What is the difference between a "Thumbprint Algorithm" "Signature Algorithm" and "Signature Hash Algorithm" for a certificate?

I'm a bit confused on the differences between `Signature Algorithm`, `Signature Hash Algorithm`, and `Thumbprint Algorithm` that are present in SSL/TLS certificates. Can someone please elaborate?

You are confused because some people (yeah I am looking at you, Microsoft) have been using the terms inconsistently.

A signature algorithm is a cryptographic algorithm such that:

• The signer owns a public/private key pair. The public key is public, the private key is private; even though both keys are mathematically linked together, it is not feasible to recompute the private key from the public key (which is why the public key could safely be made public).
• On a given input message, the signer can use his private key to compute a signature, which is specific to both the signer's key pair, and the input message.
• There is a verification algorithm that takes as input the message, the signature and the public key, and answers "true" (they match) or "false" (they don't).

The cornerstone of signature security is that it should not be feasible, without knowledge of the private key, to generate pairs message+signature that the verification algorithm will accept.

You may encounter some "explanations" that try to say that digital signatures are some kind of encryption; they usually describe it as "you encrypt with the private key". Don't believe it; these explanations are actually wrong, and confusing.

For technical reasons, signature algorithms (both for signing and for verifying) often begin with a hash function. A hash function is a completely public algorithm with no key. The point of hash functions is that they can eat up terabytes of data, and produce a "digest" (also called "fingerprint" or even "thumbprint") that has a fixed, small size. Signature algorithms need that, because they work with values in some algebraic structure of a finite size, and thus cannot accommodate huge messages. Therefore, the message is first hashed, and only the hash value is used for generating or verifying a signature.

That hash algorithm, when it is used as first step of a signature generation or verification algorithm, will be called "signature hash algorithm". When we say something like "RSA/SHA-256", we mean "RSA signature, with SHA-256 as accompanying hash function".

A "thumbprint algorithm" is another name for a hash function. It is often encountered when talking about certificates: the "thumbprint" of a certificate really is the result of a hash function applied to the certificate itself (in Windows systems, the SHA-1 hash function is used).

• Digital signatures DO involve encrypting with the private key. As the diagram in the 'signature algorithm' link shows, you take the hash of the whole certificate, then encrypt that hash with the private key. The signature is included along with the certificate. The verification procedure is to decrypt the signature with the public key to get the hash, then hash the certificate with the same hash algorithm and see if the 2 hash values are the same. Commented Mar 31, 2015 at 0:51
• Fantastic answer Tom. Really clear on the Signature Algorithm and Signature Hash Algorithm. Can you confirm that the "thumbprint algorithm" (as stated on a typical TLS certificate) isn't used as part of the signature generation/verification, and also if that's the case, what is it used for (why is it there at all?) Thanks Commented Feb 25, 2016 at 16:10
• Thumbprint is for humans. All the certificate validation is about verifying signatures from CA, themselves verified with signatures from other CA. It must start somewhere, with a "root CA" that is trusted because it is already there. When you install a new root CA in your Windows, the interface will display the root CA certificate thumbprint (in hexadecimal); you are supposed to check that it is the correct one (presumably, you compare against a reference printed document that contains the expected root CA thumbprint, or you phone the sysadmin and speak the hex digits). Commented Feb 25, 2016 at 16:18
• So the thumbprint on certificates is for validation jobs that are not covered by the generic certificate validation mechanism. In particular, installation of root CA, because root CA certificates cannot be validated otherwise (because that's what it means to be a root). Commented Feb 25, 2016 at 16:19
• Per digital signature on Cryptography: `A digital signature is not encryption... It is often explained as encryption with a private key because the RSA signature scheme uses modular exponentiation both for encryption as well as generating signatures. ...(PKCS#1 v2.2) goes out of the way to explain that signature generation is not encryption...` Commented Mar 23, 2020 at 20:18

While the accepted answer goes into some detail about how the calculation is done, it doesn't address the original question at all - Signature Algorithm, Signature Hash Algorithm, and Thumbprint Algorithm that are present in SSL/TLS certificates - which one is which piece of the equation (especially if Microsoft also mix it up)?

The answer seems fairly simple - from:

and marked as correct answer over there:

Differences between "Signature algorithm" and "Signature Hash Algorithm"

They are used to determine the signature algorithm and hash function used to sign the certificate. This information is used by certificate chaining engine to validate the signature of the certificate. Certificate chaining engine calculates a hash over a certificate (signed part). Hash method is selected from Signature Hash Algorithm field. Then certificate chaining engine decodes attached signature by using signature algorithm specified in the Signature Algorithm field and recovers signed hash. If both hashes match, then signature is valid, if they differ, the signature is considered invalid.

My addition from another source - Thumbprint Algo - simple non crypto property used to identify the cert on a given system (not authenticate it or verify its validity)

• I'm really having trouble seeing how you have added anything to what the accepted answer provides. You are stating everything that Tom does. Commented Mar 31, 2017 at 6:41
• I understand the motivation for this answer, since Tom's answer was not very clear/explicit on what the difference between "Signature algorithm" and "Signature hash algorithm" is. However, it is still not clear to me why both exist Commented Feb 23, 2018 at 21:15