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.

The mathematics of quantum-based cryptography, and why existing data is already at risk

Seeing the potential of this technology, some governments and malicious actors are thought to have already started storing encrypted data, in the hopes that quantum computers within the next 10 to 20 years will be able to decrypt it. As of 2022, the National Institute of Standards and Technology had selected four algorithms for inclusion in a quantum-resistant encryption standard.

Watch how the Enigma machine encrypts messages and learn about its critical design flaw

Using settings selected from a series of rotors and a plugboard that let operators swap letters with any of their choosing, the machine lit up the output letter for each letter typed. While this system could generate messages requiring any of 150 quintillion potential keys, the machine's reflector never encrypted a letter as itself, providing an entry point for exploiting the system.

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.

Create your own cipher by combining mathematical operations and encryption protocols

This tool allows users to build an encryption workflow in which the output of one customizable function becomes the input to another, thereby creating undiscovered methods for obscuring information via a series of ciphers.

Claude Shannon, for whom Anthropic's AI tools are named, proved the existence of an unbreakable cipher

In the one-time pad approach, the encryption key is randomly generated and of equal length to the plaintext. This randomness ensures that every possible plaintext is equally likely—RBBHN can mean HELLO, SOLVE, CODES, and so on, using a different key—while matching the message and key lengths prevents the appearance of patterns that can be exploited.

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.

How hash functions encrypt stored passwords and can be enhanced with 'salt'

Hash functions convert text into unique, random sets of characters, a process that is computationally expensive and time-consuming to undo. Because hashed outputs for many common passwords are publicly listed, some sites use a salt—additional characters appended to a password—before hashing to produce an output that is not known. When entering a password, it is salted and hashed, then checked against the database for a matching hash before permitting login.

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.

Passkeys, a new authentication method, aim to replace traditional passwords

Passkeys use cryptography keys to secure logins without passwords, resisting phishing and risks from data breaches. The FIDO Alliance developed passkeys as a combination of public and private keys for authentication.

In mathematics, a 'perfect number' is equal to the sum of the numbers it is divisible by, excluding itself, such as 28, which can be evenly divided by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 equals 28.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

View a list of the 25 most common passwords according to an analysis of 2.5 terabytes of public data breaches.

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...

A look at historical ciphers, from the Spartan 'scytale' to the Navajo Code Talkers

From the Greek word for "hidden writing," cryptography in Ancient Greece took the form of batons around which a long strip of leather or parchment was wrapped, after which messages were written. Recipients needed a scytale of the correct width to encircle received ribbons and reveal the message. Members of the Navajo Nation created a code that remained unbroken throughout its use in World War II.

Practice encrypting and decrypting text using an online library of ciphers

This tool provides users with customizable protocols, including Caesar, Affine, and Vigenère ciphers, to hide messages in transformed text and convert them back into legible formats. Mechanisms are also available to convert text into other communication languages, such as Morse code and URL encoding.

Quantum computers can simulate nature more effectively than classical computers

Ordinary computers struggle to simulate quantum fields due to entanglement, but systems built from those elements can model the complexity. Systems using particle in more than two states—qudits—have successfully modeled 2D electromagnetic fields and done so more efficiently.

Although there are 10,000 possible combinations of four-digit personal identification numbers, data suggests that about 27% of all PINs are one of 20 combinations, including 1234, 1111, and 0000.

The first algorithmic random number generator was developed in 1946

Although previous methods of randomness involved physical tools with limited outcomes, such as dice, or relied on chaotic variables for number generation, such as electronic noise, John Von Neumann's middle-square method turned the problem into a mathematical computation. It has since been replaced by methods that generate numbers from unique initial values, such as the time on a clock.

Quantum computers harness weird physics to explore unsolvable problems

Classical computers manipulate bits as either 0 or 1, but quantum computers use qubits to hold combinations of both, enabling new algorithms. This new approach could one day help with breakthroughs in medicine, chemistry, and AI.

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).

Quantum computers use qubits to solve problems much faster than classical machines

Unlike regular computers, where each bit is either 0 or 1, quantum computers use qubits, which can simultaneously be both—allowing for massive parallel processing power. Google’s quantum processor once solved a problem in seconds that would have taken a supercomputer 47 years.

Mathematicians often solve problems using wordless mental processes

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

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.

In recent years, NIST has suggested that passwords no longer be created using a mix of symbols, numbers, and capital and lowercase letters, as was once recommended.

Qubits let quantum computers process massive datasets all at once

The quantum phenomena of superposition and entanglement enable qubits to act in 0 and 1 binary states simultaneously and influence each other instantly. These features give quantum computers incredible power, though keeping them stable is a major challenge.

Calculators convert input values into a form of scientific notation—a fixed set of digits with a decimal multiplied by a base to some exponent—to store and process very large or very small quantities efficiently at the cost of rounding errors.

Trick a large language model into revealing secret passwords over multiple rounds, each with increasing safeguards.

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.

The Majorana 1 chip may provide quantum computing using geometry-based qubits

Traditional quantum computers are incredibly fragile, with encoded information easily disrupted and requiring significant error correction systems. Microsoft’s approach involves storing data on a surface, which is more stable, but involves an elusive particle first theorized in 1938.

Mathematical breakthroughs influenced Shakespeare’s stories

New ideas in math were transforming Europe at the time Shakespeare penned his plays, including newly popular conceptions of zero via philosophers Descartes and Leibniz.

See how computer memory really works—from binary bits to modern RAM

The video explains how computers store information as 1s and 0s, how RAM functions as fast working memory, and why systems utilize different types of storage. The video demonstrates how this process enables computers to operate.

Microchips in payment cards improve security by generating unique transaction codes

Magnetic strips store unchanging data that is easily cloned, but microchips act as miniature computers that encrypt sensitive data. They also resist fraud by creating data unique to each transaction, preventing duplication.

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.

A breakdown of algorithms, how they are constructed, and their applications

Algorithms are instructions for accomplishing a goal and typically take the form of a step-by-step procedure. Given the conditions, these procedures can take various paths, increasing an algorithm's complexity. While traditional computer algorithms are programmed to account for all possible scenarios within a given process, machine learning systems build their own rules from data-driven patterns.

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.

One of the first mechanical calculators—the Pascaline—was built in the 1640s by French mathematician Blaise Pascal to help his father, who worked as a tax collector.

Onion routing disguises internet traffic in layers resembling a nesting doll

Data is encrypted at least three times before being sent to a node—a server that can only decrypt one layer. Then, the underlying encrypted data is relayed to another node, and the process repeats until all layers are decrypted. Thousands of possible nodes make it extremely difficult to trace activity within the system.

Compression algorithms can reduce file sizes without losing quality by consolidating repeated data or replacing blocks of code with more compact representations, thereby reducing storage demands and improving data transmission rates.

Misunderstandings about how quantum computing works may have overhyped their ability

Quantum computers are often viewed as machines that can solve all problems instantly by trying every solution simultaneously. Instead, they amplify correct answers and cancel out wrong ones, which only works for specific kinds of problems with special structures.

Slide rules—ruler-sized mechanical calculators with a component that slides out to show relationships between numbers—were used by engineers at NASA during the Apollo missions.

An explanation of Turing machines—the foundations of digital algorithms and computers

The device reads data, overwrites or ignores it, and then moves on to the next piece of data based on a set of programmed rules. By imagining this data as zeros and ones—binary that makes up computer code—Turing machines became the theoretical models for how any algorithm could be run on a computer if the procedure could be coded into a set of digital rules.

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.

Lasers, semiconductors, medical imaging, and atomic clocks rely on quantum mechanics

Semiconductors, crucial in electronics, operate based on quantum superposition. Lasers use stimulated emission, a quantum process, and MRI machines leverage the quantum spin of hydrogen atoms for imaging. Atomic clocks measure time with unprecedented precision using quantum transitions.