Equihash is most commonly used to refer to Zcash’s (ZEC) POW algorithm, which can also be used to mine Komodo and Horizen. Equihash was introduced in 2016 by Alex Biryukov and Dmitry Khovratovich specifically as a memory hard POW algorithm that could be used to create ASIC resistant cryptocurrencies.
There are several variants of this algorithm distinguished by how it sets the “n” and “k” parameter of Equihash. ZCash sets n = 200 and k = 9. Other cryptocurrencies set the two value differently, such as BEAM which uses n = 150 and k = 5, Bitcoin Gold which uses n=144 and k=5, and AION which uses n=210 and k=9. For ease of reference, WariHash will refer to each Equihash variant by its most commonly mined coin (i.e, Equihash-Zcash, Equihash-BEAM).
The n and k parameters determine how much memory the algorithm uses and how long it takes to compute the solution. Thus, an ASIC optimized for one set of parameters cannot mine optimally on another set of parameters. The most common Equihash-Zcash ASIC miners, the Innosilicon A and the Bitmain Z series, can only mine Equihash-Zcash and no other variants.
Scrypt is the mining algorithm used to mine Litecoin (LTC). Scrypt was invented by Colin Percival in 2009 to be used as a memory hard cryptographic hash function. A “memory hard” function uses large amounts of memory in order to compute the end result. This hinders the development of custom hardware that can be used to brute force the function, commonly known as ASICs.
Scrypt was first used in Tenebrix in 2011, which Charlie Lee cloned to create Fairbrix. Both projects failed to gain any traction mostly due to their premine. Lee then created Litecoin as a response and the project has been a constant presence in the cryptocurrency scene since.
Along with Litecoin, Scrypt is also used to mine Dogecoin (merge mined with Litecoin), and Digibyte (in Digibyte Scrypt is one algo in a multi-algo POW system).
Although the initial goal of Scrypt was to hinder the development of ASICs, ASIC mining is now the norm and has been since around 2014. Currently, Bitmain’s L3 series is the most prevalent ASIC and much of Scrypt mining happens professionally alongside Bitcoin mining operations.