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Asymmetric signatures are one of the slowest operations (on relatively small sizes of data) around. RSA signature generation is particularly slow, as it requires modular exponentiation over the private exponent. It's possible to use Chinese Remainder Theorem and/or multi-prime RSA, but that won't get you close to hashing speeds. That I specify the algorithm ...

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An analogy might not be that helpful but an example for example with RSA signatures. RSA Signatures work like this: s = m^d mod N where s is the signature, m the message and d the private key. (See example below. Verification works like this: m' = s^e mod N where s is still the signature and e is the publicly known and trusted public key. If m' = m ...

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Ok, here's a toy example (which really doesn't work) of a simple signature scheme, which you can use as an analogy of a real system: Suppose the signer Alice picks three integers $b, c, p$, and computes $a = b \times c \bmod p$. She then publishes $a, b, p$ as her public key, and keeps $c, p$ as her private key. Then, when Alice wants to sign a message $M$...

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To decode from a public-key encoded message, you need the secret private key. Anyone else cannot do it. For the mathematical details how this is possible, you need to analyse the respective asymmetric cryptographic algorithms. There are several different asymmetrical encryption algorithms, including RSA and ElGamal, see the Wikipedia links for an ...

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