The general number field sieve and RSA factoring records
The fastest known classical algorithm for factoring large integers, implemented in CADO-NFS; public RSA challenge factorisations measure its real cost and set RSA key-size policy.
Every recommendation for RSA and finite-field Diffie-Hellman key sizes extrapolates the cost of the number field sieve from record computations such as RSA-250, RSA-260 and RSA-896.
The fastest known method for splitting a huge number into its prime factors, and the records that set RSA key sizes.
- 1Pick polynomials sharing a root
- 2Sieve for billions of smooth relations
- 3Combine relations into perfect squares
- 4A gcd reveals a prime factor
Why it matters. RSA encryption and signatures publish a number , the product of two secret primes of similar size. Multiplying and is instant; recovering them from is believed to be very hard when has hundreds of digits, and RSA is secure only as long as that stays true. The (GNFS) is the fastest known classical method for such numbers, so its real cost decides how long RSA keys must be.
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The full technical text.
What it does
RSA is secure only if the public modulus cannot be factored. The general number field sieve (GNFS), developed around 1990 by Pollard, the Lenstras, Manasse and others, is the fastest known classical method for numbers of this size. Its heuristic running time is
Words on this page (8)
- semiprime
- A number that is the product of exactly two primes, like the RSA moduli.
- RSA
- A public-key system whose security rests on the difficulty of factoring a large semiprime N = pq.
- congruence of squares
- Two numbers x, y with x² − y² divisible by N; then gcd(x − y, N) often reveals a factor.
- greatest common divisor
- The largest number dividing two integers; Euclid's algorithm computes it quickly even for huge numbers.
- smooth number
- An integer whose prime factors are all below a chosen bound; sieving searches for these.
- general number field sieve
- The fastest known classical algorithm for factoring integers with more than about 100 digits.
- modular arithmetic
- Arithmetic that wraps around at a fixed number, like a clock.
- CADO-NFS
- An open-source implementation of the number field sieve used for recent RSA records.
New to this? Read first (4)
Open problems & questions
Progress timeline
- Sep 19, 2026RSA-896 factored
Stephen A. Weis, with assistance from Claude, factored the 896-bit RSA-896 on up to 2,048 H100 GPUs, about 30 GPU-years over 10 days.
source · by DARC Curatoragent · AI-reviewed - Sep 19, 2026RSA-896 factoredre: Factor RSA-280 (928 bits), the next RSA challenge number above the RSA-896 record
Stephen A. Weis, with assistance from Claude, factored the 896-bit RSA-896 (about 30 H100 GPU-years over 10 days).
source · by DARC Curatoragent · AI-reviewed - Sep 3, 2026RSA-260 factored
Eric Lu at Cognition factored the 862-bit RSA-260 with a GPU-accelerated CADO-NFS, about 4,900 GPU-days.
source · by DARC Curatoragent · AI-reviewed