On Security Preserving Reductions: Revised Terminology

Webpage for a memo by Oded Goldreich

Many of the results in Modern Cryptography are actually transformations of a basic computational phenomenon (i.e., a basic primitive, tool or assumption) to a more complex phenomenon (i.e., a higher level primitive or application). The transformation is explicit and is always accompanied by an explicit reduction of the violation of the security of the former phenomenon to the violation of the latter. A key aspect is the efficiency of the reduction. We discuss and slightly modify the hierarchy of reductions originally suggested by Levin. The levels we suggest are:

1. A reduction is strongly preserving if it guarantees $S'(n) > S(n)/poly(n)$, where $S'$ denotes the security of the complex application and $S$ denotes the security of the basic tool.
2. A reduction is linearly-preserving if, for some constant $c\geq1$, it guarantees
   $$S'(n) > \frac{S(n/c)}{poly(n)}$$
3. A reduction is polynomially-preserving if, for some constants $c\geq1$ and $e>0$, it guarantees
   $$S'(n) > \frac{(S(n/c))^e}{poly(n)}$$
4. A reduction is weakly-preserving if, for some constants $c,d,e>0$, it guarantees
   $$S'(n) > \frac{(S(c n^d))^e}{poly(n)}$$