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Date: Thu, 11 Sep 2014 16:47:25 +0200
From: CodesInChaos <>
Subject: Re: [PHC] Makwa parameter generation

On Thu, Sep 11, 2014 at 3:54 PM,  <> wrote:
> To keep on with security levels as recommended in 2014, we would
> prefer prime factors of at least 1024 bits

It's not that simple. We need a modulus of at least 2048 bits which
contains no primes smaller than 550 or so bits.
NFS has cost based on the modulus size, but ECM which can take
advantage of smaller factors is less efficient, so a 550 bit factor
should match a 2048 bit modulus.
We also can use trial division, ECM etc. to eliminate all moduli with
factors smaller than 250 or so bits.

> Now a raw Makwa could run with a huge modulus;

There isn't really a reason to actually use that huge modulus. You can
use several of those 3000 bit moduli chaining or combining them such
that the result is secure if at least one of them is secure. It still
gives an attacker who factors some of them an advantage, but you can't
avoid that with this generation method.

>  - When the modulus is large, classic implementations of the squaring
>    operation (Montgomery multiplication) are no longer optimal; to
>    get the fastest code (and you want to get it, because that's what
>    the attacker will do), you have to switch to more advanced
>    algorithms such as FFT-based multiplication, and that entails
>    considerable development effort. When Makwa uses a 2048-bit RSA
>    modulus, then existing optimized code for RSA can be reused, and
>    that is a good thing; huge moduli deprive us of that advantage.

If you'd use a big modulus, you should be able to optimize it with CRT.
While you don't know individual primes, you know smaller composite factors.

>  - Delegation cost raises with the modulus size (quadratically, if you
>    use Montgomery multiplication).

Why does montgomery multiplication have larger asymptotic cost
compared to multiplication modulo a power of two?
I admit that I'm not familiar with the details of montgomery
multiplication, but doesn't it just double the number of
multiplication and add some conversion overhead?

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