Four Kinds of Can’t
Multiply two primes, each about 309 digits long, and an ordinary laptop finishes in a fraction of a second. Double their length and the work roughly quadruples. That is the flat line.
A quantum computer gains nothing worth having here. Multiplying was never the hard direction.
Run it backwards: take the 617-digit product and recover the two primes. The cost of the best known method grows faster than any fixed power of the number's length. RSA-250 took about 2,700 core-years, meaning one processor core working for 2,700 years.
Peter Shor's 1994 procedure, run on a machine obeying quantum rules, bends that curve into a slope: the cost grows roughly with the cube of the length. Craig Gidney's 2025 estimate puts RSA-2048 at fewer than a million noisy qubits, or quantum bits, running for under a week. Nobody has built that machine.
More hardware only moves you along the curve. A million computers running for a million years sit on the horizontal line, and the classical curve passes above it well before 617 digits.
Nothing about arithmetic changed. The curve moved because a different piece of physics does the work: interference, in which quantities called amplitudes add together or cancel out.
Illustrative shapes, not measurements. Work is on a logarithmic scale, so each gridline stands for ten billion times the one below it. The classical curve follows the general number field sieve, the best known classical factoring method, pinned to the 2020 RSA-250 record. The quantum curve follows the cube of the number's length. The million-computer line assumes a billion operations per second each.