Phi-hiding assumption
Open-knowledge reference entry
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, where it was used in a private information retrieval scheme.
Applications
The phi-hiding assumption has found applications in the construction of a few cryptographic primitives.
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This entry incorporates text from “Phi-hiding assumption” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.