three options can be achieved: consistency, availability, and partition tolerance

A DALL-E 2 generated “photograph of a teddy bear who is frustrated because it can’t finish a jigsaw puzzle”

Someone told me that growing from 30 to 60 is the biggest adjustment for a team or business.

I remember thinking, That's random. Each company is unique. I've seen teams of all types confront the same issues during development periods. With new enterprises starting every year, we should be better at navigating growing difficulties.

As a team grows, its processes and systems break down, requiring reorganization or declining results. Why always? Why isn't there a perfect scaling model? Why hasn't that been found?

The Three Things Productive Organizations Must Have

Any company should be efficient and productive. Three items are needed:

First, it must verify that no two team members have conflicting information about the roadmap, strategy, or any input that could affect execution. Teamwork is required.

Second, it must ensure that everyone can receive the information they need from everyone else quickly, especially as teams become more specialized (an inevitability in a developing organization). It requires everyone's accessibility.

Third, it must ensure that the organization can operate efficiently even if a piece is unavailable. It's partition-tolerant.

From my experience with the many teams I've been on, invested in, or advised, achieving all three is nearly impossible. Why a perfect organization model cannot exist is clear after analysis.

The CAP Theorem: What is it?

Eric Brewer of Berkeley discovered the CAP Theorem, which argues that a distributed data storage should have three benefits. One can only have two at once.

The three benefits are consistency, availability, and partition tolerance, which implies that even if part of the system is offline, the remainder continues to work.

This notion is usually applied to computer science, but I've realized it's also true for human organizations. In a post-COVID world, many organizations are hiring non-co-located staff as they grow. CAP Theorem is more important than ever. Growing teams sometimes think they can develop ways to bypass this law, dooming themselves to a less-than-optimal team dynamic. They should adopt CAP to maximize productivity.

Path 1: Consistency and availability equal no tolerance for partitions

Let's imagine you want your team to always be in sync (i.e., for someone to be the source of truth for the latest information) and to be able to share information with each other. Only division into domains will do.

Numerous developing organizations do this, especially after the early stage (say, 30 people) when everyone may wear many hats and be aware of all the moving elements. After a certain point, it's tougher to keep generalists aligned than to divide them into specialized tasks.

In a specialized, segmented team, leaders optimize consistency and availability (i.e. every function is up-to-speed on the latest strategy, no one is out of sync, and everyone is able to unblock and inform everyone else).

Partition tolerance suffers. If any component of the organization breaks down (someone goes on vacation, quits, underperforms, or Gmail or Slack goes down), productivity stops. There's no way to give the team stability, availability, and smooth operation during a hiccup.

Path 2: Partition Tolerance and Availability = No Consistency

Some businesses avoid relying too heavily on any one person or sub-team by maximizing availability and partition tolerance (the organization continues to function as a whole even if particular components fail). Only redundancy can do that. Instead of specializing each member, the team spreads expertise so people can work in parallel. I switched from Path 1 to Path 2 because I realized too much reliance on one person is risky.

What happens after redundancy? Unreliable. The more people may run independently and in parallel, the less anyone can be the truth. Lack of alignment or updated information can lead to people executing slightly different strategies. So, resources are squandered on the wrong work.

Path 3: Partition and Consistency "Tolerance" equates to "absence"

The third, least-used path stresses partition tolerance and consistency (meaning answers are always correct and up-to-date). In this organizational style, it's most critical to maintain the system operating and keep everyone aligned. No one is allowed to read anything without an assurance that it's up-to-date (i.e. there’s no availability).

Always short-lived. In my experience, a business that prioritizes quality and scalability over speedy information transmission can get bogged down in heavy processes that hinder production. Large-scale, this is unsustainable.

Accepting CAP

When two puzzle pieces fit, the third won't. I've watched developing teams try to tackle these difficulties, only to find, as their ancestors did, that they can never be entirely solved. Idealized solutions fail in reality, causing lost effort, confusion, and lower production.

As teams develop and change, they should embrace CAP, acknowledge there is a limit to productivity in a scaling business, and choose the best two-out-of-three path.

Vitalik

3 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

$x+5$
$x^4$
$x^3+3x^2+3x+1$
$628x^{271}+318x^{270}+530x^{269}+…+69x+381$

i.e. they are a sum of any (finite!) number of terms of the form $cx^k$

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation $A(x)+ B(x) = C(x)$ . If this equation is true, then it's also true that:

$A(0)+B(0)=C(0)$
$A(1)+B(1)=C(1)$
$A(2)+B(2)=C(2)$
$A(3)+B(3)=C(3)$

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

$12+1=13$
$10+8=18$
$15+8=23$
$15+13=28$

You can use a procedure called Lagrange interpolation to construct polynomials $A(x)$ that give $(12,10,15,15)$ as outputs at some specific set of coordinates (eg. $(0,1,2,3)$ ), $B(x)$ the outputs $(1,8,8,13)$ on thos same coordinates, and so forth. In fact, here are the polynomials:

$A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12$
$B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1$
$C(x)=5x+13$

Checking the equation $A(x)+B(x)=C(x)$ with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial $F, F(x+2)=F(x)+F(x+1)$ with the integer range ${0,1\dots89}$ (so if you also check $F(0)=F(1)=1$ , then $F(100)$ would be the 100th Fibonacci number)

As polynomials, $F(x+2)-F(x+1)-F(x)$ would not be exactly zero, as it could give arbitrary answers outside the range $x={0,1\dots98}$ . But we can do something clever. In general, there is a rule that if a polynomial $P$ is zero across some set $S=\{x_1,x_2…x_n\}$ then it can be expressed as $P(x)=Z(x)*H(x)$ , where $Z(x)=(x-x_1)*(x-x_2)*\dots*(x-x_n)$ and $H(x)$ is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing $P(x)$ by $Z(x)$ , we will get a quotient $Q(x)$ and a remainder $R(x)$ is strictly less than that of $Z(x)$ . Since we know that $P$ is zero on all of $S$ , it means that $R$ has to be zero on all of $S$ as well. So we can simply compute $R(x)$ via polynomial interpolation, since it's a polynomial of degree at most $n-1$ and we know $n$ values (the zeros at $S$ ). Interpolating a polynomial with all zeroes gives the zero polynomial, thus $R(x)=0$ and $H(x)=Q(x)$ .

Going back to our example, if we have a polynomial $F$ that encodes Fibonacci numbers (so $F(x+2)=F(x)+F(x+1)$ across $x=\{0,1…98\}$ ), then I can convince you that $F$ actually satisfies this condition by proving that the polynomial $P(x)=F(x+2)-F(x+1)-F(x)$ is zero over that range, by giving you the quotient:
$H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}$
Where $Z(x) = (x-0)*(x-1)*\dots*(x-98)$ .
You can calculate $Z(x)$ yourself (ideally you would have it precomputed), check the equation, and if the check passes then $F(x)$ satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

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