phi-hiding assumption

The phi-hiding assumption or Φ-hiding assumption is an assumption about the difficulty of finding small factors of φ(m) where m is a number whose factorization is unknown, and φ is Euler's totient function. The security of many modern cryptosystems comes from the perceived difficulty of certain problems. Since P vs. NP problem is still unresolved, cryptographers cannot be sure computationally intractable problems exist. Cryptographers thus make assumptions as to which problems are hard. It is commonly believed that if m is the product of two large primes, then calculating φ(m) is currently computationally infeasible; this assumption is required for the security of the RSA cryptosystem. The Φ-hiding assumption is a stronger assumption, namely that if p₁ and p₂ are small primes exactly one of which divides φ(m), there is no polynomial-time algorithm which can distinguish which of the primes p₁ and p₂ divides φ(m) with probability significantly greater than one-half.

This assumption was first stated in the 1999 paper titled Computationally Private Information Retrieval with Polylogarithmic Communication,{{cite book |last1=Cachin |first1=Christian |last2=Micali |first2=Silvio|last3=Stadler|first3=Markus|title=Advances in Cryptology — EUROCRYPT '99 |chapter=Computationally Private Information Retrieval with Polylogarithmic Communication |series=Lecture Notes in Computer Science |year=1999|publisher=Springer|volume= 1592|pages=402–414|doi=10.1007/3-540-48910-X_28|editor1-last=Stern |editor1-first=Jacques|isbn=978-3-540-65889-4 |s2cid=29690672 }} where it was used in a private information retrieval scheme.