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.

Cryptoland rollercoaster

I've lost money in crypto.

Unimportant.

The real issue: I didn’t understand how.

I'm surrounded with winners. To learn more, I created my own NFTs, currency, and DAO.

Web3 is a hilltop castle. Everything is valuable, decentralized, and on-chain.

The castle is Disneyland: beautiful in images, but chaotic with lengthy lines and kids spending too much money on dressed-up animals.

When the throng and businesses are gone, Disneyland still has enchantment.

The Real Story of Web3

NFTs

Scarcity. Scarce NFTs. That's their worth.

Skull. Rare-looking!

Nonsense.

Bored Ape Yacht Club vs. my NFTs?

Marketing.

BAYC is amazing, but not for the reasons people believe. Apecoin and Otherside's art, celebrity following, and innovation? Stunning.

No other endeavor captured the zeitgeist better. Yet how long did you think it took to actually mint the NFTs?

1 hour? Maybe a week for the website?

Minting NFTs is incredibly easy. Kid-friendly. Developers are rare. Think about that next time somebody posts “DevS dO SMt!?”

NFTs will remain popular. These projects are like our Van Goghs and Monets. Still, be wary. It still uses exclusivity and wash selling like the OG art market.

Not all NFTs are art-related.

Soulbound and anonymous NFTs could offer up new use cases. Property rights, privacy-focused ID, open-source project verification. Everything.

NFTs build online trust through ownership.

We just need to evolve from the apes first.

NFTs' superpower is marketing until then.

Crypto currency

What the hell is a token?

99% of people are clueless.

So I invested in both coins and tokens. Same same. Only that they are not.

Coins have their own blockchain and developer/validator community. It's hard.

Creating a token on top of a blockchain? Five minutes.

Most consumers don’t understand the difference, creating an arbitrage opportunity: pretend you’re a serious project without having developers on your payroll.

Few market sites help. Take a look. See any tokens?

There's a hint one click deeper.

Some tokens are legitimate. Some coins are bad investments.

Tokens are utilized for DAO governance and DApp payments. Still, know who's behind a token. They might be 12 years old.

Coins take time and money. The recent LUNA meltdown indicates that currency investing requires research.

DAOs

Decentralized Autonomous Organizations (DAOs) don't work as you assume.

Yes, members can vote.

A productive organization requires more.

I've observed two types of DAOs.

• Total decentralization total dysfunction

• Centralized just partially. Community-driven.

A core team executes the DAO's strategy and roadmap in successful DAOs. The community owns part of the organization, votes on decisions, and holds the team accountable.

DAOs are public companies.

Amazing.

A shareholder meeting's logistics are staggering. DAOs may hold anonymous, secure voting quickly. No need for intermediaries like banks to chase up every shareholder.

Successful DAOs aren't totally decentralized. Large-scale voting and collaboration have never been easier.

And that’s all that matters.

Scale, speed.

My Web3 learnings

Disneyland is enchanting. Web3 too.

In a few cycles, NFTs may be used to build trust, not clout. Not speculating with coins. DAOs run organizations, not themselves.

Finally, some final thoughts:

• NFTs will be a very helpful tool for building trust online. NFTs are successful now because of excellent marketing.

• Tokens are not the same as coins. Look into any project before making a purchase. Make sure it isn't run by three 9-year-olds piled on top of one another in a trench coat, at the very least.

• Not entirely decentralized, DAOs. We shall see a future where community ownership becomes the rule rather than the exception once we acknowledge this fact.

Crypto Disneyland is a rollercoaster with loops that make you sick.

Always buckle up.

Have fun!

Leonardo—The New Midjourney?

Users are comparing the new cowboy to Midjourney.

Leonardo.AI creates great photographs and has several unique capabilities I haven't seen in other AI image systems.

Midjourney's quality photographs are evident in the community feed.

Create Pictures Using Models

You can make graphics using platform models when you first enter the app (website):

Luma, Leonardo creative, Deliberate 1.1.

Clicking a model displays its description and samples:

Click Generate With This Model.

Then you can add your prompt, alter models, photos, sizes, and guide scale in a sleek UI.

Changing Pictures

Leonardo's Canvas editor lets you change created images by hovering over them:

The editor opens with masking, erasing, and picture download.

Develop Your Own Models

I've never seen anything like Leonardo's model training feature.

Upload a handful of similar photographs and save them as a model for future images. Share your model with the community.

You can make photos using your own model and a community-shared set of fine-tuned models:

Obtain Leonardo access

Leonardo is currently free.

Visit Leonardo.ai and click "Get Early Access" to receive access.

Add your email to receive a link to join the discord channel. Simply describe yourself and fill out a form to join the discord channel.

Please go to 👑│introductions to make an introduction and ✨│priority-early-access will be unlocked, you must fill out a form and in 24 hours or a little more (due to demand), the invitation will be sent to you by email.

I got access in two hours, so hopefully you can too.

Last Words

I know there are many AI generative platforms, some free and some expensive, but Midjourney produces the most artistically stunning images and art.

Leonardo is the closest I've seen to Midjourney, but Midjourney is still the leader.

It's free now.

Leonardo's fine-tuned model selections, model creation, image manipulation, and output speed and quality make it a great AI image toolbox addition.

Google announced 6% layoffs recently (or 12,000 people). This aligns it with most tech companies. A publicly contrite CEO explained that they had overhired during the COVID-19 pandemic boom and had to address it, but they were sorry and took full responsibility. I thought this was "bullshit" too. Meta, Amazon, Microsoft, and others must feel similarly. I spent 10 years at Google, and these things don't reflect well on the company's leaders.

All publicly listed companies have a fiduciary duty to act in the best interests of their shareholders. Dodge vs. Ford Motor Company established this (1919). Henry Ford wanted to reduce shareholder payments to offer cheaper cars and better wages. Ford stated.

My ambition is to employ still more men, to spread the benefits of this industrial system to the greatest possible number, to help them build up their lives and their homes. To do this we are putting the greatest share of our profits back in the business.

The Dodge brothers, who owned 10% of Ford, opposed this and sued Ford for the payments to start their own company. They won, preventing Ford from raising prices or salaries. If you have a vocal group of shareholders with the resources to sue you, you must prove you are acting in their best interests. Companies prioritize shareholders. Giving activist investors a stick to threaten you almost enshrines short-term profit over long-term thinking.

This underpins Google's current issues. Institutional investors who can sue Google see it as a wasteful company they can exploit. That doesn't mean you have to maximize profits (thanks to those who pointed out my ignorance of US corporate law in the comments and on HN), but it allows pressure. I feel for those navigating this. This is about unrestrained capitalism.

When Google went public, Larry Page and Sergey Brin knew the risks and worked hard to keep control. In their Founders' Letter to investors, they tried to set expectations for the company's operations.

Our long-term focus as a private company has paid off. Public companies do the same. We believe outside pressures lead companies to sacrifice long-term opportunities to meet quarterly market expectations.

The company has transformed since that letter. The company has nearly 200,000 full-time employees and a trillion-dollar market cap. Large investors have bought company stock because it has been a good long-term bet. Why are they restless now?

Other big tech companies emerged and fought for top talent. This has caused rising compensation packages. Google has also grown rapidly (roughly 22,000 people hired to the end of 2022). At $300,000 median compensation, those 22,000 people added $6.6 billion in salary overheads in 2022. Exorbitant. If the company still makes $16 billion every quarter, maybe not. Investors wonder if this value has returned.

Investors are right. Google uses people wastefully. However, by bluntly reducing headcount, they're not addressing the root causes and hurting themselves. No studies show that downsizing this way boosts productivity. There is plenty of evidence that they'll lose out because people will be risk-averse and distrust their leadership.

The company's approach also stinks. Finding out that you no longer have a job because you can’t log in anymore (sometimes in cases where someone is on call for protecting your production systems) is no way to fire anyone. Being with a narcissistic sociopath is like being abused. First, you receive praise and fancy perks for making the cut. You're fired by text and ghosted. You're told to appreciate the generous severance package. This firing will devastate managers and teams. This type of firing will take years to recover self-esteem. Senior management contributed to this. They chose the expedient answer, possibly by convincing themselves they were managing risk and taking the Macbeth approach of “If it were done when ’tis done, then ’twere well It were done quickly”.

Recap. Google's leadership did a stupid thing—mass firing—in a stupid way. How do we get rid of enough people to make investors happier? and "have 6% less people." Empathetic leaders should not emulate Elon Musk. There is no humane way to fire 12,000 people, but there are better ways. Why is Google so wasteful?

Ambition answers this. There aren't enough VP positions for a group of highly motivated, ambitious, and (increasingly) ruthless people. I’ve loitered around the edges of this world and a large part of my value was to insulate my teams from ever having to experience it. It’s like Game of Thrones played out through email and calendar and over video call.

Your company must look a certain way to be promoted to director or higher. You need the right people at the right levels under you. Long-term, growing your people will naturally happen if you're working on important things. This takes time, and you're never more than 6–18 months from a reorg that could start you over. Ambitious people also tend to be impatient. So, what do you do?

Hiring and vanity projects. To shape your company, you hire at the right levels. You value vanity metrics like active users over product utility. Your promo candidates get through by subverting the promotion process. In your quest for growth, you avoid performance managing people out. You avoid confronting toxic peers because you need their support for promotion. Your cargo cult gets you there.

Its ease makes Google wasteful. Since they don't face market forces, the employees don't see it as a business. Why would you do when the ads business is so profitable? Complacency causes senior leaders to prioritize their own interests. Empires collapse. Personal ambition often trumped doing the right thing for users, the business, or employees. Leadership's ambition over business is the root cause. Vanity metrics, mass hiring, and vague promises have promoted people to VP. Google goes above and beyond to protect senior leaders.

The decision-makers and beneficiaries are not the layoffees. Stock price increase beneficiaries. The people who will post on LinkedIn how it is about misjudging the market and how they’re so sorry and take full responsibility. While accumulating wealth, the dark room dwellers decide who stays and who goes. The billionaire investors. Google should start by addressing its bloated senior management, but — as they say — turkeys don't vote for Christmas. It should examine its wastefulness and make tough choices to fix it. A 6% cut is a blunt tool that admits you're not running your business properly. why aren’t the people running the business the ones shortly to be entering the job market?

This won't fix Google's wastefulness. The executives may never regain trust after their approach. Suppressed creativity. Business won't improve. Google will have lost its founding vision and us all. Large investors know they can force Google's CEO to yield. The rich will get richer and rationalize leaving 12,000 people behind. Cycles repeat.

It doesn’t have to be this way. In 2013, Nintendo's CEO said he wouldn't fire anyone for shareholders. Switch debuted in 2017. Nintendo's stock has increased by nearly five times, or 19% a year (including the drop most of the stock market experienced last year). Google wasted 12,000 talented people. To please rich people.

Prioritize big-ticket purchases

I've made several spending blunders. I get sick thinking about how much money I spent.

My financial mental model was poor back then.

Stoicism and mindfulness keep me from attaching to those feelings. It still hurts.

Until four or five years ago, I bought a new winter jacket every year.

Ten years ago, I spent twice as much. Now that I have a fantastic, warm winter parka, I don't even consider acquiring another one. No more spending. I'm not looking for jackets either.

Saving time and money by spending well is my thinking paradigm.

The philosophy is expressed in most languages. Cheap is expensive in the Netherlands. This applies beyond shopping.

In this essay, I will offer three examples of how this mental paradigm transformed my financial life.

Publishing books

In 2015, I presented and positioned my first book poorly.

I called the book Huge Life Success and made a funny Canva cover in 30 minutes. This:

That looks nothing like my present books. No logo or style. The book felt amateurish.

The book started bothering me a few weeks after publication. The advice was good, but it didn't appear professional. I studied the book business extensively.

I created a style for all my designs. Branding. Win Your Inner Wars was reissued a year later.

Title, cover, and description changed. Rearranging the chapters improved readability.

Seven years later, the book sells hundreds of copies a month. That taught me a lot.

Rushing to finish a project is enticing. Send it and move forward.

Avoid rushing everything. Relax. Develop your projects. Perform well. Perform the job well.

My first novel was underfunded and underworked. A bad book arrived. I then invested time and money in writing the greatest book I could.

That book still sells.

Traveling

I hate travel. Airports, flights, trains, and lines irritate me.

But, I enjoy traveling to beautiful areas.

I do it strangely. I make up travel rules. I never go to airports in summer. I hate being near airports on holidays. Unworthy.

No vacation packages for me. Those airline packages with a flight, shuttle, and hotel. I've had enough.

I try to avoid crowds and popular spots. July Paris? Nuts and bolts, please. Christmas in NYC? No, please keep me sane.

I fly business class behind. I accept upgrades upon check-in. I prefer driving. I drove from the Netherlands to southern Spain.

Thankfully, no lines. What if travel costs more? Thus? I enjoy it from the start. I start traveling then.

I rarely travel since I'm so difficult. One great excursion beats several average ones.

Personal effectiveness

New apps, tools, and strategies intrigue most productivity professionals.

No.

I researched years ago. I spent years investigating productivity in university.

I bought books, courses, applications, and tools. It was expensive and time-consuming.

Im finished. Productivity no longer costs me time or money. OK. I worked on it once and now follow my strategy.

I avoid new programs and systems. My stuff works. Why change winners?

Spending wisely saves time and money.

Spending wisely means spending once. Many people ignore productivity. It's understudied. No classes.

Some assume reading a few articles or a book is enough. Productivity is personal. You need a personal system.

Time invested is one-time. You can trust your system for life once you find it.

Concentrate on the expensive choices.

Life's short. Saving money quickly is enticing.

Spend less on groceries today. True. That won't fix your finances.

Adopt a lifestyle that makes you affluent over time. Consider major choices.

Are they causing long-term poverty? Are you richer?

Leasing cars comes to mind. The automobile costs a fortune today. The premium could accomplish a million nice things.

Focusing on important decisions makes life easier. Consider your future. You want to improve next year.