Knowledge Base
Diffie-Hellman Parameter
A+

3072-bit MODP Group/Oakley Group 15 (RFC 3526)

Summary

Name:
3072-bit MODP Group/Oakley Group 15 (RFC 3526)
Key size:
3072 bits
Publishers:

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Security

A+
Key Size
Name

3072

Security

The Diffie-Hellman ephemeral public key parameter[409][410][411] with prime size greater than 2048[422] is considered secure, not weakened against logjam attack[7][8][9][10].

Recommendations

Use a well-known Diffie-Hellman ephemeral public key parameter[426][427][428], or generate a custom Diffie-Hellman ephemeral public key parameter[409][410][411] with a size greater or equal than 2048 bits but less or equal than 4096 bits. In the case of custom parameters, validate that the prime is a safe prime[176] to avoid a small subgroup confinement attack[72][73].

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Prime
Name

Safe Prime

Security

Diffie-Hellman ephemeral public key parameter[409][410][411] contains a safe prime[176], so the connection is certainly not vulnerable to a small subgroup confinement attack[72][73].

A
Post-Quantum
Name

False

Security

The Diffie-Hellman[99][100][101][102][103] key exchange provides no protection against a cryptanalytic attack by a quantum computer, and no classical Diffie-Hellman does — just as no classical elliptic curve does — because a quantum computer breaks the hardness assumption they rely on. Only a hybrid key exchange (a classical algorithm combined with a post-quantum cryptography[158][159] one) or a pure post-quantum algorithm is quantum-safe.

Recommendations

Enable a hybrid key exchange or a pure post-quantum algorithm on your server, and prefer it where the configuration allows, so the connection stays secure against a future quantum computer.

Parameter Numbers

Prime (p)
0xffffffffffffffffc90fdaa22168c234c4c6628b80dc1cd129024e088a67cc74020bbea63b139b22514a08798e3404ddef9519b3cd3a431b302b0a6df25f14374fe1356d6d51c245e485b576625e7ec6f44c42e9a637ed6b0bff5cb6f406b7edee386bfb5a899fa5ae9f24117c4b1fe649286651ece45b3dc2007cb8a163bf0598da48361c55d39a69163fa8fd24cf5f83655d23dca3ad961c62f356208552bb9ed529077096966d670c354e4abc9804f1746c08ca18217c32905e462e36ce3be39e772c180e86039b2783a2ec07a28fb5c55df06f4c52c9de2bcbf6955817183995497cea956ae515d2261898fa051015728e5a8aaac42dad33170d04507a33a85521abdf1cba64ecfb850458dbef0a8aea71575d060c7db3970f85a6e1e4c7abf5ae8cdb0933d71e8c94e04a25619dcee3d2261ad2ee6bf12ffa06d98a0864d87602733ec86a64521f2b18177b200cbbe117577a615d6c770988c0bad946e208e24fa074e5ab3143db5bfce0fd108e4b82d120a93ad2caffffffffffffffff
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Generator (g)
0x2
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Representations

PEM

Default private value length
-----BEGIN DH PARAMETERS-----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-----END DH PARAMETERS-----
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Small private value length
-----BEGIN DH PARAMETERS-----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-----END DH PARAMETERS-----
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PARI/GP

powermod(x, k, m) = lift(Mod(x, m) ^ k);

g = 0x2;
p = 2^3072 - 2^3008 - 1 + 2^64 * (floor(2^2942 * Pi) + 1690314);
s = 256;

a = 2 ^ (s - 1) + random(2 ^ (s - 1));
A = powermod(g, a, p);

b = 2 ^ (s - 1) + random(2 ^ (s - 1));
B = powermod(g, b, p);

S = powermod(B, a, p);
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LaTeX

$\begin{eqnarray}
p \equiv 2^{3072} - 2^{3008} - 1 + 2^{64} * (\lfloor 2^{2942} * \pi \rfloor + 1690314) \\
g \equiv 0x2 \\
\\
a \in \mathbb{Z}_q \\
b \in \mathbb{Z}_q \\
\\
A \equiv g^a \pmod{p} \\
B \equiv g^b \pmod{p} \\
\\
S \equiv {(g^{a})}^{b}={(g^{b})}^{a}=g^{ab} \pmod{p}
\end{eqnarray}$
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