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 prime factors 3 and 5 respectively. All 5-smooth numbers are of the form 2^a × 3^b × 5^c, where a, b and c are positive integers or zero.

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 B ≥ p. 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 $\Psi (x,y)$ 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 $\Psi (x,B)$ :

\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 $\rho (u)$ is the Dickman function.

The average size of the smooth part of a number of given size is known as $\zeta (u)$ and it is known to decay much more slowly than $\rho (u)$ .^[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 $\scriptstyle p^{\nu }$ 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 $\phi :\mathbb {Z} [\theta ]\to \mathbb {Z} /n\mathbb {Z}$ .^[8]

Notes

↑ Hellman, M. E.; Reyneri, J. M. (1983). "Fast computation of discrete logarithms in GF (q)". Advances in Cryptology: Proceedings of CRYPTO '82 (eds. D. Chaum, R. Rivest, A. Sherman). New York: Plenum Press: 3–13. doi:10.1007/978-1-4757-0602-4_1.
↑ Sloane, N.J.A. (ed.). "Sequence A002473 (7-smooth numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ 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 .
↑ Longuet-Higgins, H. C. (1962), "Letter to a musical friend", Music Review (August): 244–248 .
↑ Dijkstra, Edsger W. (1981), Hamming's exercise in SASL (PDF), Report EWD792. Originally a privately circulated handwritten note .
↑ Naccache, David; Shparlinski, Igor (17 October 2008). "Divisibility, Smoothness and Cryptographic Applications" (PDF). eprint.iacr.org. Retrieved 26 July 2017. f
↑ Tanaka, Keisuke; Suga, Yuji (20 August 2015). 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.
↑ Briggs, Matthew E. (17 April 1998). "An Introduction to the General Number Field Sieve" (PDF). math.vt.edu. Blacksburg, Virginia: Virginia Polytechnic Institute and State University. Retrieved 26 July 2017.

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

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[1] Hellman, M. E.; Reyneri, J. M. (1983). "Fast computation of discrete logarithms in GF (q)". Advances in Cryptology: Proceedings of CRYPTO '82 (eds. D. Chaum, R. Rivest, A. Sherman). New York: Plenum Press: 3–13. doi:10.1007/978-1-4757-0602-4_1.

[2] Sloane, N.J.A. (ed.). "Sequence 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 handwritten note .

[6] Naccache, David; Shparlinski, Igor (17 October 2008). "Divisibility, Smoothness and Cryptographic Applications" (PDF). eprint.iacr.org. Retrieved 26 July 2017. f

[7] Tanaka, Keisuke; Suga, Yuji (20 August 2015). 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] Briggs, Matthew E. (17 April 1998). "An Introduction to the General Number Field Sieve" (PDF). math.vt.edu. Blacksburg, Virginia: Virginia Polytechnic Institute and State University. Retrieved 26 July 2017.

Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant