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Various hacking notes -*- text -*-
=======================
Taking optimized MPI code out of GMP:
-------------------------------------
I generated the pentium4/* files by glueing the existing assembler
prologues to the GMP 4.2.1 assembler files generated with the m4
tool in GMP's build process, for example:
$ m4 -DHAVE_CONFIG_H -D__GMP_WITHIN_GMP -DOPERATION_rshift -DPIC \
rshift.asm >tmp-rshift.s
Then tmp-rshift will contain the assembler instructions for the
configured platform. Unfortunately, this way the comments are lost.
For most files I re-inserted some of the comments, but this is
tedious work.
Debugging math stuff:
---------------------
While debugging the ECC code in libgcrypt, I was in need for some
computer algebra system which would allow me to verify the numbers
in the debugging easily. I found that PARI (pari-gp package in
Debian) has support for elliptic curves. The below commands shows
how they are set up and used with an example.
===8<========
hextodec(s)=local(v=Vec(s),a=10,b=11,c=12,d=13,e=14,f=15,A=10,B=11,C=12,D=13,E=14,F=15,h);if(#setunion(Set(v),Vec("0123456789ABCDEFabcdef"))>22,error);for(i=1,#v,h=shift(h,4)+eval(v[i]));h
p = hextodec("01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF")
a = hextodec("01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC")
b = hextodec("51953EB9618E1C9A1F929A21A0B68540EEA2DA725B99B315F3B8B489918EF109E156193951EC7E937B1652C0BD3BB1BF073573DF883D2C34F1EF451FD46B503F00")
/* Set up y^2 = x^3 + ax + b mod (p). */
e = ellinit(Mod(1,p)*[0,0,0,a,b]);
gx = hextodec ("00C6858E06B70404E9CD9E3ECB662395B4429C648139053FB521F828AF606B4D3DBAA14B5E77EFE75928FE1DC127A2FFA8DE3348B3C1856A429BF97E7E31C2E5BD66")
gy = hextodec ("011839296A789A3BC0045C8A5FB42C7D1BD998F54449579B446817AFBD17273E662C97EE72995EF42640C550B9013FAD0761353C7086A272C24088BE94769FD16650")
g = Mod(1,p)*[gx,gy]
n = hextodec ("01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFA51868783BF2F966B7FCC0148F709A5D03BB5C9B8899C47AEBB6FB71E91386409")
/* Verify that G is on the curve, and that n is the order. */
ellisoncurve (e,g)
isprime (n)
ellpow (e,g,n)
d = hextodec ("018F9573F25059571BDF614529953DE2540497CEDABD04F3AF78813BED7BB163A2FD919EECF822848FCA39EF55E500F8CE861C7D53D371857F7774B79428E887F81B")
qx = hextodec ("00316AAAD3E905875938F588BD9E8A4785EF9BDB76D62A83A5340F82CB8E800B25619F5C3EA02B7A4FA43D7497C7702F7DFBEAC8E8F92C3CAABD9F84182FDA391B3B")
/* Note: WRONG! (It is apparent that this is the same as X shifted by
8 bit). */
qy = hextodec ("0000316AAAD3E905875938F588BD9E8A4785EF9BDB76D62A83A5340F82CB8E800B25619F5C3EA02B7A4FA43D7497C7702F7DFBEAC8E8F92C3CAABD9F84182FDA391B")
q = Mod(1,p)*[qx,qy]
/* Calculate what Q should be given d. */
ellpow (e,g,d)
/* This is not 0 and thus shows that libgcrypt gave Q and d that do
not match. */
ellpow (e,g,d) - q
====8<=====================

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