Everything about Integer Factorization totally explained
In
number theory,
integer factorization is the process of breaking down a
composite number into smaller non-trivial
divisors, which when multiplied together equal the original integer.
When the numbers are very large, no efficient integer
factorization algorithm is publicly known; a recent effort which factored a 200-digit number (
RSA-200) took eighteen months and used over half a century of computer time. The presumed difficulty of this problem is at the heart of certain algorithms in
cryptography such as
RSA. Many areas of
mathematics and
computer science have been brought to bear on the problem, including
elliptic curves,
algebraic number theory, and
quantum computing.
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are
semiprimes, for example the product of two distinct
prime numbers. When they're both large, randomly chosen, and about the same size (but not too close), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical.
Prime decomposition
By the
fundamental theorem of arithmetic, every positive integer greater than one has a unique prime factorization. However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence.
Given an algorithm for integer factorization, one can factor any integer down to its constituent
prime factors by repeated application of this algorithm.
Practical applications
The hardness of this problem is at the heart of several important cryptographic systems. A fast integer factorization algorithm would mean that the
RSA public-key algorithm is insecure. Some cryptographic systems, such as the
Rabin public-key algorithm and the
Blum Blum Shub pseudo-random number generator can make a stronger guarantee — any means of breaking them can be used to build a fast integer factorization algorithm; if integer factorization is hard, then they're strong. In contrast, it may turn out that there are attacks on the
RSA problem more efficient than integer factorization, though none is currently published.
A similar hard problem with cryptographic applications is the
discrete logarithm problem.
Even in the absence of cryptographic systems based on its hardness, integer factorization also has many positive applications in algorithms. For example, once an integer
n is placed in its prime factorization representation, it enables the rapid computation of
multiplicative functions on
n. It can also be used to save storage, since any
multiset of prime numbers can be stored without loss of information as its product; this was exploited, for example, by the
Arecibo message.
Current state of the art
A team at the German Federal Agency for Information Technology Security (
BSI) holds the record for factorization of
semiprimes in the series proposed by the
RSA Factoring Challenge sponsored by
RSA Security. On
May 9,
2005, this team announced factorization of
RSA-200, a 663-bit number (200 decimal digits), using the
general number field sieve.
The same team later announced factorization of
RSA-640, a smaller number containing 193 decimal digits (640 bits), on
November 4,
2005.
Both factorizations required several months of computer time using the combined power of 80
AMD Opteron CPUs.
Difficulty and complexity
If a large,
b-
bit number is the product of two primes that are roughly the same size, then no
algorithm has been published that can factor in
polynomial time,
for example, that can factor it in time
O(
bk) for some constant
k. There are published algorithms that are faster than
O((1+ε)
b) for all positive ε,
for example, sub-exponential.
The best published asymptotic running time is for the
general number field sieve (GNFS) algorithm, which, for a
b-bit number n, is:
»
For an ordinary computer, GNFS is the best published algorithm for large
n (more than about 100 digits). For a
quantum computer, however,
Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. This will have significant implications for cryptography if a large quantum computer is ever built.
Shor's algorithm takes only O(
b3) time and O(
b) space on
b-bit number inputs. In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15.
When discussing what
complexity classes the integer factorization problem falls into, it's necessary to distinguish two slightly different versions of the problem:
- The function problem version: given an integer N, find an integer d with 1 < d < N that divides N (or conclude that N is prime). This problem is trivially in FNP and it's not known whether it lies in FP or not. This is the version solved by most practical implementations.
- The decision problem version: given an integer N and an integer M with 1 ≤ M ≤ N, does N have a factor d with 1 < d < M? This version is useful because most well-studied complexity classes are defined as classes of decision problems, not function problems. This is a natural decision version of the problem, analogous to those frequently used for optimization problems, because it can be combined with binary search to solve the function problem version in a logarithmic number of queries.
It isn't known exactly which
complexity classes contain the decision version of the integer factorization problem. It is known to be in both
NP and
co-NP. This is because both YES and NO answers can be trivially verified given the prime factors (we can verify their primality using the
AKS primality test, and that their product is N by multiplication). It is known to be in
BQP because of
Shor's algorithm. It is suspected to be outside of all three of the complexity classes
P,
NP-Complete, and
co-NP-Complete. If it could be proved that it's in either NP-Complete or co-NP-Complete, that would imply NP = co-NP. That would be a very surprising result, and therefore integer factorization is widely suspected to be outside both of those classes. Many people have tried to find classical polynomial-time algorithms for it and failed, and therefore it's widely suspected to be outside P.
In contrast, the decision problem "is
N a
composite number?" (or equivalently: "is
N a
prime number?") appears to be much easier than the problem of actually finding the factors of
N. Specifically, the former can be solved in polynomial time (in the number
n of digits of
N) with the
AKS primality test. In addition, there are a number of
probabilistic algorithms that can test primality very quickly if one is willing to accept the small possibility of error. The ease of
primality testing is a crucial part of the
RSA algorithm, as it's necessary to find large prime numbers to start with.
Factoring algorithms
Special-purpose
A special-purpose factoring algorithm's running time depends on the properties of its unknown factors: size, special form, etc. Exactly what the running time depends on varies between algorithms.
Trial division
Pollard's rho algorithm
Algebraic-group factorisation algorithms amongst which are Pollard's p − 1 algorithm, Williams' p+1 algorithm and Lenstra elliptic curve factorization
Fermat's factorization method
Euler's factorization method
Special number field sieve
General-purpose
A general-purpose factoring algorithm's running time depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
Dixon's algorithm
Continued fraction factorization (CFRAC)
Quadratic sieve
General number field sieve
Shanks' square forms factorization (SQUFOF)
Other notable algorithms
Shor's algorithm, for quantum computersFurther Information
Get more info on 'Integer Factorization'.
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