Curve25519

In cryptography, Curve25519 is an elliptic curve used in elliptic-curve cryptography (ECC) offering 128 bits of security (256-bit key size) and designed for use with the Elliptic-curve Diffie–Hellman (ECDH) key agreement scheme. It is one of the fastest curves in ECC, 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 ${\displaystyle 2^{255}-19}$  (hence the numeric "25519" in the name), 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, Curve25519 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.^{[citation needed]}

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] The use of the curve was eventually standardized for both key exchange and signature in 2020.^[20][21]

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.^[22] Both are described in RFC 7748.^[23] A 2019 draft of "FIPS 186-5" notes the intention to allow usage of Ed25519^[24] for digital signatures. The 2023 update of Special Publication 800-186 allows usage of Curve25519.^[25]

In 2018, DKIM specification was amended so as to allow signatures with this algorithm.^[26]

Also in 2018, RFC 8446 was published as the new Transport Layer Security v1.3 standard. It recommends support for X25519, Ed25519, X448, and Ed448 algorithms.^[27]

Libraries

• Libgcrypt^[28]
• libssh^[18][29]
• libssh2 (since version 1.9.0)
• NaCl^[30]
• GnuTLS^[31]
• mbed TLS (formerly PolarSSL)^[32]
• wolfSSL^[33]
• Botan^[34]
• Schannel^[a][35]
• Libsodium^[36]
• OpenSSL since version 1.1.0^[37]
• LibreSSL^[38]
• NSS since version 3.28^[39]
• Crypto++
• curve25519-dalek^[40]
• Bouncy Castle^[41]

Protocols

• OMEMO, a proposed extension for XMPP (Jabber)^[42]
• Secure Shell
• Signal Protocol
• Matrix (protocol)
• Tox
• Zcash
• Transport Layer Security
• WireGuard

Applications

• Conversations Android application^[b]
• Cryptocat^[43][b]
• DNSCrypt^[44]
• DNSCurve
• Dropbear^[29][45]
• Facebook Messenger ^[c][d]
• Gajim via plugin^[46][b]
• GNUnet^[47]
• GnuPG
• Google Allo^[e][d]
• I2P^[48]
• IPFS^[49]
• iOS^[50]
• Monero^[51]
• OpenBSD^[f]
• OpenSSH^[29][g]
• Peerio^[56]
• Proton Mail^[57]
• PuTTY^[58]
• Signal^[d]
• Silent Phone
• SmartFTP^[29]
• SSHJ^[29]
• SQRL^[59]
• Threema Instant Messenger^[60]
• TinySSH^[29]
• TinyTERM^[29]
• Tor^[61]
• Viber^[62]
• WhatsApp^[d][63]
• Wire
• WireGuard

Notes

1. Starting with Windows 10 (1607), Windows Server 2016
2. Via the OMEMO protocol
3. Only in "secret conversations"
4. Via the Signal Protocol
5. Only in "incognito mode"
6. Used to sign releases and packages^[52][53]
7. Exclusive key exchange in OpenSSH 6.7 when compiled without OpenSSL.^[54][55]

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 3 February 2023.
4. "[Cfrg] 25519 naming". Retrieved 2016-02-25.
5. Bernstein, Daniel J. (2006). "Curve25519: New Diffie-Hellman Speed Records" (PDF). In Yung, Moti; Dodis, Yevgeniy; Kiayias, Aggelos; et al. (eds.). Public Key Cryptography - PKC 2006. 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 2016-02-08.
7. Bernstein, Daniel J.; Lange, Tanja (2017-01-22). "SafeCurves: Introduction". SafeCurves: choosing safe curves for elliptic-curve cryptography. Retrieved 2016-02-08.
8. Bernstein, Daniel J.; Duif, Niels; Lange, Tanja; Schwabe, Peter; Yang, Bo-Yin (2017-01-22). "Ed25519: high-speed high-security signatures". Retrieved 2019-11-09.
9. Bernstein, Daniel J.; Duif, Niels; Lange, Tanja; Schwabe, Peter; Yang, Bo-Yin (2011-09-26). "High-speed high-security signatures" (PDF). Retrieved 2019-11-09.
10. Bernstein, Daniel J.; Lange, Tanja (2007). "Faster addition and doubling on elliptic curves". In Kurosawa, Kaoru (ed.). Advances in Cryptology – ASIACRYPT 2007. 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.
11. Kelsey, John (May 2014). "Dual EC in X9.82 and SP 800-90" (PDF). National Institute of Standards in Technology. Retrieved 2018-12-02.
12. Green, Matthew (2015-01-14). "A Few Thoughts on Cryptographic Engineering: The Many Flaws of Dual_EC_DRBG". blog.cryptographyengineering.com. Retrieved 2015-05-20.
13. "SafeCurves: Introduction".
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21. B. Harris; L. Velvindron (February 2020). Ed25519 and Ed448 Public Key Algorithms for the Secure Shell (SSH) Protocol. doi:10.17487/RFC8709. RFC 8709.
22. . National Institute of Standards and Technology. 2017-10-31. Archived from the original on 2018-03-11. Retrieved 2019-09-04.
23. RFC 7748. Retrieved from rfc:7748.
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33. "wolfSSL Embedded SSL/TLS Library | Products – wolfSSL".
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42. Straub, Andreas (25 October 2015). "OMEMO Encryption". conversations.im.
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45. Matt Johnston. "Dropbear SSH - Changes". Retrieved 2016-02-25.
46. Bahtiar Gadimov; et al. "Gajim plugin for OMEMO Multi-End Message and Object Encryption". GitHub. Retrieved 2016-10-01.
47. . gnunet.org. Archived from the original on 9 December 2017. Retrieved 11 December 2014.
48. zzz (2014-09-20). "0.9.15 Release - Blog". Retrieved 20 December 2014.
49. "go-ipfs_keystore.go at master". Github.com. 30 March 2022.
50. "Apple Platform Security". Apple Support.
51. (PDF). getmonero.com. Archived from the original (PDF) on 2019-05-01. Retrieved 2018-06-05.
52. Murenin, Constantine A. (2014-01-19). Soulskill (ed.). "OpenBSD Moving Towards Signed Packages — Based On D. J. Bernstein Crypto". Slashdot. Retrieved 2014-12-27.
53. Murenin, Constantine A. (2014-05-01). timothy (ed.). "OpenBSD 5.5 Released". Slashdot. Retrieved 2014-12-27.
54. Friedl, Markus (2014-04-29). "ssh/kex.c#kexalgs". BSD Cross Reference, OpenBSD src/usr.bin/. Retrieved 2014-12-27.
55. Murenin, Constantine A. (2014-04-30). Soulskill (ed.). "OpenSSH No Longer Has To Depend On OpenSSL". Slashdot. Retrieved 2014-12-26.
56. . Peerio. Archived from the original on 2017-12-09. Retrieved 2015-11-04.
57. "Proton Mail now offers elliptic curve cryptography for advanced security and faster speeds". 25 April 2019.
58. "PuTTY Change Log". www.chiark.greenend.org.uk.
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60. "Threema Cryptography Whitepaper" (PDF).
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62. "Viber Encryption Overview". Viber. 3 May 2016. Retrieved 24 September 2016.
63. Nidhi Rastogi; James Hendler (2017-01-24). "WhatsApp security and role of metadata in preserving privacy". arXiv:1701.06817 [cs.CR].

External links

• Official website