This is a response I got from their Premium Services
Hello,
This is actually a issue with the way SSL validates names containing a period, ‘.’, > character. We’ve documented this behavior here:
http://docs.amazonwebservices.com/AmazonS3/latest/dev/BucketRestrictions.html
The only straight-forward fix for this is to use a bucket name that does not contain that character. You might instead use a bucket named ‘furniture-retailcatalog-us’. This would allow you use HTTPS with
https://furniture-retailcatalog-us.s3.amazonaws.com/
You could, of course, put a CNAME DNS record to make that more friendly. For example,
images-furniture.retailcatalog.us IN CNAME furniture-retailcatalog-us.s3.amazonaws.com.
Hope that helps. Let us know if you have any other questions.
Amazon Web Services
Unfortunately your “friendly” CNAME will cause host name mismatch when validating the certificate, therefore you cannot really use it for a secure connection. A big missing feature of S3 is accepting custom certificates for your domains.
UPDATE 10/2/2012
From @mpoisot:
The link Amazon provided no longer says anything about https. I poked around in the S3 docs and finally found a small note about it on the Virtual Hosting page: http://docs.amazonwebservices.com/AmazonS3/latest/dev/VirtualHosting.html
UPDATE 6/17/2013
From @Joseph Lust:
Just got it! Check it out and sign up for an invite: http://aws.amazon.com/cloudfront/custom-ssl-domains