## Cryptology ePrint Archive: Report 2000/064

On the Security of Modular Exponentiation with Application to the Construction of Pseudorandom Generators

Oded Goldreich and Vered Rosen

Abstract: Assuming the inractability of factoring, we show that the output of the exponentiation modulo a composite function $f_{N,g}(x)=g^x\bmod N$ (where $N=P\cdot Q$) is pseudorandom, even when its input is restricted to be half the size. This result is equivalent to the simultaneous hardness of the upper half of the bits of $f_{N,g}$, proven by Hastad, Schrift and Shamir. Yet, we supply a different proof that is significantly simpler than the original one. In addition, we suggest a pseudorandom generator which is more efficient than all previously known factoring based pseudorandom generators.

Category / Keywords: foundations / Modular exponentiation, discrete logarithm, hard core predicates, simultaneous security, pseudorandom generator, factoring assumption.