I'm learning the rainbow table and hellman table and I'm curious about the difference between them, so I left a question like this.
Wikipedia describes the following sentence:
The term, "Rainbow Tables," was first used in Oechslin's initial paper. The term refers to the way different reduction functions are used to increase the success rate of the attack. The original method by Hellman uses many small tables with a different reduction function each. Rainbow tables are much bigger and use a different reduction function in each column. When colors are used to represent the reduction functions, a rainbow appears in the rainbow table. Figure 2 of Oechslin's paper contains a black-and-white graphic that illustrates how these sections are related. For his presentation at the Crypto 2003 conference, Oechslin added color to the graphic in order to make the rainbow association more clear. The enhanced graphic that was presented at the conference is shown to the right.
According to this, hellman table uses the same reduction function for each table while storing several tables of size $m\times t$.
On the other hand, the rainbow table creates one much larger table and uses a different reduction function for each column.
At this point, a few questions arise.
In the end, whichever of the two methods is used, isn't it advantageous to store a large number of elements (Of course, there will be a lot of pre-calculation.)?
Does the difference in how the reduction function is used lead to different results?
From my point of view, in general, people seem to use rainbow tables more than Hellman tables, but why?