The cloud trend brought its problems. While in the classical system your servers were near to you, so the side-channel capability was limited. They were mostly on the smart-card systems where the card reader is malicious.
There are side-channel attacks on the cloud if the attacker can co-locate with you for this you need to use a shared machine to reduce your cost. Co-locating is a real problem since you are not selecting your cloud machine when you use shared hardware where virtualization happens.
Assume the attacker located you, so both you and the attacker use the same hardware. What they can do? Cache-attack on the AES encryption. This attack works if you encrypt your files on the server and use software AES (not AES-NI) that was using T tables for the encryption.
During your encryption the attacker runs their process, fills all of the cache lines, waits, your AES runs, and they enter again, then they check the cache lines. If a line still has attacker's data this implies the cache line is not used during the encryption, if changed then it is used during the encryption. Cache hit and cache miss. After many many steps, the key is extracted. How cache works see B degnan's answer.
2007 - ACI ICMEZ, O. Yet another microarchitectural attack: exploiting
2012 - WEISS, M., HEINZ, B., AND STUMPF, F. A cache timing attack
on AES in virtualization environments
2016 - Ashokkumar C.; Ravi Prakash Giri; Bernard Menezes (2016). "Highly Efficient Algorithms for AES Key Retrieval in Cache Access Attacks"
2018 - Mehmet S. Inci; Berk Gulmezoglu; Gorka Irazoqui; Thomas Eisenbarth; Berk Sunar, Cache Attacks Enable Bulk Key Recovery on the Cloud
2018 - Gorka Irazoqui; Mehmet Sinan Inci; Thomas Eisenbarth; Berk Sunar Wait a minute! A fast, Cross-VM attack on AES
2018 - Yuval Yarom; Katrina Falkner, Flush+Reload: a High Resolution, Low Noise, L3 Cache Side-Channel Attack
Now the above works if you encrypt on the cloud and use software AES.
The aim was to eliminate copies of the data in the memory so that the cloud provider can reduce the cost, and we have another attack vector.
2011 - SUZAKI, K., IIJIMA, K., YAGI, T., AND ARTHO, C. Memory deduplication as a threat to the guest.
2015 - Gorka Irazoqui, Mehmet Sinan Inci, Thomas Eisenbarth, and Berk Sunar Know Thy Neighbor: Crypto Library Detection in Cloud. PoPETs 2015, 1 (2015)
2018 - Jens Lindemann and Mathias Fischer A Memory-Deduplication Side-Channel Attack to Detect Applications in Co-Resident Virtual Machines
This helps the attacker to determine whats is going on with the neighbors ( on the shared machine).
Spectre and Meltdown
Whereas Meltdown allows unauthorized applications to read from privileged memory to obtain sensitive data from processes running on the same cloud server, Spectre can allow malicious programs to induce a hypervisor to transmit the data to a guest system running on top of it.
Spectre is the thread to the shared hardware with hypervisor and patching the Spectre is not an easy job.
If you encrypt locally and use the cloud only as a storage service, then this means you don't process your data then you are fine from this attack.
But if you want the encryption on the server, this brings problems, you may be lost your encryption key. To mitigate this, some cloud providers provide HSM on the cloud those are useful mostly for servers and of course have some additional costs.
Encrypt locally, most CPUs have AES-NI so you have fast encryption locally. Use a password manager to manage your key and use a good password like dicewire for the master password.
The problem starts when you want to access your files. Dou you want to search on them then you need a mechanism. This is another long story. Start reading from the Song's famous article about this.
In any case, Fully-Homomorphic Encryption is the holy grail of Cryptography to solve many issues about the operation on the encrypted data. Let write some applications, waits there.
An important note should be taken; due to the business consideration, the cloud-provider considered a semi-honest adversary that is semi-honest adversary obeys the protocol however, they can use all information they observed and deduce more if they can. Over time we understood that this approach is not correct. We should consider them as a covert adversary,
adversaries have the property that they may deviate arbitrarily from the protocol specification in an attempt to cheat, but do not wish to be “caught” doing so
So if you plan to store your data, even only for storage, use authenticated encryption to produce the integrity and authentication of your data. A Merkle Tree-based approach is necessary to ensure the freshness of your data ( to detect a possible rollback).