Exponent 3 damage spreads...

[Thierry Moreau]

Jostein Tveit wrote:

> Ben Laurie <ben at algroup.co.uk> writes:
> 
> 
>>...thought this might interest people here.
> 
> 
> Anyone got a test key with a real and a forged signature to test
> other implementations than OpenSSL?
> 

If I understand the attack mathematics correctly, the following 
algorithm should give you an alleged signature value that would be 
mistakenly accepted by a flawed RSA implementation. I didn't implement 
the algorithm, and I will not make suggestions as a convenient big 
number arithmetic tool to implement it.

Note: The algorithm output value is NOT A FORGED SIGNATURE, since a 
non-flawed RSA signature verification implementation will correctly 
reject it. Nonetheless, using public exponent 3 with any use of RSA 
should be deprecated.

For the record, I am referring to
Hal Finney, "Bleichenbacher's RSA signature forgery based on 
implementation error" Wed, 30 Aug 2006
http://www.mail-archive.com/cryptography@metzdowd.com/msg06537.html

Input:

N, large public modulus (of unknown factorization)
h, hash value

Constant:

p: hex 01 FF 00 30 21 30 09 06 05 2B 0E 03 02 1A 05 00 04 14

A random binary source (e.g. large enough PRNG output)

Algorithm:

(A) find the largest value of r such that b=(p*2^20+h)*2^(8r) such that 
b+2^(8r)-1<N

(B) select random a, 0<a<N^2, then set c=a*N^2+b+2^(8r)-1

(C) using a simple binary search, find the d = integer cubic root of c

(D) if d^3<a*N^2+b, go back to step (B) -- if it occurs with a high 
probability, that's a failure of the approach proposed here, intuition 
suggests that the probability is either very close to zero, or very 
close to one

(E) set alleged signature s=d mod N (indeed, d<N, so s=d) and validate 
(merely as a software self-check) that (s^3 mod N) div 2^(8r) equals 
(p*2^20+h)

(F) output alleged signature s

Regards,

-- 

- Thierry Moreau

CONNOTECH Experts-conseils inc.
9130 Place de Montgolfier
Montreal, Qc
Canada   H2M 2A1

Tel.: (514)385-5691
Fax:  (514)385-5900

web site: http://www.connotech.com
e-mail: thierry.moreau at connotech.com


---------------------------------------------------------------------
The Cryptography Mailing List
Unsubscribe by sending "unsubscribe cryptography" to majordomo at metzdowd.com