Internet Draft W. Ladd <draft-ladd-safecurves-00.txt> Grad Student Category: Informational UC Berkeley Expires 9 July 2014 5 January 2014 Addition Elliptic Curves for IETF protocols <draft-ladd-safecurves-00.txt> Status of this Memo Distribution of this memo is unlimited. This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet- Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. This Internet-Draft will expire on date. Copyright Notice Copyright (c) 2014 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (http://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Abstract This internet draft contains curves whose Jacobians are groups over Ladd, Watson Expires 9 July 2014 [Page 1]

Internet Draft ladd-safecurves 8 January 2014 which the Decisional Diffie-Hellman problem is hard, and which have implementation advantages. Ladd, Watson Expires 9 July 2014 [Page 2]

Internet Draft ladd-safecurves 8 January 2014 Table of Contents 1. Introduction ....................................................3 2. The curves ....................................................... 1. Introduction This document contains a set of elliptic curves over prime fields with many security advantages. 2. The Curves Each curve is given by an equation and a basepoint, together with an order. All curves are elliptic. Validation information is given at [SAFECURVES]. The names given in this document indicate the family. Curve25519 is a curve over GF(2^255-19), formula y^2=x^3+486662x^2+x, basepoint (9, 147816194475895447910205935684099868872646 06134616475288964881837755586237401), order 2^252 + 27742317777372353535851937790883648493. E-382 is a curve over GF(2^382-15), formula x^2+y^2=1-6725254x^2y^2, basepoint (3914921414754292646847594472454013487047 137431784830634731377862923477302047857640522480241 298429278603678181725699, 17), order 2^380 - 1030303207694556153926491950732314247062623204330168346855 M-383 is a curve over GF(2^383-187), forumla y^2=x^3+2065150x^2+x, basepoint (12, 473762340189175399766054630037590257683961716725770372563038 9791524463565757299203154901655432096558642117242906494), order 2^380 + 166236275931373516105219794935542153308039234455761613271 Curve383187 is a curve over GF(2^383-187), formula y^2=x^3+229969x^2+x, basepoint (5, 4759238150142744228328102229734187233490253962521130945928672202 662038422584867624507245060283757321006861735839455), order 2^380 + 356080847217269887368687156533236720299699248977882517025 Curve3617 is a curve over GF(2^414-17), formula x^2+y^2=1+3617x^2y^2, basepoint (17319886477121189177719202498822615443556957307604340815256226 171904769976866975908866528699294134494857887698432266169206165, 34), order 2^411 - 33364140863755142520810177694098385178984727200411208589594759 M-511 is a curve over GF(2^511-187), formula y^2 = x^3+530438x^2+x, basepoint (5, Ladd, Watson Expires 9 July 2014 [Page 3]

Internet Draft ladd-safecurves 8 January 2014 25004106455650724233689811491392132522115686851736085900709792642 48275228603899706950518127817176591878667784247582124505430745177 116625808811349787373477), order 2^508 + 107247547596357476240445315140681218420707566274348330289655408 08827675062043 3. Security Considerations This entire document discusses methods of implementing cryptography securely. The time for an attacker to break the DLP on these curves is the square root of the group order with the best known attacks. Curves of Edwards form are best when addition is required, those of Montgomery form make excellent candidates for Diffie-Hellman key agrement on the Kummer surface. Explicit formulas are in the Explicit-Formula Database [EFD]. 4. IANA Considerations IANA should maintain a registry of these curves, calling them safecurve-XXXX where XXX is the curve identifier. 5. References [SAFECURVES] safecurves.cr.yp.to [EFD] http://www.hyperelliptic.org/EFD/g1p/index.html Author Addresses Watson Ladd watsonbladd@gmail.com Berkeley, CA Ladd, Watson Expires 9 July 2014 [Page 4]