A mathematical breakdown of how RSA encryption uses prime numbers to encrypt data

Explore a visual guide to how quantum computers can crack RSA encryption

In 1994, Peter Shor introduced an algorithm that showed a quantum computer could efficiently factor the product of two very large prime numbers, undermining the mathematics that makes RSA so secure. While using larger keys can delay the decryption time of quantum computers, this solution increases the time it takes to encrypt data, which may become impractical.

How quantum computers can empower Shor's algorithm to break digital encryption

Secure communications largely rely on numeric keys generated by multiplying two enormous prime numbers together, and it is computationally impractical to determine these primes by trying to divide by every possible one. While Shor's algorithm can provide better guesses for this brute-force approach, it relies on a number that is difficult to determine on classical computers but much easier to calculate using the simultaneous computations enabled by quantum superposition.

View how AES—the global standard for symmetric encryption—works

Established by the National Institute of Standards and Technology in 2001, the Advanced Encryption Standard transforms plaintext into ciphertext using multiple rounds of processing, including substitution, transposition, and mixing. Larger keys undergo more rounds to produce 16-byte blocks of scrambled data, with 256-bit AES keys taking 3.31 × 10⁵⁶ years to decrypt via brute force.

Visualize how passkeys, which rely on asymmetric cryptography to authenticate users, work

After using biometrics or a PIN to access the private key stored on a device, it is combined with a random number generated by the site to produce a cryptographic signature. This signature is verified using the site's public key, allowing a user to log in without entering a password. This reduces the risk of fraudulent messages—phishing attacks—and data breaches.

The science of cryptography, including methods to encrypt data and how to break encryption

Monoalphabetic ciphers use a set of rules to replace each letter with another consistently, making them susceptible to brute force attacks that try every kind of replacement. While more complex ciphers can transform text in less obvious ways, because certain letters are used more often than others, an analysis of letter frequency may reveal patterns that help uncover the rules that make up a cipher.

Understanding the Diffie-Hellman key exchange, an example of public key cryptography

Published in 1976 by Whitfield Diffie and Martin Hellman, the system involves two parties combining each of their unique private keys with a public key using a mathematical function. The combinations are exchanged and mixed to establish a common key that no third party can determine. The Diffie-Hellman system underlies many encryption protocols for internet traffic.

How end-to-end encryption protects messages, even from service providers

Using symmetric and asymmetric cryptography, data is encrypted on the sender's device before being transmitted across application servers that lack the key needed to decrypt it. The message is not decrypted until it reaches the recipient's device, which uses their private key to recover the plaintext. E2EE's effectiveness has led to its adoption by WhatsApp and Signal.

A visual guide to how bitcoin transactions work and rely on cryptography

The "crypto-" in cryptocurrencies like bitcoin refers to the mathematical functions that generate pairs of public and private keys and encrypt data in blocks. When combined, these systems secure transactions on the network, verify digital asset and token transfers, and contribute to end-user anonymity.

The history and operation of SIGABA, the US counterpart of the Nazi Enigma machine

Developed by the US Army and US Navy, over 10,000 of these machines were deployed by the end of World War II to encrypt messages for the armed forces using three series of rotors. The device would continue to be used until the 1950s, while personal computers were shown to possess enough computing power to crack it by 2021.

The most bizarre numbers in the universe

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

Crunching the numbers on the cost of ransomware

This blog is a preview of the 2022 Crypto Crime Report, offering a summary of ransomware activity in 2021, the most active strains of ransomware, and more (w/ data, charts, and illustrations).

Binary mathematics represented as switches allows silicon rocks to do math

Each switch is assigned a zero or a one based on whether it is open or closed, and logic gates are built from these switches that let the system perform mathematical operations. Modern software builds upon these operations to perform increasingly complex tasks.

An overview of 12 fundamental types of algorithms and the uses for each

These include search algorithms to quickly process and find important information, cryptographic algorithms to secure communications, randomized algorithms—such as Monte Carlo simulations for estimating probabilities of outcomes—and greedy algorithms, which make optimal choices based on available data in a step-by-step fashion, as seen in navigation apps.

The fast Fourier transform breaks down wave signals into discrete components

The algorithm was initially discovered by German mathematician Carl Friedrich Gauss in 1805, but his notes were not found until after his death. Today, FFTs form the basis for most compression software in media and are essential to technologies such as radar, sonar, and telecommunications.

A 14-year-old "human calculator" shatters six mathematical world records in one day

A teenage prodigy from Maharashtra, India named Aaryan Shukla set the record for the fastest time to mentally add 50 five-digit numbers. He did so in just 25.19 seconds–roughly equivalent to one addition every 0.5 seconds. The math whiz said he practices calculations for around five to six hours a day.

National Security Agency whistleblower Edward Snowden once leaked documents suggesting Palantir had a role in alleged surveillance

After releasing documents detailing an NSA surveillance scheme named PRISM (which is also the name of a Palantir service), further documents show that Palantir's software enabled the expansion of the NSA's widest-reaching data programs, XKEYSCORE. Palantir has frequently denied claims of supporting government surveillance and has insisted that its software is designed to maintain "civil liberties," such as privacy.

How the National Science Foundation supports the US economy

The NSF has filled funding gaps in the research and development of products developed in the private sector, particularly when the research requires more than three years to complete. These investments, including in startups such as Qualcomm and Symantec, have supported job creation, generated economic returns, and helped build the technologies many industries rely upon, such as the internet.

It took 70 years for computer scientists to determine how to cut a cake fairly

The cake-cutting problem explores how to divide something fairly, which is not solely about cutting evenly-sized pieces. A protocol that accounted for personal preferences (e.g., a corner piece) between just three people took 16 years to develop.

Explore this deep dive into how SSRIs work

This 101-style explainer on selective serotonin reuptake inhibitors explains how they work in the brain—complete with classroom-style whiteboard drawings. It covers common SSRIs, why someone might try this medication, and a deeper discussion about potential side effects.

One of the most important pieces of Amazon’s corporate empire? Its Prime subscription business

Prime drives customer loyalty and boosts Amazon’s revenue through its monthly and yearly fees. The key components included with a Prime subscription include free two-day shipping on eligible items, as well as access to services like Amazon Music and Prime Video.

The science of symmetry | The Royal Society

Nature is full of symmetries, from spiral galaxies to florets of broccoli. So why do complex creatures break the pattern?This short video discusses why symmetry came to dominate life on Earth, and how many complex life forms, such as humans, transitioned from radial to lateral symmetry.

Understanding R0, a measure of how contagious a pathogen is

The R0 value is a basic measure of how many people an infected person will transmit a pathogen to, on average. If R0 is less than 1, transmission dies out; if greater than 1, an outbreak is likely. Measles has one of the highest known R0 values at almost 18, meaning that after four transmission steps, more than 100,000 people could be infected.

Radioactive elements can introduce software errors because the ionizing radiation they produce can detach electrons from atoms, and computer code relies on the distribution of electrons.

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

Private schools are generally smaller than public schools

As of 2021, 44% of private schools enrolled fewer than 300 students. By comparison, over 60% of public schools enroll more than 500 students.

Today's massive data centers evolved from single-room mainframes

Punch card computers from the late 1940s were used to store and process a company's data locally before the advent of desktop computers. As the amount of data and company sizes increased alongside the miniaturization of technology, mainframes were replaced by local server rooms and external facilities with interconnected servers running cloud-based and AI software.

Fractional reserve banking creates deposits from loans, not the other way around

The traditional view of banking is that customers deposit money into the bank and the bank lends out those deposits. That's partly true, but most banks work by making loans first, which creates an asset on their balance sheet as a corresponding deposit. Banks often operate under a money multiplier ratio, lending out several times their total reserves.

Some coin denominations have ridges etched on their sides, partly to prevent counterfeiting, in a practice called 'reeding.'

Clever Hans was a 1900s horse that was believed to be able to solve math equations, tell time, and read and comprehend German, until a psychology student discovered how he got his answers.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.

Art theft is the third-largest source of criminal revenues worldwide, thanks in part to the subjective nature of valuing a work of art.

Hedge funds, the private investment firms that trade a wide swath of financial assets, make money by imposing steep fees on investors and from the fund's investment performance.

Watch how algorithms can be designed to train themselves and self-improve without human input, to the point where the optimized code is too complex for programmers to understand.

Calculate how much of your tax dollars go to each federal program based on your income, filing status, and number of children

Larger animals like giraffes, elephants, and blue whales have developed a unique defense against cancer, a phenomenon known as Peto's Paradox

SpaceX's S-1 or 'prospectus' revealed many previously unknown details about the company, including its unprofitability despite doing $18.7B in revenue in 2025, exactly how many voting shares Elon Musk will have in the company (roughly 85%), and more.

According to the physics of black holes, our 4D reality may be a hologram—a projection of information coded on a 2D surface on the edge of the universe.

Elk Cloner, the first computer virus to spread in public, was written as a practical joke by Richard Skrenta, a Pittsburgh high school student.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

Trace how the Bill of Rights, inspired by centuries of political theory, took shape between 1774 and 1789.

Waves off the coast of Nazaré, Portugal, can reach over 100 feet and are caused by the area's deep underwater canyon geography

Fireworks make a loud booming sound after exploding due to air expanding faster than the speed of sound.

Alzheimer's can lead to brain shrinkage, with an average of around 2% of volume reduction per year

Scientists can determine Earth's past climate conditions by analyzing air trapped in microscopic bubbles within ice cores—columns of compressed snow layers that build up over hundreds of thousands of years.

The first steps to rebuilding your credit score include getting to know exactly what's on your credit report and figuring out how far you are from the '700 club,' or 'good score' range.

Up to a million Girl Scouts sell about 200 million cookie boxes every spring, generating $800 million, largely because refusing a child is frowned upon

In 2025, four executives from large tech companies were given positions in the US Army Reserves—including Palantir Chief Technology Officer Shyam Sankar, which reinforced Palantir's ties with the US military and its role as a leading defense partner.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

Studies suggest when you sleep is as important as how much you get, and syncing with your circadian rhythms may lead to less fatigue and improved long-term health outcomes

During the French Revolution, a botanist's letter asking about "dear Rhus" (poison ivy) was intercepted, leading authorities to suspect he was conspiring with the Russians.