In mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. By "succinct", we usually mean that we wish for the proof to be at most polynomially larger than the number of digits in the number itself (for example, if the number has b bits, the proof might contain roughly b2 bits).
Primality certificates lead directly to proofs that problems such as primality testing and the complement of integer factorization lie in NP, the class of problems verifiable in polynomial time given a solution. These problems already trivially lie in co-NP. This was the first strong evidence that these problems are not NP-complete, since if they were it would imply NP = co-NP, a result widely believed to be false; in fact, this was the first demonstration of a problem in NP intersect co-NP not known (at the time) to be in P.
Producing certificates for the complement problem, to establish that a number is composite, is straightforward; it suffices to give a nontrivial divisor. Standard probabilistic primality tests such as the Baillie-PSW primality test, the Fermat primality test, and the Miller-Rabin primality test also produce compositeness certificates in the event where the input is composite, but do not produce certificates for prime inputs.
The concept of primality certificates was historically introduced by the Pratt certificate, conceived in 1975 by Vaughan Pratt, who described its structure and proved it to have polynomial size and to be verifiable in polynomial time. It is based on the Lucas primality test, which is essentially the converse of Fermat's little theorem with an added condition to make it true:
- Suppose we have an integer a such that:
- a is coprime to n;
- an −1 ≡ 1 (mod n)
- For every prime factor q of n −1, it is not the case that a(n −1)/q ≡ 1 (mod n).
- Then, n is prime.
Given such an a (called a witness) and the prime factorization of n−1, it's simple to verify the above conditions quickly: we only need to do a linear number of modular exponentiations, since every integer has fewer prime factors than bits, and each of these can be done by exponentiation by squaring in O(log n) multiplications (see big-O notation). Even with grade-school integer multiplication, this is only O((log n)4) time; using the multiplication algorithm with best-known asymptotic running time, the Schönhage–Strassen algorithm, we can lower this to O((log n)3(log log n)(log log log n)) time, or using soft-O notation Õ((log n)3).
However, it is possible to trick a verifier into accepting a composite number by giving it a "prime factorization" of n−1 that includes composite numbers. For example, suppose we claim that n=85 is prime, supplying a=4 and n−1=6×14 as the "prime factorization". Then (using q=6 and q=14):
- 4 is coprime to 85
- 485−1 ≡ 1 (mod 85)
- 4(85−1)/6 ≡ 16 (mod 85), 4(85−1)/14 ≡ 16 (mod 85)
We would falsely conclude that 85 is prime. We don't want to just force the verifier to factor the number so a better way to avoid this issue is to give primality certificates for each of the prime factors of n−1 as well, which are just smaller instances of the original problem. We continue recursively in this manner until we reach a number known to be prime, such as 2. We end up with a tree of prime numbers, each associated with a witness a. For example, here is a complete Pratt certificate for the number 229:
- 229 (a=6, 229−1 = 22×3×19)
- 2 (known prime)
- 3 (a=2, 3−1 = 2)
- 2 (known prime)
- 19 (a=2, 19−1 = 2×32)
- 2 (known prime)
- 3 (a=2, 3−1 = 2)
- 2 (known prime)
This proof tree can be shown to contain at most values other than 2 by a simple inductive proof (based on Theorem 2 of Pratt). The result holds for 3; in general, take p > 3 and let its children in the tree be p1,...,pk. By the inductive hypothesis the tree rooted at pi contains at most values, so the entire tree contains at most:
since k ≥ 2 and p1...pk = p−1. Since each value has at most log n bits, this also demonstrates that the certificate has a size of O((log n)2) bits.
Since there are O(log n) values other than 2 and each requires at most one exponentiation to verify (and exponentiations dominate the running time), the total time is O((log n)3(log log n)(log log log n)) or Õ((log n)3), which is quite feasible for numbers in the range that computational number theorists usually work with.
However, while useful in theory and easy to verify, actually generating a Pratt certificate for n requires factoring n−1 and other potentially large numbers. This is simple for some special numbers such as Fermat primes, but currently much more difficult than simple primality testing for large primes of general form.
To address the problem of efficient certificate generation for larger numbers, in 1986 Shafi Goldwasser and Joe Kilian described a new type of certificate based on the theory of elliptic curves. This was in turn used by A. O. L. Atkin and François Morain as the basis for Atkin-Goldwasser-Kilian-Morain certificates, which are the type of certificates generated and verified by elliptic curve primality proving systems. Just as Pratt certificates are based on Lehmer's theorem, Atkin-Goldwasser-Kilian-Morain certificates are based on the following theorem of Goldwasser and Kilian (Lemma 2 of "Almost All Primes Can Be Quickly Certified"):
- Theorem: Suppose we are given:
- a positive integer n not divisible by 2 or 3;
- Mx,My,A,B in (the integers mod n) satisfying My2 = Mx3 + AMx + B and with 4A3 + 27B2 coprime to n;
- a prime .
- Then M=(Mx,My) is a non-identity point on the elliptic curve y2 = x3 + Ax + B. Let kM be M added to itself k times using standard elliptic curve addition. Then, if qM is the identity element I, then n is prime.
Technically, an elliptic curve can only be constructed over a field, and is only a field if n is prime, so we seem to be assuming the result we're trying to prove. The difficulty arises in the elliptic curve addition algorithm, which takes inverses in the field that may not exist in . However, it can be shown (Lemma 1 of "Almost All Primes Can Be Quickly Certified") that if we merely perform computations as though the curve were well-defined and do not at any point attempt to invert an element with no inverse, the result is still valid; if we do encounter an element with no inverse, this establishes that n is composite.
To derive a certificate from this theorem, we first encode Mx, My, A, B, and q, then recursively encode the proof of primality for q < n, continuing until we reach a known prime. This certificate has size O((log n)2) and can be verified in O((log n)4) time. Moreover, the algorithm that generates these certificates can be shown to be expected polynomial time for all but a small fraction of primes, and this fraction exponentially decreases with the size of the primes. Consequently, it's well-suited to generating certified large random primes, an application that is important in cryptography applications such as generating provably valid RSA keys.
Impact of PRIMES in P
Because primality testing can now be done deterministically in polynomial time using the AKS primality test, a prime number could itself be considered a certificate of its own primality. This test runs in Õ((log n)6) time. In practice this method of verification is more expensive than the verification of Pratt certificates, but does not require any computation to determine the certificate itself.
- Vaughan Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, vol.4, pp.214–220. 1975. Citations, Full-text
- Goldwasser, S. and Kilian, J. "Almost All Primes Can Be Quickly Certified." Proc. 18th STOC. pp. 316-329, 1986. Full text
- Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primality Proving." Math. Comput. 61, 29-68, 1993. At Citeseer
- Mathworld: Primality Certificate
- Mathworld: Pratt Certificate
- Mathworld: Atkin-Goldwasser-Kilian-Morain Certificate
- Vašek Chvátal. Lecture notes on Pratt's Primality Proofs. Department of Computer Science. Rutgers University. PDF version at Concordia University.
- Wim van Dam. Proof of Pratt's Theorem. (Lecture notes, PDF)