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Earlier Perry asked for references to blinding methods. I can provide
one, but the mathematics is fairly simple and straightforward, so I'll
just talk about it here. Any good intro text on number theory, or even
some crypto books (Denning, Seberry & Peiprzyk) will have all the math
you need to know.
Conceptually, when you blind a message, nobody else can read it. A
property about blinding is that under the right circumstances if another
party digitally signs a blinded message, the unblinded message will
contain a valid digital signature.
So if Alice blinds the message "I owe Alice $1000" so that it reads (say)
"a;dfafq)(*&" or whatever, and Bob agrees to sign this message, later
Alice can unblind the message Bob signed to retrieve the original. And
Bob's digital signature will appear on the original, although he didn't
sign the original directly.
Mathematically, blinding a message means multiplying it by a number (think
of the message as being a number). Unblinding is simply dividing the
original blinding factor out.
One thing protocols that involve signing blinded messages have to watch
for are messages that you don't really want to sign. If someone asks
you to digitally sign a random stream of symbols, remember that what you
sign may be unblinded to reveal a contract, etc. Techniques of getting
around this seem to be cut-and-choose protocols, which I won't get into
Judy Moore's paper "Protocol Failures in Cryptosystems" - appeared in
IEEE Proceedings, May 1988, Vol. 76, No.5, and also appears in the big
IEEE Crypto book Simmons edited - discusses this as the "Notory Protocol."
I'll excerpt from her paper:
1) To setup the notary protocol, Alice chooses RSA parameters p, q, e, d
She publishes her public key e, and n = pq
Bob now wants to trick Alice into signing a message which says she owes
him some money.
2) Bob now chooses an arbitrary number x. He computes y = x^e mod n.
e is Alice's public key and everyone knows it.
Bob can now use y (bliding factor) to obtain forgeries on another
3) Bob forms the messages he wants, and muliplies in the blinding factor
y. He calulates m' = ym
4) Alice agrees to sign a message m' which is total gibberish. She
computes s = m'^d mod n and returns the result s to Bob.
5) Bob then calculates s' = s x^-1.
A valid signature on m' is m'^d mod n = (ym)^d mod n
= y^d m^d mod n
= x m^d mod n
so all Bob has to do is remove x from what Alice signed and he has
m^d mod n, Alice's digital signature on message m.
An example with numbers (I'm currently learning Scheme so I will give
the Scheme code I used):
Alice chooses p = 43, q = 47, thus n = 2021
d = 5, gcd(d,phi(n)) = 1, so e = 773 (* see note below)
Bob chooses x = 314, y = x^e mod n
= 314^773 mod 2021
= (expt-mod 314 773 2021)
Bob creates the message m = 99, which means Alice owes him money.
Bob blinds the message by calculating m' = ym
= 1271 99
= (modulo (* 1271 99) 2021)
Alice agrees to sign 527, a message which is possible unintelligible.
She calculates s = m'^d mod n
= 527^5 mod 2021
= (expt-mod 527 5 2021)
Bob takes Alice's signed message s = 360 and unblinds it.
He calculates s' = s x^-1
First he calculates x^-1 = 354 (*see note below)
Then, s' = 360 364
= (modulo (* 360 354) 2021)
As a check, say Alice decides to sign the original message m=99.
Then, the signed message would be m^d mod n
= 99^5 mod 2021
So Bob does indeed have a message which is his original message with
Alice's digital signature on it. Be more careful next time, Alice.
There are two ways I know of to calculate the inverse of a number.
First, x = a^((phi(n)-1) mod n will yeild x, the inverse of a mod n.
But, sharp eyed people will note you need phi(n) to calculate this way
- - and only Alice knows the factorization of n. So she can calculate
e, the inverse of d mod phi(n) as:
e = d^((phi(phi(n))-1) mod phi(n)
= 5^(phi(1932)-1) mod 1932
= 5^(264-1) mod 1932
= 5^263 mod 1932
= (expt-mod 5 263 1932)
Now how does Bob calculate the inverse of x mod n? He does not know
the factorization of n.
Well, for the purposes of this problem I did now how n factors so I
used it :-)
BUT, there is a way you can calculate the inverse of a number mod n
without knowing how n factors. The algorithm is related to Euclid's,
the one that you can use to tell if two numberse a relatively prime.
Essentially, you run through Euclid's forwards, and then in the reverse
direction, grouping and substituting, and the inverse will pop out.
Once you do it by hand it will be clear, and you won't ever want to do
it by hand again :-)
If you use Mathematica, it will let you do PowerMod[x, -1, n] to
calculate the inverse of x mod n. But Scheme won't since the three
integers for expt-mod must be positive.
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