Assignment 18

Sections 7.3-7.5 elaborated on applications of discrete logorathims just like the ensuing sections in chapter six continued to elaborate on the different approaches to the RSA algorithm. The key part to all three sections was the difficulty in computing discrete logarithms just like the fundamental part of the RSA applications was the difficulty factoring large prime numbers. What struck me as interesting in this chapter was that, like the section on factoring large prime numbers, the computational diffie-hellman problem and decision diffie-hellman problem and eventually the elGamal cryptosystem all could be fundamentally boiled down to discrete logarithms but were different applications of that idea. The RSA algorithm on the other hand was much more straightforward in its application and the different approaches weren't different cryptosystems but instead were different approaches to computing the compositeness of a number (non primality). Probably the most difficult part of this reading, which was largely conceptual, was trying to find differences in security and application between the diffie-hellman and elGamal cryptosystem. I also had to reread the section on bit-commitment to convince myself that even if Eve had access to all of the communication between Alice and Bob, she would still have to compute discrete logs to find the key K.

These sections were important in a broad sense largely because they illustrated the idea that the RSA system is not the only application of a sort of one-way function for a public-key cryptosystem and that there are potentially many others. It's very possible that once technology, whether quantum computers or otherwise, can factor large primes or compute discrete logarithms for large primes, other one-way functions for public-key cryptosystems may be developed. On a side note, by far my favorite part of this reading was the mentioning basically that "this is not yet known" or "has not been developed" to describe the feasibility of computing discrete logarithms, basically implying that mathematics is still hard at work and can give us a "busy" signal every once in awhile.