ChatMost cryptographic systems only guarantee that a mathematical lock is hard to pick on average; Miklós Ajtai proved that breaking a random lattice lock requires an algorithm capable of solving the hardest geometric problem in the entire universe. Published in 1996, Ajtai’s worst-case to average-case reduction is the foundational mathematical theorem behind all post-quantum cryptography protecting global cybersecurity from quantum computers.
In classical cryptography, algorithms like RSA rely on factoring prime numbers, but mathematicians could never prove that every single RSA key was hard to crack—some randomly generated keys might secretly contain easy shortcuts that hackers could exploit.
Hungarian mathematician Miklós Ajtai constructed a mathematical fortress based on high-dimensional lattices. He proved a theorem: if a hacker invents an algorithm that cracks even a random, everyday lattice lock, that exact same algorithm can be used to solve the single most difficult geometric puzzle in the universe.
Ajtai’s reduction became the foundation for post-quantum security. By guaranteeing that lattice encryption contains no easy backdoor shortcuts, by resisting Shor’s quantum factoring algorithms, and by anchoring the new NIST post-quantum encryption standards (ML-KEM and Dilithium), Ajtai’s proof defends the future internet.
Generating hard instances of lattice problems (extended abstract)
. We give a random class of lattices in Z n so that, if there is a probabilistic polynomial time algorithm which finds a short vector in a random lattice with a probability of at least 1 2 then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in every lattice in Z n with a probability exponentially close to one. (1) Find the length of a shortest nonzero vector in an n-dimensional lattice, approximately, up to a polynomial factor. (2) Find the shortest nonzero vector in an n-dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length is at most n c kvk is parallel to v, where c is a sufficiently large absolute constant. (3) Find a basis b 1 ; :::; b n in the n-dimensional lattice L whose length, defined as max n i=1 kb i k, is the smallest possible up to a polynomial factor. A large number of the existing techniques of cryptography include the generation of a specific ins...
Telecommunication engineers believed that static noise permanently corrupted electrical signals; Claude Shannon proved that information can be transmitted with zero errors across noisy channels by converting meaning into binary bits. Published in 1948, Shannon’s masterwork founded information theory and created the digital language of the modern world.
Physical sensors in rockets and satellites are constantly corrupted by noise, vibration, and sensor drift; Rudolf Kalman developed an optimal mathematical filter that predicts true position by fusing imperfect sensor readings with equations of motion in real time. Published in 1960, the Kalman Filter guided the Apollo spacecraft to the Moon and is the mathematical brain powering every GPS receiver, autonomous drone, and commercial jetliner on Earth.
Computer scientists believed that mastering the ancient game of Go required human intuition that machines would not achieve for decades; DeepMind’s AlphaGo paired deep neural networks with Monte Carlo tree search to defeat the world champion. Published in 2016, AlphaGo proved that deep reinforcement learning could exhibit genuine creative intuition, sparking the modern Artificial General Intelligence revolution.
For thousands of years, secret communication required sending a physical key via a trusted courier before a message could be sent; Whitfield Diffie and Martin Hellman proved that two total strangers can agree on a shared secret over a compromised public channel. Published in 1976, this mathematical breakthrough invented public-key cryptography and made secure global e-commerce possible.
Multiplying two massive prime numbers takes a fraction of a second on a computer; factoring that product back into its original primes takes millions of years. By anchoring asymmetric encryption to the asymmetry of prime numbers, Ron Rivest, Adi Shamir, and Leonard Adleman created RSA—the cryptographic algorithm that encrypts every browser connection, banking transfer, and software update on Earth.
Albert Einstein spent his final decades insisting that quantum entanglement was an incomplete illusion masking hidden common-sense variables; John Stewart Bell devised a single mathematical inequality that allowed laboratory experiments to test whether Einstein or quantum mechanics was right. Published in an obscure, short-lived journal in 1964, Bell’s Theorem proved that the physical universe is fundamentally non-local—founding modern quantum computing, quantum cryptography, and quantum teleportation.
Physical clocks drift out of sync across distributed computer networks because electrical signals cannot travel faster than the speed of light; Leslie Lamport defined time through cause-and-effect causality rather than ticking seconds. Published in 1978, Lamport’s logical clock paper is the most cited work in distributed systems, establishing the foundational physics of cloud computing.
Storing millions of search queries in computer memory wastes vast amounts of expensive hardware; Bloom filters verify whether a record exists using a tiny fraction of a single bit per item. Published in 1970 to speed up hyphenation in typesetting machines, Burton Bloom’s compact data structure now speeds up every Google search query and Bitcoin transaction on Earth.
Ask this paper your own questions, or keep browsing the verified research catalogue.
