One common way to design invertible block ciphers is using a Feistel network used in for example DES and Blowfish. Another common way is a substitution-permutation network used in for example Rijndael (AES).
Would there be any advantage in combining both of these in a single block cipher, for example having rounds use them alternatively, or a number of rounds first using one and a number of rounds using the other? Are there examples of such ciphers that would combine both?
This idea is not widely used, neither analyzed in detail. However I have found a recent paper which seems to implement it here with some preliminary and very basic testing. Since the paper is new, and the venue of publication is not one that is considered a mainstream crypto outlet, I don't think the paper has gone through enough scrutiny yet.
Check out their references as well, but as far as I can tell the previous work they refer to as Hybrid schemes do not mix SPN and Feistel.
Reference:
C. P. Arya, R. Ratan and N. Verma, Secure Hybrid Encryption Scheme based on SPN and Feistel Structures, Advances and Applications in Mathematical Sciences Volume 21, Issue 3, January 2022, Pages 1287-1301.
I posit that your scheme champions democracy and freedom against autocracy. Congratulations :-)
Last year’s publicly released US intelligence funding was approximately $100B (https://irp.fas.org/budget/index.html), excluding dark money. That was spread across all of its decryption efforts. And as far as we know, and with an abundance of caution, its programmes have been successful against all common encryptions. Can any one prove otherwise? See NOBUS (https://en.wikipedia.org/wiki/NOBUS).
A portion of that $100B is probably also spent running dis-information campaigns nudging social media sites like this one to suggest the US’s favoured encryption to their naive readers. The list of posts recommending AES-GCM is vast. And don’t roll your own! But just for fun, imagine a Feistel network with SHA-512 as a 256 bit compression function with 100 rounds...
I’ve typed it over and over till my fingers bled (https://crypto.stackexchange.com/a/112555/23115), but heroically I’ll type it again. Introducing novel/wacky/DIY/hybrids is a sure way to save us from cryptographic mono-culturalism, potato blight and death by MRSA (https://www.ncbi.nlm.nih.gov/books/NBK482221/). If AES is broken it all falls down. If the (e.g.) Slutterbug3 primitive is broken, only (e.g.) Tenerife golf holiday bookings fall down. Wasn’t CloudStrike a learning moment?
Multiple encryption algorithms fragment the Man’s finite decryption resources. So the state’s antithesis becomes: The more the merrier. From the people’s perspective. And you are people aren’t you? And you do want to be heard?
