Re: [Cfrg] Compression of tori in draft-irtf-cfrg-pairing-friendly-curves-02

Yumi Sakemi <yumi.sakemi@lepidum.co.jp> Sat, 21 March 2020 07:29 UTC

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From: Yumi Sakemi <yumi.sakemi@lepidum.co.jp>
Date: Sat, 21 Mar 2020 16:29:36 +0900
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To: Watson Ladd <watson=40cloudflare.com@dmarc.ietf.org>
Cc: cfrg@irtf.org, Tetsutaro Kobayashi <tetsutaro.kobayashi.dr@hco.ntt.co.jp>, SAITO Tsunekazu <tsunekazu.saito.hg@hco.ntt.co.jp>
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Subject: Re: [Cfrg] Compression of tori in draft-irtf-cfrg-pairing-friendly-curves-02
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Dear Dr. Watson,

Thank you for your comments.
I'm glad you to read our draft!!

The compress techniques are very interesting for us.
According to your comments, I will discuss with co-authors and
update our draft.

Best regards,
Yumi

---
Yumi Sakemi, Ph. D.
Lepidum Co. Ltd.
E-Mail: yumi.sakemi@lepidum.co.jp

2020年3月20日(金) 6:09 Watson Ladd <watson=40cloudflare.com@dmarc.ietf.org>rg>:

> Dear all,
>
> In the latest draft I see that F_{p^k} is represented directly as
> polynomials modulo an irreducible of degree k, and GT is encoded
> accordingly. This is space inefficient.
>
> In most cases k is even. Let q=p^(k/2) Consider F_q[x]/(x^2+d), the
> quadratic extension of F_{q }, and now consider the result of a
> pairing in it. As in https://eprint.iacr.org/2004/032.pdf, it is clear
> that the result satisfies a quadratic equation: a^2-db^2=1, where the
> element is a+bx.
>
> Therefore we can send only a, which is much smaller, and ecode b as a
> single bit.
>
> This technique is very useful in any protocol where an element of GT
> must be sent.
>
> Sincerely,
> Watson Ladd
>
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