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In public-key cryptography, Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based on twisted Edwards curves.[1] It is designed to be faster than existing digital signature schemes without sacrificing security. It was developed by a team including Daniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe, and Bo-Yin Yang.[2] The reference implementation is public domain software.[3]

DesignersDaniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe, Bo-Yin Yang, et al.
First published26 September 2011
StructureElliptic-curve cryptography


The following is a simplified description of EdDSA, ignoring details of encoding integers and curve points as bit strings; the full details are in the papers and RFC.[4][2][1]

An EdDSA signature scheme is a choice:

  • of finite field   over odd prime power  ;
  • of elliptic curve   over   whose group   of  -rational points has order  , where   is a large prime and   is called the cofactor;
  • of base point   with order  ; and
  • of cryptographic hash function   with  -bit outputs, where   so that elements of   and curve points in   can be represented by strings of   bits.

These parameters are common to all users of the EdDSA signature scheme. The security of the EdDSA signature scheme depends critically on the choices of parameters, except for the arbitrary choice of base point—for example, Pollard's rho algorithm for logarithms is expected to take approximately   curve additions before it can compute a discrete logarithm,[5] so   must be large enough for this to be infeasible, and is typically taken to exceed 2200.[6] The choice of   is limited by the choice of  , since by Hasse's theorem,   cannot differ from   by more than  . The hash function   is normally modelled as a random oracle in formal analyses of EdDSA's security. In the HashEdDSA variant, an additional collision-resistant hash function   is needed.

Within an EdDSA signature scheme,

Public key
An EdDSA public key is a curve point  , encoded in   bits.
An EdDSA signature on a message   by public key   is the pair  , encoded in   bits, of a curve point   and an integer   satisfying the following verification equation.   denotes concatenation.

Private key
An EdDSA private key is a  -bit string   which should be chosen uniformly at random. The corresponding public key is  , where   is the least significant   bits of   interpreted as an integer in little-endian. The signature on a message   is   where   for  , and
This satisfies the verification equation:



Ed25519 is the EdDSA signature scheme using SHA-512 (SHA-2) and Curve25519[2] where

  •   is the twisted Edwards curve

  •   and  
  •   is the unique point in   whose   coordinate is   and whose   coordinate is positive.
    "positive" is defined in terms of bit-encoding:
    • "positive" coordinates are even coordinates (least significant bit is cleared)
    • "negative" coordinates are odd coordinates (least significant bit is set)
  •   is SHA-512, with  .

The curve   is birationally equivalent to the Montgomery curve known as Curve25519. The equivalence is[2][7]



The original team has optimized Ed25519 for the x86-64 Nehalem/Westmere processor family. Verification can be performed in batches of 64 signatures for even greater throughput. Ed25519 is intended to provide attack resistance comparable to quality 128-bit symmetric ciphers.[8] Public keys are 256 bits long and signatures are 512 bits long.[9]

Secure codingEdit

As security features, Ed25519 does not use branch operations and array indexing steps that depend on secret data, so as to defeat many side channel attacks.

Like other discrete-log-based signature schemes, EdDSA uses a secret value called a nonce unique to each signature. In the signature schemes DSA and ECDSA, this nonce is traditionally generated randomly for each signature—and if the random number generator is ever broken and predictable when making a signature, the signature can leak the private key, as happened with the Sony PlayStation 3 firmware update signing key.[10][11][12] In contrast, EdDSA chooses the nonce deterministically as the hash of a part of the private key and the message. Thus, once a private key is generated, EdDSA has no further need for a random number generator in order to make signatures, and there is no danger that a broken random number generator used to make a signature will reveal the private key.

Standardization and implementation inconsistenciesEdit

Note that there are two standardization efforts for EdDSA, one from IETF, an informational RFC 8032 and one from NIST as part of FIPS 186-5 (2019).[13] The differences between the standards have been analyzed,[14][15] and test vectors are available.[16]


Notable uses of Ed25519 include OpenSSH,[17] GnuPG[18] and various alternatives, and the signify tool by OpenBSD.[19] Usage of Ed25519 (and Ed448) in the SSH protocol has been standardized.[20] In 2019 a draft version of the FIPS 186-5 standard included deterministic Ed25519 as an approved signature scheme.[13]


Ed448 is the EdDSA signature scheme using SHAKE256 and Curve448 defined in RFC 8032. It has also been approved in the draft of the FIPS 186-5 standard.[13]


  1. ^ a b Josefsson, S.; Liusvaara, I. (January 2017). Edwards-Curve Digital Signature Algorithm (EdDSA). IRTF. doi:10.17487/RFC8032. ISSN 2070-1721. RFC 8032. Retrieved 2022-07-11.
  2. ^ a b c d Bernstein, Daniel J.; Duif, Niels; Lange, Tanja; Schwabe, Peter; Bo-Yin Yang (2012). "High-speed high-security signatures" (PDF). Journal of Cryptographic Engineering. 2 (2): 77–89. doi:10.1007/s13389-012-0027-1. S2CID 945254.
  3. ^ "Software". 2015-06-11. Retrieved 2016-10-07. The Ed25519 software is in the public domain.
  4. ^ Daniel J. Bernstein; Simon Josefsson; Tanja Lange; Peter Schwabe; Bo-Yin Yang (2015-07-04). EdDSA for more curves (PDF) (Technical report). Retrieved 2016-11-14.
  5. ^ Daniel J. Bernstein; Tanja Lange; Peter Schwabe (2011-01-01). On the correct use of the negation map in the Pollard rho method (Technical report). IACR Cryptology ePrint Archive. 2011/003. Retrieved 2016-11-14.
  6. ^ Daniel J. Bernstein; Tanja Lange. "ECDLP Security: Rho". SafeCurves: choosing safe curves for elliptic-curve cryptography. Retrieved 2016-11-16.
  7. ^ 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.
  8. ^ Daniel J. Bernstein (2017-01-22). "Ed25519: high-speed high-security signatures". Retrieved 2019-09-27. This system has a 2^128 security target; breaking it has similar difficulty to breaking NIST P-256, RSA with ~3000-bit keys, strong 128-bit block ciphers, etc.
  9. ^ Daniel J. Bernstein (2017-01-22). "Ed25519: high-speed high-security signatures". Retrieved 2020-06-01. Signatures fit into 64 bytes. […] Public keys consume only 32 bytes.
  10. ^ Johnston, Casey (2010-12-30). "PS3 hacked through poor cryptography implementation". Ars Technica. Retrieved 2016-11-15.
  11. ^ fail0verflow (2010-12-29). Console Hacking 2010: PS3 Epic Fail (PDF). Chaos Communication Congress. Archived from the original (PDF) on 2018-10-26. Retrieved 2016-11-15.
  12. ^ "27th Chaos Communication Congress: Console Hacking 2010: PS3 Epic Fail" (PDF). Retrieved 2019-08-04.
  13. ^ a b c Regenscheid, Andrew (October 2019). "FIPS 186-5 (Draft): Digital Signature Standard (DSS)". NIST. doi:10.6028/NIST.FIPS.186-5-draft. S2CID 241055751. Retrieved 2022-07-11. {{cite journal}}: Cite journal requires |journal= (help)
  14. ^ Konstantinos Chalkias, Francois Garillot and Valeria Nikolaenko (2020-10-01). Taming the many EdDSAs. Security Standardisation Research Conference (SSR 2020). Retrieved 2021-02-15.
  15. ^ Jacqueline Brendel, Cas Cremers, Dennis Jackson, and Mang Zhao (2020-07-03). The provable security of ed25519: Theory and practice. IEEE Symposium on Security and Privacy (S&P 2021). Retrieved 2021-02-15.{{cite conference}}: CS1 maint: multiple names: authors list (link)
  16. ^ "ed25519-speccheck". GitHub. Retrieved 2021-02-15.
  17. ^ "Changes since OpenSSH 6.4". 2014-01-03. Retrieved 2016-10-07.
  18. ^ "What's new in GnuPG 2.1". 2016-07-14. Retrieved 2016-10-07.
  19. ^ "Things that use Ed25519". 2016-10-06. Retrieved 2016-10-07.
  20. ^ Harris, B.; Velvindron, L. (February 2020). Ed25519 and Ed448 Public Key Algorithms for the Secure Shell (SSH) Protocol. IETF. doi:10.17487/RFC8709. ISSN 2070-1721. RFC 8709. Retrieved 2022-07-11.
  21. ^ "System security for watchOS". Retrieved 2021-06-07.
  22. ^ Matt Johnston (2013-11-14). "DROPBEAR_2013.61test".
  23. ^ "Heuristic Algorithms and Distributed Computing" (PDF). Èvrističeskie Algoritmy I Raspredelennye Vyčisleniâ (in Russian): 55–56. 2015. ISSN 2311-8563. Archived from the original (PDF) on 2016-10-20. Retrieved 2016-10-07.
  24. ^ Frank Denis. "Minisign: A dead simple tool to sign files and verify signatures". Retrieved 2016-10-07.
  25. ^ minisign-misc on GitHub
  26. ^ Frank Denis (2016-06-29). "libsodium/ChangeLog". GitHub. Retrieved 2016-10-07.
  27. ^ "OpenSSL CHANGES". July 31, 2019.
  28. ^ "python/ed25519.py: the main subroutines". 2011-07-06. Retrieved 2016-10-07.
  29. ^ "Software: Alternate implementations". 2015-06-11. Retrieved 2016-10-07.
  30. ^ "eBACS: ECRYPT Benchmarking of Cryptographic Systems: SUPERCOP". 2016-09-10. Retrieved 2016-10-07.
  31. ^ "Virgil Security Crypto Library for C: Library: Foundation". GitHub. Retrieved 2019-08-04.
  32. ^ "wolfSSL Embedded SSL Library (formerly CyaSSL)". Retrieved 2016-10-07.

External linksEdit