Blum integer

Quality 0.50 · 2 views · Updated 2 months ago

Product of two distinct primes ≡ 3 (mod 4)

title: "Blum integer" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integer-sequences"] description: "Product of two distinct primes ≡ 3 (mod 4)" topic_path: "general/integer-sequences" source: "https://en.wikipedia.org/wiki/Blum_integer" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Product of two distinct primes ≡ 3 (mod 4) ::

In mathematics, a natural number n is a Blum integer if is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t + 3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are :21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ...

The integers were named for computer scientist Manuel Blum.

Properties

Given a Blum integer, Q**n the set of all quadratic residues modulo n and coprime to n and a ∈ Q**n. Then:

  • a has four square roots modulo n, exactly one of which is also in Q**n
  • The unique square root of a in Q**n is called the principal square root of a modulo n
  • The function f : Q**n → Q**n defined by f(x) = x2 mod n is a permutation. The inverse function of f is: f(x) = x((p−1)(q−1)+4)/8 mod n.
  • For every Blum integer n, −1 has a Jacobi symbol mod n of +1, although −1 is not a quadratic residue of n: :: \left(\frac{-1}{n}\right)=\left(\frac{-1}{p}\right)\left(\frac{-1}{q}\right)=(-1)^2=1

No Blum integer is the sum of two squares.

History

Before modern factoring algorithms, such as MPQS and NFS, were developed, it was thought to be useful to select Blum integers as RSA moduli. This is no longer regarded as a useful precaution, since MPQS and NFS are able to factor Blum integers with the same ease as RSA moduli constructed from randomly selected primes.

References

References

  1. Joe Hurd, Blum Integers (1997), retrieved 17 Jan, 2011 from http://www.gilith.com/research/talks/cambridge1997.pdf
  2. [[Shafi Goldwasser. Goldwasser, S.]] and [[Mihir Bellare. Bellare, M.]] [http://cseweb.ucsd.edu/~mihir/papers/gb.html "Lecture Notes on Cryptography"] {{Webarchive. link. (2012-04-21 . Summer course on cryptography, MIT, 1996-2001)
  3. {{Cite OEIS. A016105. Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4)
  4. (1997). "Handbook of applied cryptography". CRC Press.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

integer-sequences