30 termsWords

Every term used on the site, in a sentence or two, in the order you are likely to meet them.

prime: A whole number bigger than 1 that nothing divides except 1 and itself. 2, 3, 5, 7, 11. There is no biggest one.
semiprime: A number that is exactly two primes multiplied together. 15 is one (3 × 5). So is every RSA key.
factor: As a verb: to split a number into the primes that multiply to make it. As a noun: one of those primes.
bits and digits: How long a number is. A 2048-bit number has 617 decimal digits. Every extra bit doubles the number; every extra ten bits, roughly, adds three digits.
RSA: The lock from 1977, named for Rivest, Shamir and Adleman. The public key is a big semiprime N; the private key is its two primes. Used for most of the web's certificates and a lot of email and software signing. Bitcoin does not use it.
elliptic curve: The shape y² = x³ + ax + b, and a rule for 'adding' two points on it to get a third. Bitcoin's curve is y² = x³ + 7, over the integers modulo a 256-bit prime. Not an ellipse; the name is an accident of history.
discrete logarithm: The wallet's lock. Start at a fixed point G on the curve, add it to itself k times, land on P. Given P, find k. Adding is quick; un-adding is the hard part. k is your private key and P is your public key.
public key and private key: The public key is the part you can show anyone: N for RSA, the point P for a wallet. The private key is the part that opens the lock: the two primes, or the count k.
address: In Bitcoin, usually a hash of your public key, not the key itself. The key is shown only when you spend. That one layer is most of what stands between old coins and a quantum machine.
hash: A one-way mash of data into a fixed-size fingerprint. Quantum computers speed up reversing a hash only a little (square root), which is why hashes are the safe side of Bitcoin.
sieve: Any method that tests a whole row of numbers at once for a property, instead of one number at a time. Eratosthenes did it for primes; the quadratic and number field sieves do it for the smooth values modern factoring needs.
smooth: A number is smooth if all its prime factors are small. 720 = 2⁴ × 3² × 5 is smooth; 719 is prime. Factoring methods since 1926 hunt for smooth values, because those are the ones you can combine.
number field sieve (NFS): The fastest known way to split a number of no special form. GNFS is the general version; SNFS is the special version for numbers like 2^n − 1. Every RSA record since 1996 used it.
core-year: One processor core running for one year. RSA-250 took 2,700 of them; RSA-896 took about 30 GPU-years, which is a different unit — a GPU is many cores.
L(1/3): The shape of the number field sieve's cost curve: exp(c · (ln N)^(1/3) · (ln ln N)^(2/3)). Sub-exponential: it grows slower than 2^bits but faster than any power of bits. The 1/3 is what makes 1024 bits a thousand times harder than 768 rather than 2^256 times.
Shor's algorithm: Peter Shor's 1994 quantum method. It finds the period of a^x mod N — the point where the sequence starts repeating — by looking at all x at once. The period gives the factors in a few lines of Euclid. Runs in a small power of the number's length instead of exponentially.
qubit: A quantum bit: something that can be in a blend of 0 and 1 until read. 'Physical' qubits are the noisy ones a machine has; 'logical' qubits are the reliable ones built out of many physical ones by error correction.
error correction, threshold: Spreading one logical qubit across many physical ones so that errors can be caught. Below a certain physical error rate — the threshold — adding more physical qubits makes the logical one better; above it, worse. Google's Willow was the first chip clearly below it (December 2024).
surface code, qLDPC: Two families of error-correcting codes. The surface code needs only neighbour-to-neighbour wiring and about a thousand physical qubits per logical one. qLDPC codes need fewer but want long-range wiring nobody has built at scale yet.
Toffoli gate: The basic expensive operation in a fault-tolerant quantum computer — a three-qubit AND. Papers count them the way a builder counts bricks. A 2048-bit RSA break is a few billion; a wallet key about 70–90 million.
Q-Day: Press shorthand for the day a quantum machine can break a real key. Not a date anyone has; the surveys give odds per decade.
harvest now, decrypt later: Recording encrypted traffic today to read it once a machine exists. It threatens secrets; it does not threaten a Bitcoin wallet, because a wallet has no encrypted message to store — only a key that is either exposed or not.
long-range and short-range attack: Long-range: the public key is already on the chain, so the attacker has all the time in the world. Short-range: the key appears only when a transaction is broadcast, so the attacker has the ten minutes or so before it is confirmed.
P2PK, P2PKH, P2TR: Bitcoin output types. P2PK (2009–2010, Satoshi-era) puts the public key in the open. P2PKH and its SegWit cousin P2WPKH show only a hash until spent. P2TR (Taproot, 2021) shows a key from the moment coins arrive.
soft fork: A change to Bitcoin's rules that old software still accepts. Every post-quantum proposal for Bitcoin is one, and none has been activated.
Riemann Hypothesis: Riemann's 1859 guess that every non-trivial zero of ζ(s) has real part exactly one half. Equivalent to the primes being spread as evenly as they could possibly be. Unproved; checked for the first ten trillion zeros; one of the seven million-dollar problems.
zeta function, ζ(s): 1 + 1/2ˢ + 1/3ˢ + 1/4ˢ + …, extended to the whole complex plane. Euler showed it equals a product over the primes, which is why its zeros know where the primes are.
critical line: The vertical line where the real part of s is one half. All hundred zeros this site computed are on it, at heights from 14.13 up to about 236.
explicit formula: Riemann's exact recipe for the count of primes in terms of the zeros: a smooth main term, minus one wave per zero. Add enough waves and the staircase of primes appears. The demo on the Riemann page does exactly that.
GRH: The Generalised Riemann Hypothesis — the same claim for a whole family of zeta-like functions. Some proofs of running time assume it. Miller's 1976 primality test is the famous one.