Re: [TLS] Twist security for brainpoolp256r1
Oleg Gryb <oleg_gryb@yahoo.com> Thu, 13 November 2014 06:37 UTC
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Date: Thu, 13 Nov 2014 06:35:21 +0000
From: Oleg Gryb <oleg_gryb@yahoo.com>
To: Johannes Merkle <johannes.merkle@secunet.com>, Oleg Gryb <oleg@gryb.info>, "tls@ietf.org" <tls@ietf.org>
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Subject: Re: [TLS] Twist security for brainpoolp256r1
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----- Original Message ----- > From: Johannes Merkle <johannes.merkle@secunet.com> > To: Oleg Gryb <oleg@gryb.info>; "tls@ietf.org" <tls@ietf.org> > Cc: > Sent: Wednesday, November 12, 2014 9:12 AM > Subject: Re: [TLS] Twist security for brainpoolp256r1 > > As Watson Ladd just pointed out to me, the terms "quadratic twists" > and "non-quadratic twists" are not common in math > textbooks. For a curve in Weierstrass form E: y^2 = x^3 + a*x + b mod p, the > term twist denotes a curve E': y^2 = x^3 > + v^2*a*x + v^3* b mod p. > - If v is a quadratic residue, i.e., if there is a w with w^2 = v mod p, then E > and E' are isomorphic mod p and thus > have equivalent security. > - If v is a quadratic non-residue, E' is not isomorphic to E and the curve > orders satisfy |E| + |E'| = p+2. > Text books often discuss only the case of quadratic non-residues (e.g. Blake, > Seroussi, Smart). > > In the case of the Brainpool curves, the twists mentioned in RFC 5639 are twists > via a quadratic residue and have the > same security as the respective random curves. Thanks, very helpful. Just to summarize yours and Manuel's notes in regard of quadratic non-residue, or non-quadratic twists (as they are called in other emails), they can be used as a source of malicious EC points to run invalid-curve attacks on the original curves, but since all implementations compliant with X9* standards including openssl must have point-on-curve validation, invalid-curve attacks and small-group attacks become irrelevant when it comes to brainpoool's openssl implementation. Note - small group attacks are not possible, because of a different reason: cofactor for all brainpool curves is equal to 1. Efficiency is still an issue for me. Since curves such as NIST P-256 do have optimized EC arithmetic, but nothing like that is available for brainpool curves, it would be nice to know what the delta is. If it's up to 30% percent, I would probably ignore, if it's essentially more than that, I might need to re-consider my choice. If no quantitative data is available, I would probably need to write/run my own tests. > > -- > Johannes >
- [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Johannes Merkle
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Manuel Pégourié-Gonnard
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Johannes Merkle
- Re: [TLS] Twist security for brainpoolp256r1 Johannes Merkle
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Manuel Pégourié-Gonnard
- Re: [TLS] Twist security for brainpoolp256r1 Watson Ladd
- Re: [TLS] Twist security for brainpoolp256r1 Manuel Pégourié-Gonnard
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Ilari Liusvaara
- Re: [TLS] Twist security for brainpoolp256r1 Alyssa Rowan
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb
- Re: [TLS] Twist security for brainpoolp256r1 Oleg Gryb