Smooth number

In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers. For example, a 7-smooth number is a number whose prime factors are all at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not. The term seems to have been coined by Leonard Adleman.[1] Smooth numbers are especially important in cryptography relying on factorization. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers.

Definition

A positive integer is called B-smooth if none of its prime factors is greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5 respectively. All 5-smooth numbers are of the form 2a × 3b × 5c, where a, b and c are non-negative integers.

5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are also called humble numbers, and sometimes called highly composite,[2] although this conflicts with another meaning of highly composite numbers.

Note that B itself is not required to appear among the factors of a B-smooth number. If the largest prime factor of a number is p then the number is B-smooth for any Bp. Usually B is prime, but composite numbers are permitted as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.

Applications

An important practical application of smooth numbers is for fast Fourier transform (FFT) algorithms (such as the Cooley–Tukey FFT algorithm) that operate by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)

5-smooth or regular numbers play a special role in Babylonian mathematics.[3] They are also important in music theory,[4] (see Limit (music)) and the problem of generating these numbers efficiently has been used as a test problem for functional programming.[5]

Smooth numbers have a number of applications to cryptography.[6] Although most applications involve cryptanalysis (e.g. the fastest known integer factorization algorithms), the VSH hash function is one example of a constructive use of smoothness to obtain a provably secure design.

Distribution

Let ${\displaystyle \Psi (x,y)}$  denote the number of y-smooth integers less than or equal to x (the de Bruijn function).

If the smoothness bound B is fixed and small, there is a good estimate for ${\displaystyle \Psi (x,B)}$ :

${\displaystyle \Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}.}$

where ${\displaystyle \scriptstyle {\pi (B)}}$  denotes the number of primes less than or equal to ${\displaystyle \scriptstyle B}$ .

Otherwise, define the parameter u as u = log x / log y: that is, x = yu. Then,

${\displaystyle \Psi (x,y)=x\cdot \rho (u)+O\left({\frac {x}{\log y}}\right)}$

where ${\displaystyle \rho (u)}$  is the Dickman function.

The average size of the smooth part of a number of given size is known as ${\displaystyle \zeta (u)}$  and it is known to decay much more slowly than ${\displaystyle \rho (u)}$ .[7]

For any k, almost all natural numbers will not be k-smooth.

Powersmooth numbers

Further, m is called B-powersmooth (or B-ultrafriable) if all prime powers ${\displaystyle \scriptstyle p^{\nu }}$  dividing m satisfy:

${\displaystyle p^{\nu }\leq B.\,}$

For example, 720 (243251) is 5-smooth but is not 5-powersmooth (because there are several prime powers greater than 5, e.g., ${\displaystyle 3^{2}=9\nleq 5}$  and ${\displaystyle 2^{4}=16\nleq 5}$ ). It is 16-powersmooth since its greatest prime factor power is 24 = 16. The number is also 17-powersmooth, 18-powersmooth, etc.

B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm and ECM. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-powersmooth for some unspecified small number B; as B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O(B1/2) for groups of B-smooth order.

Smooth over a set A

Moreover, m is said to be smooth over a set A if there exists a factorization of m where the factors are powers of elements in A. For example, 12=4*3, 12 is smooth over sets such as A1={4,3}, A2={2,3}, and ${\displaystyle \mathbb {Z} }$ ; however it would not be smooth over the set A3={3,5} as 12 contains the factor 4 or 22 which is not in A3.

Note the set A does not have to be a set of prime factors, but it is typically a proper subset of the primes as seen in the factor base of Dixon's factorization method and the Quadratic sieve. Likewise it is what the General number field sieve uses to build its notion of smoothness under the homomorphism ${\displaystyle \phi :\mathbb {Z} [\theta ]\to \mathbb {Z} /n\mathbb {Z} }$ .[8]