Zero-Knowledge Proofs for Beginners

Published here originally.

Introduction

I Spy—did you play as a kid? One person chose a room object, and the other had to guess it by answering yes or no questions. I Spy was entertaining, but did you know it could teach you cryptography?

Zero Knowledge Proofs let you show your pal you know what they picked without exposing how. Math replaces electronics in this secret spy mission. Zero-knowledge proofs (ZKPs) are sophisticated cryptographic tools that allow one party to prove they have particular knowledge without revealing it. This proves identification and ownership, secures financial transactions, and more. This article explains zero-knowledge proofs and provides examples to help you comprehend this powerful technology.

What is a Proof of Zero Knowledge?

Zero-knowledge proofs prove a proposition is true without revealing any other information. This lets the prover show the verifier that they know a fact without revealing it. So, a zero-knowledge proof is like a magician's trick: the prover proves they know something without revealing how or what. Complex mathematical procedures create a proof the verifier can verify.

Want to find an easy way to test it out? Try out with tis awesome example! ZK Crush

Describe it as if I'm 5

Alex and Jack found a cave with a center entrance that only opens when someone knows the secret. Alex knows how to open the cave door and wants to show Jack without telling him.

Alex and Jack name both pathways (let’s call them paths A and B).

1. In the first phase, Alex is already inside the cave and is free to select either path, in this case A or B.

2. As Alex made his decision, Jack entered the cave and asked him to exit from the B path.

3. Jack can confirm that Alex really does know the key to open the door because he came out for the B path and used it.

To conclude, Alex and Jack repeat:

1. Alex walks into the cave.

2. Alex follows a random route.

3. Jack walks into the cave.

4. Alex is asked to follow a random route by Jack.

5. Alex follows Jack's advice and heads back that way.

What is a Zero Knowledge Proof?

At a high level, the aim is to construct a secure and confidential conversation between the prover and the verifier, where the prover convinces the verifier that they have the requisite information without disclosing it. The prover and verifier exchange messages and calculate in each round of the dialogue.

The prover uses their knowledge to prove they have the information the verifier wants during these rounds. The verifier can verify the prover's truthfulness without learning more by checking the proof's mathematical statement or computation.

Zero knowledge proofs use advanced mathematical procedures and cryptography methods to secure communication. These methods ensure the evidence is authentic while preventing the prover from creating a phony proof or the verifier from extracting unnecessary information.

ZK proofs require examples to grasp. Before the examples, there are some preconditions.

Criteria for Proofs of Zero Knowledge

1. Completeness: If the proposition being proved is true, then an honest prover will persuade an honest verifier that it is true.

2. Soundness: If the proposition being proved is untrue, no dishonest prover can persuade a sincere verifier that it is true.

3. Zero-knowledge: The verifier only realizes that the proposition being proved is true. In other words, the proof only establishes the veracity of the proposition being supported and nothing more.

The zero-knowledge condition is crucial. Zero-knowledge proofs show only the secret's veracity. The verifier shouldn't know the secret's value or other details.

Example after example after example

To illustrate, take a zero-knowledge proof with several examples:

Initial Password Verification Example

You want to confirm you know a password or secret phrase without revealing it.

Use a zero-knowledge proof:

1. You and the verifier settle on a mathematical conundrum or issue, such as figuring out a big number's components.

2. The puzzle or problem is then solved using the hidden knowledge that you have learned. You may, for instance, utilize your understanding of the password to determine the components of a particular number.

3. You provide your answer to the verifier, who can assess its accuracy without knowing anything about your private data.

4. You go through this process several times with various riddles or issues to persuade the verifier that you actually are aware of the secret knowledge.

You solved the mathematical puzzles or problems, proving to the verifier that you know the hidden information. The proof is zero-knowledge since the verifier only sees puzzle solutions, not the secret information.

In this scenario, the mathematical challenge or problem represents the secret, and solving it proves you know it. The evidence does not expose the secret, and the verifier just learns that you know it.

My simple example meets the zero-knowledge proof conditions:

1. Completeness: If you actually know the hidden information, you will be able to solve the mathematical puzzles or problems, hence the proof is conclusive.

2. Soundness: The proof is sound because the verifier can use a publicly known algorithm to confirm that your answer to the mathematical conundrum or difficulty is accurate.

3. Zero-knowledge: The proof is zero-knowledge because all the verifier learns is that you are aware of the confidential information. Beyond the fact that you are aware of it, the verifier does not learn anything about the secret information itself, such as the password or the factors of the number. As a result, the proof does not provide any new insights into the secret.

Explanation #2: Toss a coin.

One coin is biased to come up heads more often than tails, while the other is fair (i.e., comes up heads and tails with equal probability). You know which coin is which, but you want to show a friend you can tell them apart without telling them.

Use a zero-knowledge proof:

1. One of the two coins is chosen at random, and you secretly flip it more than once.

2. You show your pal the following series of coin flips without revealing which coin you actually flipped.

3. Next, as one of the two coins is flipped in front of you, your friend asks you to tell which one it is.

4. Then, without revealing which coin is which, you can use your understanding of the secret order of coin flips to determine which coin your friend flipped.

5. To persuade your friend that you can actually differentiate between the coins, you repeat this process multiple times using various secret coin-flipping sequences.

In this example, the series of coin flips represents the knowledge of biased and fair coins. You can prove you know which coin is which without revealing which is biased or fair by employing a different secret sequence of coin flips for each round.

The evidence is zero-knowledge since your friend does not learn anything about which coin is biased and which is fair other than that you can tell them differently. The proof does not indicate which coin you flipped or how many times you flipped it.

The coin-flipping example meets zero-knowledge proof requirements:

1. Completeness: If you actually know which coin is biased and which is fair, you should be able to distinguish between them based on the order of coin flips, and your friend should be persuaded that you can.

2. Soundness: Your friend may confirm that you are correctly recognizing the coins by flipping one of them in front of you and validating your answer, thus the proof is sound in that regard. Because of this, your acquaintance can be sure that you are not just speculating or picking a coin at random.

3. Zero-knowledge: The argument is that your friend has no idea which coin is biased and which is fair beyond your ability to distinguish between them. Your friend is not made aware of the coin you used to make your decision or the order in which you flipped the coins. Consequently, except from letting you know which coin is biased and which is fair, the proof does not give any additional information about the coins themselves.

Figure out the prime number in Example #3.

You want to prove to a friend that you know their product n=pq without revealing p and q. Zero-knowledge proof?

Use a variant of the RSA algorithm. Method:

1. You determine a new number s = r2 mod n by computing a random number r.

2. You email your friend s and a declaration that you are aware of the values of p and q necessary for n to equal pq.

3. A random number (either 0 or 1) is selected by your friend and sent to you.

4. You send your friend r as evidence that you are aware of the values of p and q if e=0. You calculate and communicate your friend's s/r if e=1.

5. Without knowing the values of p and q, your friend can confirm that you know p and q (in the case where e=0) or that s/r is a legitimate square root of s mod n (in the situation where e=1).

This is a zero-knowledge proof since your friend learns nothing about p and q other than their product is n and your ability to verify it without exposing any other information. You can prove that you know p and q by sending r or by computing s/r and sending that instead (if e=1), and your friend can verify that you know p and q or that s/r is a valid square root of s mod n without learning anything else about their values. This meets the conditions of completeness, soundness, and zero-knowledge.

Zero-knowledge proofs satisfy the following:

1. Completeness: The prover can demonstrate this to the verifier by computing q = n/p and sending both p and q to the verifier. The prover also knows a prime number p and a factorization of n as p*q.

2. Soundness: Since it is impossible to identify any pair of numbers that correctly factorize n without being aware of its prime factors, the prover is unable to demonstrate knowledge of any p and q that do not do so.

3. Zero knowledge: The prover only admits that they are aware of a prime number p and its associated factor q, which is already known to the verifier. This is the extent of their knowledge of the prime factors of n. As a result, the prover does not provide any new details regarding n's prime factors.

Types of Proofs of Zero Knowledge

Each zero-knowledge proof has pros and cons. Most zero-knowledge proofs are:

1. Interactive Zero Knowledge Proofs: The prover and the verifier work together to establish the proof in this sort of zero-knowledge proof. The verifier disputes the prover's assertions after receiving a sequence of messages from the prover. When the evidence has been established, the prover will employ these new problems to generate additional responses.

2. Non-Interactive Zero Knowledge Proofs: For this kind of zero-knowledge proof, the prover and verifier just need to exchange a single message. Without further interaction between the two parties, the proof is established.

3. A statistical zero-knowledge proof is one in which the conclusion is reached with a high degree of probability but not with certainty. This indicates that there is a remote possibility that the proof is false, but that this possibility is so remote as to be unimportant.

4. Succinct Non-Interactive Argument of Knowledge (SNARKs): SNARKs are an extremely effective and scalable form of zero-knowledge proof. They are utilized in many different applications, such as machine learning, blockchain technology, and more. Similar to other zero-knowledge proof techniques, SNARKs enable one party—the prover—to demonstrate to another—the verifier—that they are aware of a specific piece of information without disclosing any more information about that information.

5. The main characteristic of SNARKs is their succinctness, which refers to the fact that the size of the proof is substantially smaller than the amount of the original data being proved. Because to its high efficiency and scalability, SNARKs can be used in a wide range of applications, such as machine learning, blockchain technology, and more.

Uses for Zero Knowledge Proofs

ZKP applications include:

1. Verifying Identity ZKPs can be used to verify your identity without disclosing any personal information. This has uses in access control, digital signatures, and online authentication.

2. Proof of Ownership ZKPs can be used to demonstrate ownership of a certain asset without divulging any details about the asset itself. This has uses for protecting intellectual property, managing supply chains, and owning digital assets.

3. Financial Exchanges Without disclosing any details about the transaction itself, ZKPs can be used to validate financial transactions. Cryptocurrency, internet payments, and other digital financial transactions can all use this.

4. By enabling parties to make calculations on the data without disclosing the data itself, Data Privacy ZKPs can be used to preserve the privacy of sensitive data. Applications for this can be found in the financial, healthcare, and other sectors that handle sensitive data.

5. By enabling voters to confirm that their vote was counted without disclosing how they voted, elections ZKPs can be used to ensure the integrity of elections. This is applicable to electronic voting, including internet voting.

6. Cryptography Modern cryptography's ZKPs are a potent instrument that enable secure communication and authentication. This can be used for encrypted messaging and other purposes in the business sector as well as for military and intelligence operations.

Proofs of Zero Knowledge and Compliance

Kubernetes and regulatory compliance use ZKPs in many ways. Examples:

1. Security for Kubernetes ZKPs offer a mechanism to authenticate nodes without disclosing any sensitive information, enhancing the security of Kubernetes clusters. ZKPs, for instance, can be used to verify, without disclosing the specifics of the program, that the nodes in a Kubernetes cluster are running permitted software.

2. Compliance Inspection Without disclosing any sensitive information, ZKPs can be used to demonstrate compliance with rules like the GDPR, HIPAA, and PCI DSS. ZKPs, for instance, can be used to demonstrate that data has been encrypted and stored securely without divulging the specifics of the mechanism employed for either encryption or storage.

3. Access Management Without disclosing any private data, ZKPs can be used to offer safe access control to Kubernetes resources. ZKPs can be used, for instance, to demonstrate that a user has the necessary permissions to access a particular Kubernetes resource without disclosing the details of those permissions.

4. Safe Data Exchange Without disclosing any sensitive information, ZKPs can be used to securely transmit data between Kubernetes clusters or between several businesses. ZKPs, for instance, can be used to demonstrate the sharing of a specific piece of data between two parties without disclosing the details of the data itself.

5. Kubernetes deployments audited Without disclosing the specifics of the deployment or the data being processed, ZKPs can be used to demonstrate that Kubernetes deployments are working as planned. This can be helpful for auditing purposes and for ensuring that Kubernetes deployments are operating as planned.

ZKPs preserve data and maintain regulatory compliance by letting parties prove things without revealing sensitive information. ZKPs will be used more in Kubernetes as it grows.

I read 1k+ page 3AC liquidation documentation so you don't have to. Also sharing revised Celsius recovery plans.

3AC's liquidation documents:

Someone disclosed 3AC liquidation records in the BVI courts recently. I'll discuss the leak's timeline and other highlights.

Three Arrows Capital began trading traditional currencies in emerging markets in 2012. They switched to equities and crypto, then purely crypto in 2018.

By 2020, the firm had $703mm in net assets and $1.8bn in loans (these guys really like debt).

The firm's net assets under control reached $3bn in April 2022, according to the filings. 3AC had $600mm of LUNA/UST exposure before May 9th 2022, which put them over.

LUNA and UST go to zero quickly (I wrote about the mechanics of the blowup here). Kyle Davies, 3AC co-founder, told Blockchain.com on May 13 that they have $2.4bn in assets and $2.3bn NAV vs. $2bn in borrowings. As BTC and ETH plunged 33% and 50%, the company became insolvent by mid-2022.

3AC sent $32mm to Tai Ping Shen, a Cayman Islands business owned by Su Zhu and Davies' partner, Kelly Kaili Chen (who knows what is going on here).

3AC had borrowed over $3.5bn in notional principle, with Genesis ($2.4bn) and Voyager ($650mm) having the most exposure.

Genesis demanded $355mm in further collateral in June.

Deribit (another 3AC investment) called for $80 million in mid-June.

Even in mid-June, the corporation was trying to borrow more money to stay afloat. They approached Genesis for another $125mm loan (to pay another lender) and HODLnauts for BTC & ETH loans.

Pretty crazy. 3AC founders used borrowed money to buy a $50 million boat, according to the leak.

Su requesting for $5m + Chen Kaili Kelly asserting they loaned $65m unsecured to 3AC are identified as creditors.

Celsius:

This bankruptcy presentation shows the Celsius breakdown from March to July 14, 2022. From $22bn to $4bn, crypto assets plummeted from $14.6bn to $1.8bn (ouch). $16.5bn in user liabilities dropped to $4.72bn.

In my recent post, I examined if "forced selling" is over, with Celsius' crypto assets being a major overhang. In this presentation, it looks that Chapter 11 will provide clients the opportunity to accept cash at a discount or remain long crypto. Provided that a fresh source of money is unlikely to enter the Celsius situation, cash at a discount or crypto given to customers will likely remain a near-term market risk - cash at a discount will likely come from selling crypto assets, while customers who receive crypto could sell at any time. I'll share any Celsius updates I find.

Conclusion

Only Celsius and the Mt Gox BTC unlock remain as forced selling catalysts. While everything went through a "relief" pump, with ETH up 75% from the bottom and numerous alts multiples higher, there are still macro dangers to equities + risk assets. There's a lot of wealth waiting to be deployed in crypto ($153bn in stables), but fund managers are risk apprehensive (lower than 2008 levels).

We're hopefully over crypto's "bottom," with peak anxiety and forced selling behind us, but we may chop around.

To see the full article, click here.

While our genes can't be changed easily, you can avoid some dementia risk factors. Today we discuss dementia and five drugs that may increase risk.

Memory loss appears to come with age, but we're not talking about forgetfulness. Sometimes losing your car keys isn't an indication of dementia. Dementia impairs the capacity to think, remember, or make judgments. Dementia hinders daily tasks.

Alzheimers is the most common dementia. Dementia is not normal aging, unlike forgetfulness. Aging increases the risk of Alzheimer's and other dementias. A family history of the illness increases your risk, according to the Mayo Clinic (USA).

Given that our genes are difficult to change (I won't get into epigenetics), what are some avoidable dementia risk factors? Certain drugs may cause cognitive deterioration.

Today we look at four drugs that may cause cognitive decline.

Dementia and benzodiazepines

Benzodiazepine sedatives increase brain GABA levels. Example benzodiazepines:

• Diazepam (Valium) (Valium)

• Alprazolam (Xanax) (Xanax)

• Clonazepam (Klonopin) (Klonopin)

Addiction and overdose are benzodiazepine risks. Yes! These medications don't raise dementia risk.

USC study: Benzodiazepines don't increase dementia risk in older adults.

Benzodiazepines can produce short- and long-term amnesia. This memory loss hinders memory formation. Extreme cases can permanently impair learning and memory. Anterograde amnesia is uncommon.

2. Statins and dementia

Statins reduce cholesterol. They prevent a cholesterol-making chemical. Examples:

• Atorvastatin (Lipitor) (Lipitor)

• Fluvastatin (Lescol XL) (Lescol XL)

• Lovastatin (Altoprev) (Altoprev)

• Pitavastatin (Livalo, Zypitamag) (Livalo, Zypitamag)

• Pravastatin (Pravachol) (Pravachol)

• Rosuvastatin (Crestor, Ezallor) (Crestor, Ezallor)

• Simvastatin (Zocor) (Zocor)

This finding is contentious. Harvard's Brigham and Womens Hospital's Dr. Joann Manson says:

“I think that the relationship between statins and cognitive function remains controversial. There’s still not a clear conclusion whether they help to prevent dementia or Alzheimer’s disease, have neutral effects, or increase risk.”

This one's off the dementia list.

3. Dementia and anticholinergic drugs

Anticholinergic drugs treat many conditions, including urine incontinence. Drugs inhibit acetylcholine (a brain chemical that helps send messages between cells). Acetylcholine blockers cause drowsiness, disorientation, and memory loss.

First-generation antihistamines, tricyclic antidepressants, and overactive bladder antimuscarinics are common anticholinergics among the elderly.

Anticholinergic drugs may cause dementia. One study found that taking anticholinergics for three years or more increased the risk of dementia by 1.54 times compared to three months or less. After stopping the medicine, the danger may continue.

4. Drugs for Parkinson's disease and dementia

Cleveland Clinic (USA) on Parkinson's:

Parkinson's disease causes age-related brain degeneration. It causes delayed movements, tremors, and balance issues. Some are inherited, but most are unknown. There are various treatment options, but no cure.

Parkinson's medications can cause memory loss, confusion, delusions, and obsessive behaviors. The drug's effects on dopamine cause these issues.

A 2019 JAMA Internal Medicine study found powerful anticholinergic medications enhance dementia risk.

Those who took anticholinergics had a 1.5 times higher chance of dementia. Individuals taking antidepressants, antipsychotic drugs, anti-Parkinson’s drugs, overactive bladder drugs, and anti-epileptic drugs had the greatest risk of dementia.

Anticholinergic medicines can lessen Parkinson's-related tremors, but they slow cognitive ability. Anticholinergics can cause disorientation and hallucinations in those over 70.

5. Antiepileptic drugs and dementia

The risk of dementia from anti-seizure drugs varies with drugs. Levetiracetam (Keppra) improves Alzheimer's cognition.

One study linked different anti-seizure medications to dementia. Anti-epileptic medicines increased the risk of Alzheimer's disease by 1.15 times in the Finnish sample and 1.3 times in the German population. Depakote, Topamax are drugs.

A fascinating business conversation.

This week at Web Summit, I had mentor hour.

Mentor hour connects startups with experienced entrepreneurs.

The YC-selected founder who mentored me had grown his company to $100 million in 18 months.

I had 45 minutes to question him.

I've compiled this.

Context

Founder's name is Zack.

After working in private equity, Zack opted to acquire an MBA.

Surrounded by entrepreneurs at a prominent school, he decided to become one himself.

Unsure how to proceed, he bet on two horses.

On one side, he received an offer from folks who needed help running their startup owing to lack of time. On the other hand, he had an idea for a SaaS to start himself.

He just needed to validate it.

Validating

Since Zack's proposal helped companies, he contacted university entrepreneurs for comments.

He contacted university founders.

Once he knew he'd correctly identified the problem and that people were willing to pay to address it, he started developing.

He earned $100k in a university entrepreneurship competition.

His plan was evident by then.

The other startup's founders saw his potential and granted him $400k to launch his own SaaS.

Hiring

He started looking for a tech co-founder because he lacked IT skills.

He interviewed dozens and picked the finest.

As he didn't want to wait for his program to be ready, he contacted hundreds of potential clients and got 15 letters of intent promising they'd join up when it was available.

YC accepted him by then.

He had enough positive signals to raise.

Raising

He didn't say how many VCs he called, but he indicated 50 were interested.

He jammed meetings into two weeks to generate pressure and encourage them to invest.

Seed raise: $11 million.

Selling

His objective was to contact as many entrepreneurs as possible to promote his product.

He first contacted startups by scraping CrunchBase data.

Once he had more money, he started targeting companies with ZoomInfo.

His VC urged him not to hire salespeople until he closed 50 clients himself.

He closed 100 and hired a CRO through a headhunter.

Scaling

Three persons started the business.

1. He primarily works in sales.

2. Coding the product was done by his co-founder.

3. Another person performing operational duties.

He regretted recruiting the third co-founder, who was ineffective (could have hired an employee instead).

He wanted his company to be big, so he hired two young marketing people from a competing company.

After validating several marketing channels, he chose PR.

$100 Million and under

He developed a sales team and now employs 30 individuals.

He raised a $100 million Series A.

Additionally, he stated

• He’s been rejected a lot. Like, a lot.

• Two great books to read: Steve Jobs by Isaacson, and Why Startups Fail by Tom Eisenmann.

• The best skill to learn for non-tech founders is “telling stories”, which means sales. A founder’s main job is to convince: co-founders, employees, investors, and customers. Learn code, or learn sales.

Conclusion

I often read about these stories but hardly take them seriously.

Zack was amazing.

Three things about him stand out:

1. His vision. He possessed a certain amount of fire.

2. His vitality. The man had a lot of enthusiasm and spoke quickly and decisively. He takes no chances and pushes the envelope in all he does.

3. His Rolex.

He didn't do all this in 18 months.

Not really.

He couldn't launch his company without private equity experience.

These accounts disregard entrepreneurs' original knowledge.

Hormozi will tell you how he founded Gym Launch, but he won't tell you how he had a gym first, how he worked at uni to pay for his gym, or how he went to the gym and learnt about fitness, which gave him the idea to open his own.

Nobody knows nothing. If you scale quickly, it's probable because you gained information early.

Lincoln said, "Give me six hours to chop down a tree, and I'll spend four sharpening the axe."

Sharper axes cut trees faster.

DATA MINING WITH SUPERALGORES

Old pots produce the best soup.

Science has always inspired indicator design. From physics to signal processing, many indicators use concepts from mechanical engineering, electronics, and probability. In Superalgos' Data Mining section, we've explored using thermodynamics and information theory to construct indicators and using statistical and probabilistic techniques like reduced normal law to take advantage of low probability events.

An asset's price is like a mechanical object revolving around its moving average. Using this approach, we could design an indicator using the oscillator's Total Energy. An oscillator's energy is finite and constant. Since we don't expect the price to follow the harmonic oscillator, this energy should deviate from the perfect situation, and the maximum of divergence may provide us valuable information on the price's moving average.

Definition of the Harmonic Oscillator in Few Words

Sinusoidal function describes a harmonic oscillator. The time-constant energy equation for a harmonic oscillator is:

With

Time saves energy.

In a mechanical harmonic oscillator, total energy equals kinetic energy plus potential energy. The formula for energy is the same for every kind of harmonic oscillator; only the terms of total energy must be adapted to fit the relevant units. Each oscillator has a velocity component (kinetic energy) and a position to equilibrium component (potential energy).

The Price Oscillator and the Energy Formula

Considering the harmonic oscillator definition, we must specify kinetic and potential components for our price oscillator. We define oscillator velocity as the rate of change and equilibrium position as the price's distance from its moving average.

Price kinetic energy:

It's like:

With

and

L is the number of periods for the rate of change calculation and P for the close price EMA calculation.

Total price oscillator energy =

Given that an asset's price can theoretically vary at a limitless speed and be endlessly far from its moving average, we don't expect this formula's outcome to be constrained. We'll normalize it using Z-Score for convenience of usage and readability, which also allows probabilistic interpretation.

Over 20 periods, we'll calculate E's moving average and standard deviation.

We calculated Z on BTC/USDT with L = 10 and P = 21 using Knime Analytics.

The graph is detrended. We added two horizontal lines at +/- 1.6 to construct a 94.5% probability zone based on reduced normal law tables. Price cycles to its moving average oscillate clearly. Red and green arrows illustrate where the oscillator crosses the top and lower limits, corresponding to the maximum/minimum price oscillation. Since the results seem noisy, we may apply a non-lagging low-pass or multipole filter like Butterworth or Laguerre filters and employ dynamic bands at a multiple of Z's standard deviation instead of fixed levels.

Kinetic Detrender Implementation in Superalgos

The Superalgos Kinetic detrender features fixed upper and lower levels and dynamic volatility bands.

The code is pretty basic and does not require a huge amount of code lines.

It starts with the standard definitions of the candle pointer and the constant declaration :

let candle = record.current
let len = 10
let P = 21
let T = 20
let up = 1.6
let low = 1.6

Upper and lower dynamic volatility band constants are up and low.

We proceed to the initialization of the previous value for EMA :

if (variable.prevEMA === undefined) {
    variable.prevEMA = candle.close
}

And the calculation of EMA with a function (it is worth noticing the function is declared at the end of the code snippet in Superalgos) :

variable.ema = calculateEMA(P, candle.close, variable.prevEMA)
//EMA calculation
function calculateEMA(periods, price, previousEMA) {
    let k = 2 / (periods + 1)
    return price * k + previousEMA * (1 - k)
}

The rate of change is calculated by first storing the right amount of close price values and proceeding to the calculation by dividing the current close price by the first member of the close price array:

variable.allClose.push(candle.close)
if (variable.allClose.length > len) {
    variable.allClose.splice(0, 1)
}
if (variable.allClose.length === len) {
    variable.roc = candle.close / variable.allClose[0]
} else {
    variable.roc = 1
}

Finally, we get energy with a single line:

variable.E = 1 / 2 * len * variable.roc + 1 / 2 * P * candle.close / variable.ema

The Z calculation reuses code from Z-Normalization-based indicators:

variable.allE.push(variable.E)
if (variable.allE.length > T) {
    variable.allE.splice(0, 1)
}
variable.sum = 0
variable.SQ = 0
if (variable.allE.length === T) {
    for (var i = 0; i < T; i++) {
        variable.sum += variable.allE[i]
    }
    variable.MA = variable.sum / T
for (var i = 0; i < T; i++) {
        variable.SQ += Math.pow(variable.allE[i] - variable.MA, 2)
    }
    variable.sigma = Math.sqrt(variable.SQ / T)
variable.Z = (variable.E - variable.MA) / variable.sigma
} else {
    variable.Z = 0
}
variable.allZ.push(variable.Z)
if (variable.allZ.length > T) {
    variable.allZ.splice(0, 1)
}
variable.sum = 0
variable.SQ = 0
if (variable.allZ.length === T) {
    for (var i = 0; i < T; i++) {
        variable.sum += variable.allZ[i]
    }
    variable.MAZ = variable.sum / T
for (var i = 0; i < T; i++) {
        variable.SQ += Math.pow(variable.allZ[i] - variable.MAZ, 2)
    }
    variable.sigZ = Math.sqrt(variable.SQ / T)
} else {
    variable.MAZ = variable.Z
    variable.sigZ = variable.MAZ * 0.02
}
variable.upper = variable.MAZ + up * variable.sigZ
variable.lower = variable.MAZ - low * variable.sigZ

We also update the EMA value.

variable.prevEMA = variable.EMA

Conclusion

We showed how to build a detrended oscillator using simple harmonic oscillator theory. Kinetic detrender's main line oscillates between 2 fixed levels framing 95% of the values and 2 dynamic levels, leading to auto-adaptive mean reversion zones.

Superalgos' Normalized Momentum data mine has the Kinetic detrender indication.

All the material here can be reused and integrated freely by linking to this article and Superalgos.

This post is informative and not financial advice. Seek expert counsel before trading. Risk using this material.