Smooth number

In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers. 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.

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, despite the fact that they miss out prime factors 3 and 5 respectively. 5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are also called humble, and sometimes called highly composite,^[2] although this conflicts with another meaning of highly composite numbers.

Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work 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 ${\displaystyle \Psi (x,B)}$:

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

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

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

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

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]

Powersmooth numbers

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

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

For example, 720 (2⁴3²5¹) is 5-smooth but is not 5-powersmooth (because there are several prime powers greater than 5, e.g., $3^{2}=9\nleq 5$ or $2^{3}>5$). 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. 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(B^1/2) 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 $\mathbb {Z}$; 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]

See also

• Rough number
• Størmer's theorem
• Highly composite number

Notes

1. M. E. Hellman, J. M. Reyneri, "Fast computation of discrete logarithms in GF (q)", in Advances in Cryptology: Proceedings of CRYPTO '82 (eds. D. Chaum, R. Rivest, A. Sherman), New York: Plenum Press, 1983, p. 3–13, online quote at Google Scholar: "Adleman refers to integers which factor completely into small primes as “smooth” numbers."
2. "Sloane's A002473 : 7-smooth numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Aaboe, Asger (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", Journal of Cuneiform Studies, 19 (3): 79–86, doi:10.2307/1359089, MR 0191779.
4. Longuet-Higgins, H. C. (1962), "Letter to a musical friend", Music Review (August): 244–248.
5. Dijkstra, Edsger W. (1981), Hamming's exercise in SASL (PDF), Report EWD792. Originally a privately circulated handwitten note.
6. David Naccache, Igor Shparlinski, "Divisibility, Smoothness and Cryptographic Applications", http://eprint.iacr.org/2008/437.pdf
7. Tanaka, Keisuke; Suga, Yuji (2015-08-20). Advances in Information and Computer Security: 10th International Workshop on Security, IWSEC 2015, Nara, Japan, August 26-28, 2015, Proceedings. Springer. pp. 49–51. ISBN 9783319224251.
8. https://www.math.vt.edu/people/brown/doc/briggs_gnfs_thesis.pdf

References

• G. Tenenbaum, Introduction to analytic and probabilistic number theory, (AMS, 2015) ISBN 978-0821898543
• A. Granville, Smooth numbers: Computational number theory and beyond, Proc. of MSRI workshop, 2008

External links

• Weisstein, Eric W. "Smooth Number". MathWorld.

The On-Line Encyclopedia of Integer Sequences (OEIS) lists B-smooth numbers for small Bs:

• 2-smooth numbers: A000079 (2ⁱ)
• 3-smooth numbers: A003586 (2ⁱ3^j)
• 5-smooth numbers: A051037 (2ⁱ3^j5^k)
• 7-smooth numbers: A002473 (2ⁱ3^j5^k7^l)
• 11-smooth numbers: A051038 (etc...)
• 13-smooth numbers: A080197
• 17-smooth numbers: A080681
• 19-smooth numbers: A080682
• 23-smooth numbers: A080683