Factoring In Np And Co-Np: Understanding The Consequences

Factoring Integers in Polynomial Time

The problem of factoring large integers, or finding the prime divisors of a given composite number, has long captivated mathematicians and computer scientists. The difficulty of integer factorization is closely tied to the security of popular public-key cryptography schemes. The breakthrough discovery of efficient quantum algorithms for factorization could render schemes like RSA obsolete.

Integer factorization belongs to the complexity class NP – problems for which a solution can be verified in polynomial time. However, factoring large semi-primes (products of two primes) appears intrinsically harder than NP-complete problems which have verifiable polynomial-time solutions. Factoring is also in co-NP, the class containing problems with verifiable polynomial-time proofs of no solution. This hints that the problem may lie in an intermediate region between P and the hardest problems in NP.

The significance of factoring for cryptography

The presumed difficulty of factoring large integers underpins the security of cryptosystems like RSA, used to secure communication channels and encrypt sensitive data. Finding efficient algorithms for factorization could allow adversaries to compromise keys and decrypt secret information.

RSA’s security relies on the practical infeasibility of factoring large semi-primes – numbers with exactly two prime factors. The public and private keys are generated from prime numbers hundreds of digits long which are extremely difficult brute-force factor. Therefore, scalable quantum computers that can efficiently factor large semi-primes are an existential threat to RSA encryption.

Integer factorization and the P vs NP problem

The P vs NP problem asks whether every problem with verifiable solutions also has efficiently computable solutions. Factoring’s perceived hardness places it in the territory between problems known to have polynomial-time solutions (class P), NP-complete problems with verifiable but not known efficiently computable solutions, and co-NP problems where solutions can be efficiently verified to not exist.

Factoring is in NP since given a number’s prime divisors, verifying the factorization is straightforward. But the best known classical algorithms have superpolynomial runtimes, suggesting factoring capabilities exceed conventional computation. Factoring is also in co-NP – given factors, one can check they do not divide a number. Its presence in both NP and co-NP indicates factoring may be harder than NP-complete problems.

Integer factorization algorithms

A variety of algorithms for integer factorization have been devised, employing mathematical insights about properties of primes and integers. We briefly survey some main approaches.

Trial division

Trial division tests divisions by prime numbers up to some bound. It has very low complexity for small numbers but runtime grows exponentially with number size. Useful for numbers under 100 digits or combined with more advanced algorithms.

Fermat’s factorization method

If p divides n, then n modulo (p²-1) will be 0. We can test values of this polynomial modulo successive integers searching for a factor. Performs better than trial division but still exponential runtime in general.

Pollard’s rho algorithm

Pollard’s rho method for factoring is based on finding cycles in a pseudorandom sequence defined by a polynomial. It can factor reasonably large numbers efficiently but may sometimes fail. Pseudocode:

x = 2 # Starting value 
y = 2 # Another starting position
d = 1 # GCD result

while d == 1:
    x = f(x) # Apply iteration function
    y = f(f(y)) # Apply twice
    d = gcd(abs(x-y), n) # Check GCD with difference

if d == n:
    retry with new y 
else:
    output d

Pollard’s rho can successfully factor numbers up to 120 decimal digits on a normal desktop computer. For larger composites, more advanced algorithms are needed.

Shor’s quantum algorithm for factoring integers

Peter Shor’s 1994 quantum algorithm for integer factorization provably achieves polynomial time complexity by exploiting quantum parallelism. While current quantum computers are not sufficiently advanced to run Shor’s algorithm for large semi-primes, it underscores fears that RSA encryption will eventually fall to quantum attacks.

Shor’s insight was realizing the problem of finding periods in quantum superpositions could enable efficient factorization. The algorithm cleverly reduces factoring to finding the period of a modular exponential function relying on properties of quantum Fourier transforms. Executed on a large quantum computer, this approach would break RSA almost instantaneously.

Consequences of efficient integer factorization

The advent of practical integer factorization through Shor’s quantum algorithm or an unexpected classical breakthrough would carry profound implications for cryptography, complexity theory, and beyond.

Breaking RSA encryption

Efficient integer factorization would completely compromise RSA public-key encryption, relied upon globally across the internet for secure communication channels. All standard key lengths would become instantly breakable if factoring programs matched theoretical runtimes.

Such an advance might also enable attacks on other cryptosystems such as elliptic curve cryptography through quantum algorithms for discrete logarithms. Essentially, much current public-key infrastructure could fail catastrophically. Crypto systems backed by quantum-resistant primitives like lattices may emerge as replacements.

Implications for computational complexity theory

The longstanding inability to find polynomial-time factorization algorithms, despite enormous effort, strongly hints the problem may lie between P and NP-complete status. But an unexpected breakthrough could imply P=NP, collapsing complexity classes, and profoundly reshaping computer science.

Similarly, a mathematical proof that efficient classical factoring is impossible would indicate integer factorization is neither NP-complete nor in P. This would formally cement its intermediary complexity status in some sense “harder” than NP-complete problems whose efficient solvability remains unknown.

Open problems related to integer factoring

While quantum factorization threatens cryptography, many open questions around integer factorization remain fascinating for number theorists and computer scientists, including:

• Can integer factorization be formally shown to possess intermediate complexity between P and NP-complete problems?
• Are there classical algorithms able to achieve subexponential runtimes for factoring large semi-primes?
• Can number theory yield deeper insights into the hardness of factoring, for example through properties of modular functions?
• When will quantum computers reach sufficiently advanced states to run Shor’s algorithm and break current RSA key sizes?

The enduring mystery around the true complexity of efficient factorization continues to stimulate progress across mathematics, quantum physics, and computer science after many decades exploring this deep question.