Rules of the game
Summer School
Number Theory for Cryptography
k
Y
N=
i
pα
i .
i=1
• What do we do in practice? Which size is doable?
Warwick University, june 2013
Factorization : number field sieve
O(exp(c(log N)1/3 (log log N)2/3 )) ; 768 bits (a lot of people, 2010).
Primality: hopefully without too much factoring, past some easy
trial division; 30,000 decimal digits.
Integer factorization
F. Morain
• Complexity question: to which class does isPrime? belong?
Best : P (e.g., integer multiplication).
At least : RP.
And: what about proofs?
morain@lix.polytechnique.fr
F. Morain – École polytechnique – Warwick Summer School – June 2013
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Complexity classes
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How difficult is factoring?
Solovay-Strassen
(1974)
Lehman (1982)
co-NP
co-RP
•
ZPP
P
•
NP ∩ co-NP
230
220
200
PSPACE
180
•
BPP
Agrawal
Kayal
Saxena
(2002)
F. Morain – École polytechnique – Warwick Summer School – June 2013
Adleman
Huang
(1986)
140
120
100
RP
•
•
NP
•
• 2773 + 1
•
•RSA-140
• •
•
• p(11887)
F9
•
•
n
• • RSA-155
RSA-129 Internet
Fn = 22 + 1
•
• •
•
••
•
•
•
•
RSA-130
11104 + 1 email
••
•
•
••
Cray 1S
40 •
−1
(215 − 135)41 − 1
Cray X-MP
60
Pratt (1974)
10
CFRAC: •
QS: •
SNFS: •
GNFS: •
SUN 3 cluster
80
F7 IBM 360/91
••
•
••
•
RSA-120
MPP
70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00
Thanks to E. Thomé
F. Morain – École polytechnique – Warwick Summer School – June 2013
160
•
211
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And alsoi: 03/1991: 2,463+ (c101) on a Cray Y-MP4/464; 04/1992: RSA-110 on a MasPar (16K nodes).
F. Morain – École polytechnique – Warwick Summer School – June 2013
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The reign of clusters
310
280
CFRAC: •
QS: •
SNFS: •
GNFS: •
240
•
•
2, 1039−
6, 353−
2, 773+
260
220
Plan
•
•
2, 751−
• Today: primality.
•
RSA-768
• Tomorrow: elementary factoring algorithms.
2, 1642M
RSA-200
•
• Last but not least: powerful factoring.
•
10, 211−
•
200
RSA-160
•
RSA-640
15
41
•
(2 − 135) − 1
180
•
11, 281
•
RSA-140
•
160
•
•
RSA-576
•
140
98 99 00 01 02 03 04 05 06 07 08 09 10
F. Morain – École polytechnique – Warwick Summer School – June 2013
You must practice with numbers!
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F. Morain – École polytechnique – Warwick Summer School – June 2013
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