Add rsa_recover_private_exponent function (#11193)

Given the RSA public exponent (`e`), and the RSA primes (`p`, `q`), it is possible
to calculate the corresponding private exponent `d = e⁻¹ mod λ(n)` where
`λ(n) = lcm(p-1, q-1)`.

With this function added, it becomes possible to use the library to reconstruct an RSA
private key given *only* `p`, `q`, and `e`:

    from cryptography.hazmat.primitives.asymmetric import rsa

    n = p * q
    d = rsa.rsa_recover_private_exponent(e, p, q)  # newly-added piece
    iqmp = rsa.rsa_crt_iqmp(p, q)                  # preexisting
    dmp1 = rsa.rsa_crt_dmp1(d, p)                  # preexisting
    dmq1 = rsa.rsa_crt_dmq1(d, q)                  # preexisting

    assert rsa.rsa_recover_prime_factors(n, e, d) in ((p, q), (q, p))  # verify consistency

    privk = rsa.RSAPrivateNumbers(p, q, d, dmp1, dmq1, iqmp, rsa.RSAPublicNumbers(e, n)).private_key()

Older RSA implementations, including the original RSA paper, often used the
Euler totient function `ɸ(n) = (p-1) * (q-1)` instead of `λ(n)`.  The
private exponents generated by that method work equally well, but may be
larger than strictly necessary (`λ(n)` always divides `ɸ(n)`).  This commit
additionally implements `_rsa_recover_euler_private_exponent`, so that tests
of the internal structure of RSA private keys can allow for either the Euler
or the Carmichael versions of the private exponents.

It makes sense to expose only the more modern version (using the Carmichael
totient function) for public usage, given that it is slightly more
computationally efficient to use the keys in this form, and that some
standards like FIPS 186-4 require this form.  (See
https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf#page=63)

CHANGELOG.rst

@@ -58,6 +58,8 @@ Changelog
   :meth:`~cryptography.x509.ocsp.OCSPSingleResponse.next_update_utc`,
   These are timezone-aware variants of existing properties that return naïve
   ``datetime`` objects.
+* Added
+  :func:`~cryptography.hazmat.primitives.asymmetric.rsa.rsa_recover_private_exponent`
 
 .. _v42-0-8:
 

docs/hazmat/primitives/asymmetric/rsa.rst

@@ -554,6 +554,23 @@ this without having to do the math themselves.
     Computes the ``dmq1`` parameter from the RSA private exponent (``d``) and
     prime ``q``.
 
+.. function:: rsa_recover_private_exponent(e, p, q)
+
+    .. versionadded:: 43.0.0
+
+    Computes the RSA private_exponent (``d``) given the public exponent (``e``)
+    and the RSA primes ``p`` and ``q``.
+
+    .. note::
+
+        This implementation uses the Carmichael totient function to return the
+        smallest working value of ``d``. Older RSA implementations, including the
+        original RSA paper, often used the Euler totient function, which results
+        in larger but equally functional private exponents. The private exponents
+        resulting from the Carmichael totient function, as returned here, are
+        slightly more computationally efficient to use, and some modern standards
+        require them.
+
 .. function:: rsa_recover_prime_factors(n, e, d)
 
     .. versionadded:: 0.8

docs/spelling_wordlist.txt

@@ -15,6 +15,7 @@ Botan
 Brainpool
 Bullseye
 Capitan
+Carmichael
 CentOS
 changelog
 Changelog
@@ -51,6 +52,7 @@ Docstrings
 El
 Encodings
 endian
+Euler
 extendable
 facto
 fallback
@@ -128,6 +130,7 @@ Thawte
 timestamp
 timestamps
 toolchain
+totient
 Trixie
 tunable
 Ubuntu

src/cryptography/hazmat/primitives/asymmetric/rsa.py

@@ -190,6 +190,27 @@ def rsa_crt_dmq1(private_exponent: int, q: int) -> int:
     return private_exponent % (q - 1)
 
 
+def rsa_recover_private_exponent(e: int, p: int, q: int) -> int:
+    """
+    Compute the RSA private_exponent (d) given the public exponent (e)
+    and the RSA primes p and q.
+
+    This uses the Carmichael totient function to generate the
+    smallest possible working value of the private exponent.
+    """
+    # This lambda_n is the Carmichael totient function.
+    # The original RSA paper uses the Euler totient function
+    # here: phi_n = (p - 1) * (q - 1)
+    # Either version of the private exponent will work, but the
+    # one generated by the older formulation may be larger
+    # than necessary. (lambda_n always divides phi_n)
+    #
+    # TODO: Replace with lcm(p - 1, q - 1) once the minimum
+    # supported Python version is >= 3.9.
+    lambda_n = (p - 1) * (q - 1) // gcd(p - 1, q - 1)
+    return _modinv(e, lambda_n)
+
+
 # Controls the number of iterations rsa_recover_prime_factors will perform
 # to obtain the prime factors. Each iteration increments by 2 so the actual
 # maximum attempts is half this number.

tests/hazmat/primitives/utils.py

@@ -522,13 +522,42 @@ def rsa_verification_test(backend, params, hash_alg, pad_factory):
         public_key.verify(signature, msg, pad, hash_alg)
 
 
+def _rsa_recover_euler_private_exponent(e: int, p: int, q: int) -> int:
+    """
+    Compute the RSA private_exponent (d) given the public exponent (e)
+    and the RSA primes p and q, following the usage of the original
+    RSA paper.
+
+    As in the original RSA paper, this uses the Euler totient function
+    instead of the Carmichael totient function, and thus may generate a
+    larger value of the private exponent than necessary.
+
+    See cryptography.hazmat.primitives.asymmetric.rsa_recover_private_exponent
+    for the public-facing version of this function, which uses the
+    preferred Carmichael totient function.
+    """
+    phi_n = (p - 1) * (q - 1)
+    return rsa._modinv(e, phi_n)
+
+
 def _check_rsa_private_numbers(skey):
     assert skey
     pkey = skey.public_numbers
     assert pkey
     assert pkey.e
     assert pkey.n
-    assert skey.d
+
+    # Historically there have been two ways to calculate valid values of the
+    # private_exponent (d) given the public exponent (e):
+    # - using the Carmichael totient function (gives smaller and more
+    #   computationally-efficient values, and is required by some standards)
+    # - using the Euler totient function (matching the original RSA paper)
+    # Allow for either here.
+    assert skey.d in (
+        rsa.rsa_recover_private_exponent(pkey.e, skey.p, skey.q),
+        _rsa_recover_euler_private_exponent(pkey.e, skey.p, skey.q),
+    )
+
     assert skey.p * skey.q == pkey.n
     assert skey.dmp1 == rsa.rsa_crt_dmp1(skey.d, skey.p)
     assert skey.dmq1 == rsa.rsa_crt_dmq1(skey.d, skey.q)