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When it comes to encrypting some data (where data being something of a large size), symmetric key cryptosystem is predominant. Symmetric key algorithms like AES, DES, Blowfish, IDEA etc. are explained in greater lengths, on how they function and work and why is one better then the other.

But the same han't been done in the case of asymmetric cryptosystems. In general asymmetric cryptosystem is referenced, in a hybrid cryptosystem. Where the Symmetric key is used the encrypt the actual data, and then the Symmetric key is encrypted via the public key.

Even in the wikipedia page of RSA the emphasis is only given to the key exchange process of RSA, rather then how the actual data is encrypted/decrypted via it.

The most I know about Asymmetric key Cryptographic algorithms is that:-

  • They are far slower then symmetric cryptographic algorithms;
  • They produce cipher texts of sizes larger then the original one.

This is the only thing that is available when it comes to actual encryption / decryption by asymmetric algorithms. And even this hasn't been explained, why do they possess such behavior?

Even though they may not be a good choice, when compared with symmetric key algorithms, but still at least some information about their working should be provided.

EX:- In RSA, this is the description of the encryption process:-

    CT = (PT^E) % N

    # where CT = Cipher Text, PT = Plain Text, N = Product of Two Large Primes, E = Public Key

But it has not be explained what happens in the encryption process.

I would like to get some intel on some asymmetric cryptographic algorithms used for encryption / decryption, so that I can analyze their internal working.

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You are mistaken that there aren't any cryptographic standards that describe asymmetric encryption / decryption. You name RSA as an example and then you just show the modular exponentiation. However, in PKCS#1 - the predominant RSA standard - there are methods described to encrypt with a public key and decrypt with the private key. Currently it contains PKCS#1 v1.5 compatible padding (official name RSAES-PKCS1-v1_5) and OAEP (official name RSAES-OAEP). In the standard there are also some links to older standards that these schemes have been derived from.

In principle you can simply split up the blocks and encrypt each block. However, the input block size is indeed smaller than the output block size. Furthermore, modular exponentiation is way too expensive for the private key operation. There is simply not much reason to choose just RSA over hybrid encryption. You could say that you could rearrange the RSA blocks as they always would decrypt even in the wrong order, but AES-CBC doesn't provide any integrity or message authentication either, so I've always seen this as unfair. If you require integrity or message authentication then you'll have to sign the encrypted blocks (or sign before encryption, depending on the use case).

For DSA and therefore ECDSA there is no good encryption method. There is El Gamal encryption, but because the sizes of the keys are relatively small for ECDSA, it is even less efficient than RSA encryption.

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  • $\begingroup$ Thanks!! Can you please explain, why aren't the asymmetric key cryptographic algorithms explained to the level of detail as their symmetric key counterpart does (except for the obvious reasons like increase output size etc). Are they too difficult to understand, as compared to symmetic key ones? $\endgroup$
    – Vasu Deo.S
    Commented Jul 7, 2019 at 15:06
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    $\begingroup$ @VasuDeo.S It is not clear what you mean by why aren't the asymmetric key cryptographic algorithms explained to the level of detail as their symmetric key counterpart - did you read the standards linked by Maarten? If so, what part of it caused you to conclude that there is insufficient detail? What details are you asking about that are not present? $\endgroup$
    – Ella Rose
    Commented Jul 7, 2019 at 15:23
  • $\begingroup$ @EllaRose I think you misunderstood my point. my question was from a beginner point of view, that is, in most beginner cryptographic guides/books there exists elaborate explanations on working of AES, DES etc. but such treatment isn't shown towards asymmetric cryptographic algorithms. By that I mean, almost everyone in cryptography is known to AES, DES, IDEA etc, and there exists good amount of intel on them on sites like wikipedia. But i can't find the same done for asymmetric Encryption/decryption algorithms. So why aren't these algorithms that much renowned. $\endgroup$
    – Vasu Deo.S
    Commented Jul 7, 2019 at 15:53
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    $\begingroup$ Take for instance TLS. In that case the data records in the session need to be encrypted. In that case RSA would be a terribly bad idea. So what you do is that you establish the session keys. Initially this was indeed performed by encrypting a master secret. However there are many more methods to establish the session keys, for instance ECDHE as used in TLS 1.2 and 1.3. This is also better in the sense that it provides forward security. In that sense symmetric encryption is more versatile. $\endgroup$
    – Maarten Bodewes ♦
    Commented Jul 7, 2019 at 16:00
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    $\begingroup$ Another issue is maybe that the padding used for RSA is security sensitive and that the number of "modes" is therefore more limited. RSA is also harder to understand, so it received limited amount of time. You'd rather spend it on authentication / signatures and PKI, I suppose. In the end these kind of "why" questions can only be answered fully by the authors of the text books of course, this is all just educated guesswork. $\endgroup$
    – Maarten Bodewes ♦
    Commented Jul 7, 2019 at 16:02

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