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**RSA** public-key cryptography algorithm.

Messages are checked not to exceed the key size, and this is signalled via the `Insufficient_key`

exception.

Private-key operations are optionally protected through RSA blinding.

Raised if the key is too small to transform the given message, i.e. if the numerical interpretation of the (potentially padded) message is not smaller than the modulus.

The public portion of the key.

* Sexplib convertible*.

`pub ~e ~n`

validates the public key: `1 < e < n`

, `n > 0`

, `is_odd n`

, and `numbits n >= 89`

(a requirement for PKCS1 operations).

`type priv = private {`

`e : Z.t;`

(*Public exponent

*)`d : Z.t;`

(*Private exponent

*)`n : Z.t;`

(*Modulus (

*)`p q`

)`p : Z.t;`

(*Prime factor

*)`p`

`q : Z.t;`

(*Prime factor

*)`q`

`dp : Z.t;`

(*

*)`d mod (p-1)`

`dq : Z.t;`

(*

*)`d mod (q-1)`

`q' : Z.t;`

(*

*)`q^(-1) mod p`

`}`

Full private key (two-factor version).

**Note** The key layout assumes that `p > q`

, which affects the quantity `q'`

(sometimes called `u`

), and the computation of the private transform. Some systems assume otherwise. When using keys produced by a system that computes `u = p^(-1) mod q`

, either exchange `p`

with `q`

and `dp`

with `dq`

, or re-generate the full private key using `priv_of_primes`

.

* Sexplib convertible*.

```
val priv :
e:Z.t ->
d:Z.t ->
n:Z.t ->
p:Z.t ->
q:Z.t ->
dp:Z.t ->
dq:Z.t ->
q':Z.t ->
(priv, [> `Msg of string ]) Stdlib.result
```

`priv ~e ~d ~n ~p ~q ~dp ~dq ~q'`

validates the private key: `e, n`

must be a valid `pub`

, `p`

and `q`

valid prime numbers `> 0`

, `odd`

, probabilistically prime, `p <> q`

, `n = p * q`

, `e`

probabilistically prime and coprime to both `p`

and `q`

, `q' = q ^ -1 mod p`

, `1 < d < n`

, `dp = d mod (p - 1)`

, `dq = d mod (q - 1)`

, and `d = e ^ -1 mod (p - 1) (q - 1)`

.

`priv_of_primes ~e ~p ~q`

is the private key derived from the minimal description `(e, p, q)`

.

```
val priv_of_exp :
?g:Mirage_crypto_rng.g ->
?attempts:int ->
e:Z.t ->
d:Z.t ->
n:Z.t ->
unit ->
(priv, [> `Msg of string ]) Stdlib.result
```

`priv_of_exp ?g ?attempts ~e ~d n`

is the unique private key characterized by the public (`e`

) and private (`d`

) exponents, and modulus `n`

. This operation uses a probabilistic process that can fail to recover the key.

`~attempts`

is the number of trials. For triplets that form an RSA key, the probability of failure is at most `2^(-attempts)`

. `attempts`

defaults to an unspecified number that yields a very high probability of recovering valid keys.

Note that no time masking is done for the computations in this function.

Either an `'a`

or its digest, according to some hash algorithm.

Masking (cryptographic blinding) mode for the RSA transform with the private key. Masking does not change the result, but it does change the timing profile of the operation.

``No`

disables masking. It is slightly faster but it**exposes the private key to timing-based attacks**.``Yes`

uses random masking with the global RNG instance. This is the sane option.``Yes_with g`

uses random masking with the generator`g`

.

`decrypt ~crt_hardening ~mask key ciphertext`

is the decrypted `ciphertext`

, left-padded with `0x00`

up to `key`

size.

`~crt_hardening`

defaults to `false`

. If `true`

verifies that the result is correct. This is to counter Chinese remainder theorem attacks to factorize primes. If the computed signature is incorrect, it is again computed in the classical way (c ^ d mod n) without the Chinese remainder theorem optimization. The deterministic PKCS1 signing, which is at danger, uses `true`

as default.

`~mask`

defaults to ``Yes`

.

`val generate : ?g:Mirage_crypto_rng.g -> ?e:Z.t -> bits:bits -> unit -> priv`

`generate ~g ~e ~bits ()`

is a new private key. The new key is guaranteed to be well formed, see `priv`

.

`e`

defaults to `2^16+1`

.

**Note** This process might diverge if there are no keys for the given bit size. This can happen when `bits`

is extremely small.

`module PKCS1 : sig ... end`

**PKCS v1.5** operations, as defined by **PKCS #1 v1.5**.

`module OAEP (H : Mirage_crypto.Hash.S) : sig ... end`

**OAEP**-padded encryption, as defined by **PKCS #1 v2.1**.

`module PSS (H : Mirage_crypto.Hash.S) : sig ... end`

**PSS**-based signing, as defined by **PKCS #1 v2.1**.

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