Ask Mr. Smarty Plants

The University of Arizona has a very good article, Problems and Pests of Agave, Aloe, Cactus and Yucca, that you can read to determine what is on your cactus, but I think this sounds like cochineal bugs (Dactylopius coccus).  You can test this by using a stick (a pencil will work) and pushing it into the white areas and pressing.  If there are cochineal bugs there, you should see a red spot where you crushed one of the bugs.  Here is another article from Cacti Guide with descriptions of Cacti Pests & Diseases.

This turns out to be a question we have answered several times before.  Here is what I wrote on the subject of cochineal bugs on cactus several years ago:

"Your cactus sounds as if it is infested with cochineal bugs (Dactylopius sp.).  They are small scale insects that feed on the cactus.  They produce fluffy white wax that hides their bodies as they feed on the cactus and protects them from the elements (drying out, in particular) and from predation.  The fluffy wax also serves as a sail or balloon to float on the wind and take the bugs to a new patch of cactus.  The bugs produce carminic acid that also helps protect them from predation, especially from ants. This carminic acid in the bugs has been used by indigenous peoples of southwestern North America, Central America and sub-tropical South America to make a brilliant red dye for centuries, perhaps millennia, to produce beautifully colored textiles. Originally, the cochineal bugs were limited to the New World.  When the European explorers visited and saw the beautiful red cloth of the natives, they took the cochineal bugs back with them and now they occur all over the world.  When a synthetic red dye was produced the demand for cochineal bugs decreased, although they have also been used to help control cactus populations.  Recently, however, after it was determined that the synthetic red dyes can have adverse health side effects, there has been a renewed  interest in growing cochineal bugs for red dye.  The dye made from the bugs is currently used in cosmetics and as food coloring.  Because of this, controlling cochineal bugs hasn't really been a priority and, therefore, there isn't a lot of information that I have been able to find for controlling them.  However, click here to read what one grower recommends.  If your infestation is small, I suggest scraping them off (carefully, to avoid the sharp cactus spines) and disposing of them.  You might also be able to wash them off with a water under pressure.  Test a small area first to be sure that you don't injure your cactus and gather up and dispose of any of the insects that you wash off the cactus.  Wayne's World has excellent photos of the bugs."

The Cactus Doctor has an article called Cochineal Eradication and here is more information about the bug from PestProducts.com.

Interestingly, the cochineal insects are used in southern Africa to control invasive cactus plants so there interest is preserving and increasing the number of cochineal bugs on the cactus—probably NOT what you are interested in doing.