Trapdoor permutations can be used for digital signature schemes, where computing the reverse direction with the secret key is required for signing, and computing the forward direction is used to verify signatures.
Used directly, this type of signature scheme is vulnerable to key-only existential forgery attack.
One (of many) digital signature schemes is based on RSA.
To create signature keys, generate a RSA key pair containing a modulus, N, that is the product of two random secret distinct large primes, along with integers, e and d, such that e d ≡ 1 (mod φ(N)), where φ is the Euler phi-function.
This forgery attack, then, only produces the padded hash function output that corresponds to σ, but not a message that leads to that value, which does not lead to an attack.
In the random oracle model, hash-then-sign (an idealized version of that practice where hash and padding combined have close to N possible outputs), this form of signature is existentially unforgeable, even against a chosen-plaintext attack.
A digital signature is an authentication mechanism that enables the creator of the message to attach a code that acts as a signature.
Note that we require any adversary cannot directly query the string, x, on S.
An example signature that's been verified (note the green check mark) For your convenience, you can send your certificate to [email protected]
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Several early signature schemes were of a similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite, N.
A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction without already knowing the private key ("trapdoor").