Mysql – Proper use of FULLTEXT index for a column storing unique hash value

We have an InnoDB table called user, where we have a column called token (varchar(128)) where a unique md5-like value that represents whether a user is authenticated or not, is stored. This column can be filled or empty/NULL, depending on whether the user has authenticated yet or not.

Whenever an API is being called to make a change to other data, we always check whether the token sent with the API is valid or not. The query would be something like:

select * 
from `user` 
where token ='96e1e3c72ce665bc7aeb93a357801fa3296e500b1aed8016e5342e744990cd0094c98857ae5f4b0a73cc6dfebd11a8491a5ba8448e6ddff8a631eb8213c254d0'

As you can imagine, this query runs into seconds, because it is a query on a varchar field. Now I'm trying to fix the query speed.

I tried adding a FULLTEXT on the token field, but it won't help in the = query. So I'm trying to use MATCH() AGAINST() query as follows:

select * 
from `user` 
where MATCH(token) AGAINST('96e1e3c72ce665bc7aeb93a357801fa3296e500b1aed8016e5342e744990cd0094c98857ae5f4b0a73cc6dfebd11a8491a5ba8448e6ddff8a631eb8213c254d0')

But this query does not return any result. When I check use EXPLAIN, I get:

id   1
select_type  SIMPLE
table    user
type fulltext
possible_keys    token
key  token
key_len  0
ref  NULL
rows 1
Extra    Using where

Here, the key_len being 0, and rows saying 1, even though no results were returned, are suspicious to me.

Now I'm thinking even FULLTEXT isn't the solution – it doesn't seem to be working for completely unique values with no common words – not usual text.

Can you help me understand:

1. Why FULLTEXT index with MATCH() AGAINST() is not bringing any results?
2. What is the best way to index the token field and reduce query time?