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Let  n{\mathbb{G}_{n}}    be the subgroup of elements of odd order in the groupℤ  n  ⋆{\mathbb{Z}^{\star}_{n}}   , and let  ⁢   (n   ){\mathcal{U}(\mathbb{G}_{n})}    be the uniform probability distribution on  n{\mathbb{G}_{n}}   . In this paper, we establish a probabilistic polynomial-time reduction from finding a nontrivial divisor of a composite number  n  to finding a nontrivial relation between  l  elements chosen independently and uniformly at random from  n{\mathbb{G}_{n}}   , wherel  ≥  1{l\geq 1}    is given in unary as a part of the input. Assume that finding a nontrivial divisor of a random number in some set  N  of composite numbers (for a given security parameter) is a computationally hard problem. Then, using the above-mentioned reduction, we prove that the family(   (n   ,(n   )   )   ∣  n  ∈  N  ){((\mathbb{G}_{n},\mathcal{U}(\mathbb{G}_{n}))\mid n\in N)}    of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble((n   )   ∣  n  ∈  N  ){(\mathcal{U}(\mathbb{G}_{n})\mid n\in N)}    is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable functionν  :   D  →  N{\nu\colon D\to N}    (whereD  ⊆    {  0  ,  1  }   *{D\subseteq\{0,1\}^{*}}   ) and a polynomial-time samplable probability ensemble(d   ∣  d  ∈  D  ){(\mathcal{G}_{d}\mid d\in D)}    (where  d{\mathcal{G}_{d}}    is a distribution on   ν  ⁢   (  d  ){\mathbb{G}_{\nu(d)}}    for eachd  ∈  D{d\in D}   ) such that the family(   (ν  ⁢   (  d  ),d   )   ∣  d  ∈  D  ){((\mathbb{G}_{\nu(d)},\mathcal{G}_{d})\mid d\in D)}    of computational abelian groups is weakly pseudo-free.

A certain family of subgroups of ℤ 𝑛 ⋆ is weakly pseudo-free under the general integer factoring intractability assumption