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Why don't we glue the private key in an asymmetric cryptographic system?

With the standard use of an asymmetric cryptographic system, encryption is performed using the public key, decryption with the private key.

Turning to the process, “private key encryption” is called “signature”.

Standard tools, despite the terminology and the lack of direct tools, allow you to implement an encryption system that will use the private key for encryption.

Can someone explain why such a solution is vulnerable?

User Case:

Keep in mind that Alice wants to send Bob some things in an unconventional way:

Alice and Bob once met, and Alice gave Bob the "public key" created using the private key she created, but she warned Bob to keep it secret. And she kept secret the secret key and had not yet given anyone the public key.

Can Bob make sure that the messages he receives from Alice (provided that they are encrypted with Alice’s private key) can only be read by him (if he really saved his copy of Alice’s secret key)?

And how does he compare this hardness of encryption with the traditional way, which in our case would be Bob sending messages to Alice (encrypted with Alice’s public key)?

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Source: https://habr.com/ru/post/1716767/

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