Exponent 3 damage spreads...

Tue Sep 19 23:55:38 EDT 2006

I tried coming up with my own forged signature that could be validated with
OpenSSL (which I intended to use to test other libraries).  I haven't
succeeded, either because in the particular example I came up with OpenSSL
does something that catches the invalid signature, or I messed up somewhere
(the likelihood of which is far from negligible).  Unfortunately, I don't
have much more time to play with this.  I decided to share the methodology I
used with those of you who are interested in case the info is helpful to
anyone, or someone can tell me why the signature I produced doesn't get
validated by OpenSSL.

I followed the instructions of Hal Finney's excellent post:
http://www.mail-archive.com/cryptography@metzdowd.com/msg06537.html

I started out by generating 3072 RSA key pair, with public exponent e = 3.

openssl genrsa -des3 -3 -out my.key 3072

the resulting key can be found bellow, the passwords is "test" if you ever
want to use it.

Then I created the corresponding public key certificate:

openssl req -new -x509 -days 1001 -key my.key -out my.cer

The public key certificate can be found bellow as well.  You can import this
in Windows key store, for example.

I then created a plaintext file, messageAunicode.txt, for which I computed a
signature on (a valid signature).  The idea was then to forge a signature on
an alternate messageBunicode.txt, without using the private key of course.
The two files can be found in attachment, they are in Unicode because I
wanted to also try this out with a VBscript implementing a signature
function using CAPICOM. (you can get a file in Unicode by opening it with a
simple text editor that allows you to save as Unicode, such as notepad, and
erase any extra bytes (header) with a hex editor such as XVI32).

The hashes of these files are

openssl dgst -sha1 messageAunicode.txt
SHA1(messageAunicode.txt)= eb8302606217ae549fe6ab1345f0b4c804195367

openssl dgst -sha1 messageBunicode.txt
SHA1(messageBunicode.txt)= 5d89b46034e0f41a920b2fa964e230ebb2d040b0

Now, create the valid signature over messageAunicode.txt to see what the
output looks like:

openssl dgst -hex -sha1 -sign my.key messageAunicode.txt
Enter pass phrase for my.key:
SHA1(messageAunicode.txt)=00d3cda91b578b6df29aeb140272bd9198759f79fa10dc410b
5d10362048ac7abe5df7fe0d94a6646e791c5b95b29f2c6384a570769dc888ed0b7ad510ccd3
c758cebeb648511620490e0fd54162badb1ed05411acc853509b62a4c1b242e1e2f737a1e7e4
340f5a79b05ec3475d7ba6fc73b3302f1258abac1079f11e8dfb9fc09d42716ba4054add460b
b12fc1b0b8d5d32db50395374aeb3c215c2bc566328d2f03bf043068c5c9abc649ba1767e97d
f32b6aa734594ee22fffe7fb06ea3b77030e79bd6fe7683ab7ffce462abfba5777b3914de466
5b86c1ec203feb6fccb3dadb8ba51fd87a7457c62385418e65d17809c4256e3d27dc2017d7a0
93c8bd193a09168f34d522dd7d3afb95fc61c9f4339091cf25d78bf461b4ea5620eed722ab7d
3eff99cea4a4f546bff6ce338d7763aff20a9b61452da07179590d3316bbce63b06b43d996d7
75d6843f46633ff107a3c866e3b0a8aaaea31f4a2048c9fcb448958287f8e961c9f3393e18fc
9a05460d51a286737aec14a1a7b27a51

Now, let's do some bignumber math.
I wanted to look at the value obtained when you verify a signature (RSA
encrypt the signature with exponent 3).  I use BC, a bignumber calculator I
like allot:
http://www.gnu.org/software/bc/
A version that can be installed on Windows:
http://gnuwin32.sourceforge.net/packages/bc.htm

I use a fast modexp function I implemented for BC, available at
http://crypto.cs.mcgill.ca/~stiglic/Programming/modgroup.bc

You can load it by simply calling bc like this:

bc modgroup.bc

I did the calculations all in hex, so typed the following in bc

obase=16
ibase=16

Now, denote by s the signature given above, e = 3 and m is the modulus in
the public key certificate I generated.  When pasting the values into BC,
the hex digits need to be in capital letters.  You can get the value of the
modulus by using a ASN.1 viewer, such as ASN.1 Editor or Peter Gutmann's
ASN1dump.  Here are the BC calculations:

s=00D3CDA91B578B6DF29AEB140272BD9198759F79FA10DC410B5D10362048AC7ABE5DF7FE0D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m=01D851D5148345606F586935D227CD5CF7F04F890AC5024178BA5F4EE85D7796918C3DC7A5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modexp(s, 3, m)
1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFF003021300906052B0E03021A05000414EB8302606217AE549FE6AB134\
5F0B4C804195367

You can see the 01 FF FF FF .. FF pattern followed by 
003021300906052B0E03021A05000414
Which is the ASN.1 header (with 00 in front of it) followed by 
EB8302606217AE549FE6AB1345F0B4C804195367
Which is the hash of messageAunicode.txt as we calculated above.

Now, let's do the calculations in Hal's post, using the above ASN.1 header
(which stays the same for all signatures using the above key and the SHA1
algorithm), with the hash of messageBunicode.txt in order to forge a
signature over the content of messageBunicode.txt 

Since we are calculating in hex, here are the equivalences of the exponents
which are given in decimal in Hal's post:
288d = 120h
1019d = 3FBh
34d = 22h

Now, in bc, using the notations of Hal's post :

d=003021300906052B0E03021A050004145D89B46034E0F41A920B2FA964E230EBB2D040B0
n = 2^120 - d
n
FFCFDECFF6F9FAD4F1FCFDE5FAFFFBEBA2764B9FCB1F0BE56DF4D0569B1DCF144D2F\
BF50
s = 2^3FB - (n*2^22)/3
s
7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEAAEAD6EAB6B2B18EBD595822B1555\
AC5D20CF08046814578C2B994E1DBD8413A43C0564000000000

Now let's look at s^3
1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF003021300906052B0E03021A05000\
4145D89B46034E0F41A920B2FA964E230EBB2D040B00000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000\
0000000002A9AA11CBB60CB35CB569DDD576C272967D774B02AE385C6EE43238C8C9\
1477DBD0ED06ECF8BC4B8D3DC4D566FA65939092D09D13E0ED8F8BE5D5CB9E72C47C\
743B52BBFA7B9697FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFDA285694CD9347AB7528\
D15F9D0DBF0C82C967D1C7CA3CCF69D2E09519FEAD7B96F1FCCB6D7D78AC9B244C2D\
85C08FEE0982D080AB2250A546F64BF15B1C540EA5655A36E52756CC57BBB11BBA3B\
81D72CE1FB7EBFB784027F3087CA7078541278C45764E6F2B1F3E532400000000000\
00000000000000000

This has the form we are looking for, the 01 FF FF ... FF header that ends
with 00, and then we have 
03021300906052B0E03021A050004145D89B46034E0F41A920B2FA964E230EBB2D040B0
which is the d we started out with, and the rest is the GARBAGE part.

Only one problem, s^3 is larger than m, so if we computed modexp(s, 3, m)
the result would be rounded out modulo m and we would loose the above
structure.
So what I did (and this might be a mistake, but I believe it works because
of the explanation I will give), is took s as the signature but cutting out
00 at the end in order to produce s':

sp=s/100

sp^3 will start with the same digits as s^3, but will be smaller.

Let's look at the modexp calculation

modexp(sp, 3, m)
1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF003021300906052B0E03021A05000\
4145D89B46034E0F41A920B2FA964E230EBB2D040B00000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000\
0000000002A9AA11CBB60CB35CB569DDD576C272967D774B02AE385C6EE43238C8C9\
1477DBD0ED06ECF8BC4B8D3DC4D566FA65939092D09D13E0ED8F8BE5D5CB9E72C47C\
743B52BBFA7B9697FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFDA285694CD9347AB7528\
D15F9D0DBF0C82C967D1C7CA3CCF69D2E09519FEAD7B96F1FCCB6D7D78AC9B244C2D\
85C08FEE0982D080AB2250A546F64BF15B1C540EA5655A36E52756CC57BBB11BBA3B\
81D72CE1FB7EBFB784027F3087CA7078541278C45764E6F2B1F3E532400000000000\
00000000000

Good!
So sp is my tentative forged signature.  I saved it to a file in order to
try to verify it with OpenSSL.  I don't know how to make openssl command
line to verify a signature that is represented in hex (it can output a
signature in hex, but I can't seem to find how to verify one in this format,
seems to only accept binary).  I look for a hex2bin converter, which is
harder to come by than you would think.  Finally, not wanting to code
something (to lazy) I used XVI32 hex editor in some unorthodox way to
transform the hex signature in binary.  The result can be found in
attachment.  Unfortunately (or fortunately), I can't get OpenSSL to validate
this signature.

openssl dgst -prverify my.key -sha1 -signature opensslB-fake.sig
messageBunicode.txt
Enter pass phrase for my.key:
Verification Failure

(I copied here the verification command that uses the private key, you can
of course use the verification command with just the public key, at the
moment of writing this I simply couldn't remember how to make it work).

If you know what is happening, please tell me.  Hope the info is useful to
someone!

--Anton

-----BEGIN RSA PRIVATE KEY-----
Proc-Type: 4,ENCRYPTED
DEK-Info: DES-EDE3-CBC,162F46975BB386BE

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3BlWtFuV0KsNymLUFHQ9gbwjImY8eGyaA2Dys0iFIC0=
-----END RSA PRIVATE KEY-----

-----BEGIN CERTIFICATE-----
MIIFaTCCA9KgAwIBAgIJAObO4xl/ruKUMA0GCSqGSIb3DQEBBQUAMIGAMQswCQYD
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-----END CERTIFICATE-----
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