Programmatic access to LucidWorks Cloud uses "Basic Authentication" as its security mechanism.
The URL circled below is your access URL, and you'll use that as the base URL for all API calls. If you follow any of the examples in the LucidWorks REST API documentation, you'll replace http://localhost:8888 shown in examples with the access string for your instance.

Authentication is required, and uses your API access key. To find your API access key, click the 'My Account' link to see your account settings. Note, this link is only available from the My Search Server page. The API access key is listed first in the menu on the left side of the screen. The password is always 'x', and cannot be changed. If you believe your account has been compromised, please contact us to request a new access key.

As an example, let's try out the Collections API. Given the above instance URL and API access key circled above, the following curl command will return a list of the collections in the instance (indented).
Input:
curl -u v69aqeg5YbsxyxMk:x -s "https://s-18850c1d.azure.lucidworks.io/api/collections"
Output:
[
{
"instance_dir": "collection1_0",
"name": "collection1"
},
{
"instance_dir": "LucidWorksLogs",
"name": "LucidWorksLogs"
}
]
Solr API access for queries is authenticated this same way. This simple query (to return all documents) against an empty collection gives an empty query result:
Input:
curl -u v69aqeg5YbsxyxMk:x -s "https://s-18850c1d.azure.lucidworks.io/solr/collection1/select?q=*:* &indent=true&wt=json"
Output:
{
"responseHeader":{
"status":0,
"QTime":2,
"params":{
"indent":"true",
"wt":"json",
"q":"*:*"}},
"response":{"numFound":0,"start":0,"docs":[]
}}
Solr Admin and logs access
If you wish to use the Solr admin or /logs feature, the browser will give a popup which requires a name and password. Use the API access key as the login name, and 'x' as the password.