Blum integer

In mathematics, more specifically in number theory, a natural number "n" is a Blum integer if "n = pq" 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 4"t"+3, for some integer "t". 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, ... OEIS|id=A016105

Properties of Blum integers

Given "n" = "pq" a Blum integer, "Q"_"n" the set of all quadratic residues modulo n, and "a" ∈ "Q"_"n". Then:

*"a" has precisely 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 ^-1(x) = x^{((p-1)(q-1)+4)/8}" mod "n".A.J. Menezes, P.C. van Oorschot, and S.A. Vanstone, [http://cacr.math.uwaterloo.ca/hac/ Handbook of Applied Cryptography] ISBN 0-8493-8523-7.]
*For every Blum integer "n", -1 has a Jacobi symbol mod "n" of +1, although -1 is not a quadratic residue of "n":
:$left(frac\{-1\}\{n\}\; ight)=left(frac\{-1\}\{p\}\; ight)left(frac\{-1\}\{q\}\; ight)=(-1)^2=1$

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