Re: [Cfrg] Question about A=6 Montgomery over 2^89-1
"Grigory Marshalko" <marshalko_gb@tc26.ru> Sat, 12 December 2015 18:21 UTC
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From: Grigory Marshalko <marshalko_gb@tc26.ru>
To: Dan Brown <dbrown@certicom.com>, cfrg@ietf.org
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Subject: Re: [Cfrg] Question about A=6 Montgomery over 2^89-1
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Yes, exactly 13, see for example http://www2.warwick.ac.uk/fac/cross_fac/complexity/people/students/dtc/students2013/klaise/janis_klaise_ug_report.pdf Regards, Grigory Marshalko, expert, Technical committee for standardisation "Cryptography and security mechanisms" (ТC 26) www.tc26.ru 12 декабря 2015 г., 18:23, "Dan Brown" <dbrown@certicom.com> написал: > So 66^3 is one of the few (13???) integral j-invariants with CM over the rationals (the > Baker-Stark-Heegner theorem?). And yes Elkies shows that half the primes will give a supersingular. > I'll look at how j=66^3 corresponds to A=6 in 2016, and eventually try to sort out whether there's > much to say about small |A|. > > Original Message > From: Grigory Marshalko > Sent: Friday, December 11, 2015 4:09 PM > To: Dan Brown; cfrg@ietf.org > Subject: Re: [MASSMAIL][Cfrg] Question about A=6 Montgomery over 2^89-1 > > This seems to be a better answer > > http://alexricemath.com/wp-content/uploads/2013/07/EC2.pdf > > Regards, > Grigory Marshalko, > expert, > Technical committee for standardisation "Cryptography and security mechanisms" (ТC 26) > www.tc26.ru > 11 декабря 2015 г., 23:20, "Grigory Marshalko" <marshalko_gb@tc26.ru> написал: > >> Hi, >> >> May be this is the case: >> from wiki: >> If an elliptic curve over the rationals has complex multiplication then the set of primes for which >> it is supersingular has density 1/2. If it does not have complex multiplication then Serre showed >> that the set of primes for which it is supersingular has density zero. Elkies (1987) showed that >> any elliptic curve defined over the rationals is supersingular for an infinite number of primes. >> >> and this is also may be useful http://pages.cs.wisc.edu/~cdx/ComplexMult.pdf >> >> Regards, >> Grigory Marshalko, >> expert, >> Technical committee for standardisation "Cryptography and security mechanisms" (ТC 26) >> www.tc26.ru >> 11 декабря 2015 г., 00:22, "Dan Brown" <dbrown@certicom.com> написал: >> >>> Hi, >>> >>> I stumbled upon something surprising (to me), using Sage (while searching >>> for something else). >>> >>> The Montgomery curve y^2 = x^3 + 6x^2 + x over the field of size 2^89-1, has >>> order 2^89, so it is maximally vulnerable to Pohlig-Hellman. (Other >>> details: it has order p+1, so is also vulnerable to MOV. I haven't checked >>> yet, but I'd _bet_ it's supersingular. It has j-invariant 66^3.) >>> >>> As is well-known, the supersingular curve y^2 = x^3 + x also has order 2^89 >>> (it has j-invariant 1728=12^3). But I recall a result of Koblitz saying >>> that curves over F_p with order p+1 are very rare (among isomorphism >>> classes). Naively, I would think that finding two such curves so close >>> together (A=0 and A= 6) has negligible chance, unless these weak curves are >>> distributed towards small |A|. >>> >>> Nonetheless, I still hope that this does _not_ indicate some general _weak_ >>> correlation between Montgomery curves with a small coefficient and known >>> attacks. >>> >>> To that end, I'd be curious if somebody here could explain the theory behind >>> this example curve. For example, it would be re-assuring to explain this as >>> a mere one-time coincidence, rather than a higher chance of a known attack >>> (e.g. MOV or PH) on smaller-coefficient curves. (Purely speculating: maybe >>> there's a good theory of supersingular j-invariants for each prime p, then a >>> way to deduce A from j, such that p=2^89-1 and j=66^3 formed a superstorm to >>> arrive at a small A=6.) >>> >>> Absent such an explanation, the worry is that if known attacks more >>> generally exhibit this kind of correlation with coefficient size, then how >>> wise is it to suggest small-coefficient curve as a remedy against secret >>> attacks? >>> >>> I am aware that there are other worries of a different nature >>> ("manipulation") involved with methods that generate larger coefficients, >>> but maybe there's a good way to balance both concerns. >>> >>> Best regards, >>> >>> Daniel Brown >>> >>> _______________________________________________ >>> Cfrg mailing list >>> Cfrg@irtf.org >>> https://www.irtf.org/mailman/listinfo/cfrg
- [Cfrg] Question about A=6 Montgomery over 2^89-1 Dan Brown
- Re: [Cfrg] [MASSMAIL] Question about A=6 Montgome… Grigory Marshalko
- Re: [Cfrg] [MASSMAIL] Question about A=6 Montgome… Grigory Marshalko
- Re: [Cfrg] Question about A=6 Montgomery over 2^8… Dan Brown
- Re: [Cfrg] Question about A=6 Montgomery over 2^8… Grigory Marshalko
- Re: [Cfrg] Question about A=6 Montgomery over 2^8… Paterson, Kenny
- Re: [Cfrg] Question about A=6 Montgomery over 2^8… Ben Laurie
- Re: [Cfrg] Question about A=6 Montgomery over 2^8… Dan Brown