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Curve25519

## Summary

In cryptography, Curve25519 is an elliptic curve offering 128 bits of security (256 bits key size) and designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme. It is one of the fastest ECC curves and is not covered by any known patents.[1] The reference implementation is public domain software.[2][3]

The original Curve25519 paper defined it as a Diffie–Hellman (DH) function. Daniel J. Bernstein has since proposed that the name Curve25519 be used for the underlying curve, and the name X25519 for the DH function.[4]

## Mathematical properties

The curve used is ${\displaystyle y^{2}=x^{3}+486662x^{2}+x}$, a Montgomery curve, over the prime field defined by the prime number ${\displaystyle 2^{255}-19}$, and it uses the base point ${\displaystyle x=9}$. This point generates a cyclic subgroup whose order is the prime ${\displaystyle 2^{252}+27742317777372353535851937790883648493}$, this subgroup has a co-factor of ${\displaystyle 8}$, meaning the number of elements in the subgroup is ${\displaystyle 1/8}$ that of the elliptic curve group. Using a prime order subgroup prevents mounting a Pohlig–Hellman algorithm attack.[5]

The protocol uses compressed elliptic point (only X coordinates), so it allows efficient use of the Montgomery ladder for ECDH, using only XZ coordinates.[6]

Curve25519 is constructed such that it avoids many potential implementation pitfalls.[7] By design, it is immune to timing attacks and it accepts any 32-byte string as a valid public key and does not require validating that a given point belongs to the curve, or is generated by the base point.

The curve is birationally equivalent to a twisted Edwards curve used in the Ed25519[8][9] signature scheme.[10]

## History

In 2005, Curve25519 was first released by Daniel J. Bernstein.[5]

In 2013, interest began to increase considerably when it was discovered that the NSA had potentially implemented a backdoor into the P-256 curve based Dual_EC_DRBG algorithm.[11] While not directly related,[12] suspicious aspects of the NIST's P curve constants[13] led to concerns[14] that the NSA had chosen values that gave them an advantage in breaking the encryption.[15][16]

"I no longer trust the constants. I believe the NSA has manipulated them through their relationships with industry."

— Bruce Schneier, The NSA Is Breaking Most Encryption on the Internet (2013)

Since 2013, Curve25519 has become the de facto alternative to P-256, being used in a wide variety of applications.[17] Starting in 2014, OpenSSH[18] defaults to Curve25519-based ECDH and GnuPG adds support for Ed25519 keys for signing and encryption.[19] Behavior for general SSH protocol is still being standardized as of 2018.[20]

In 2017, NIST announced that Curve25519 and Curve448 would be added to Special Publication 800-186, which specifies approved elliptic curves for use by the US Federal Government.[21] Both are described in RFC 7748.[22] A 2019 draft of "FIPS 186-5" confirms this claim.[23]

In 2018, DKIM specification was amended so as to allow signatures with this algorithm.[24]

Also in 2018, RFC 8446 was published as the new Transport Layer Security v1.3 standard. It requires mandatory support for X25519, Ed25519, X448, and Ed448 algorithms.[25]

## Notes

1. ^ Starting with Windows 10 (1607), Windows Server 2016
2. ^ a b c Via the OMEMO protocol
3. ^ Only in "secret conversations"
4. ^ a b c d Via the Signal Protocol
5. ^ Only in "incognito mode"
6. ^ Used to sign releases and packages[51][52]
7. ^ Exclusive key exchange in OpenSSH 6.7 when compiled without OpenSSL.[53][54]

## References

1. ^ Bernstein. "Irrelevant patents on elliptic-curve cryptography". cr.yp.to. Retrieved 2016-02-08.
2. ^ A state-of-the-art Diffie-Hellman function by Daniel J. Bernstein"My curve25519 library computes the Curve25519 function at very high speed. The library is in the public domain."
3. ^ "X25519". Crypto++. 5 March 2019. Archived from the original on 29 August 2020. Retrieved 29 August 2020.
4. ^ "[Cfrg] 25519 naming". Retrieved 2016-02-25.
5. ^ a b Bernstein, Daniel J. (2006). Yung, Moti; Dodis, Yevgeniy; Kiayias, Aggelos; et al. (eds.). Curve25519: New Diffie-Hellman Speed Records (PDF). Public Key Cryptography. Lecture Notes in Computer Science. Vol. 3958. New York: Springer. pp. 207–228. doi:10.1007/11745853_14. ISBN 978-3-540-33851-2. MR 2423191.
6. ^ Lange, Tanja. "EFD / Genus-1 large-characteristic / XZ coordinates for Montgomery curves". EFD / Explicit-Formulas Database. Retrieved 8 February 2016.
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8. ^ Daniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe, Bo-Yin Yang (2017-01-22). "Ed25519: high-speed high-security signatures". Retrieved 2019-11-09.`{{cite web}}`: CS1 maint: multiple names: authors list (link)
9. ^ Daniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe, Bo-Yin Yang (2011-09-26). "High-speed high-security signatures" (PDF). Retrieved 2019-11-09.`{{cite web}}`: CS1 maint: multiple names: authors list (link)
10. ^ Bernstein, Daniel J.; Lange, Tanja (2007). Kurosawa, Kaoru (ed.). Faster addition and doubling on elliptic curves. Advances in cryptology—ASIACRYPT. Lecture Notes in Computer Science. Vol. 4833. Berlin: Springer. pp. 29–50. doi:10.1007/978-3-540-76900-2_3. ISBN 978-3-540-76899-9. MR 2565722.
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24. ^ John Levine (September 2018). A New Cryptographic Signature Method for DomainKeys Identified Mail (DKIM). IETF. doi:10.17487/RFC8463. RFC 8463.
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