# Bleichenbacher 1998 “Million message attack” on RSA

I have been reading Bleichenbacher's 1998 paper on a forged message attack on RSA. The paper assumes access to an Oracle that takes a ciphertext $c$ and will check the decrypted text for valid PKCS #1 padding and returns the validity of the padding. This side-channel can be used for an attack since we can send a forged ciphertext by selecting random integers $s$ and compute:

$c' = cs^{e}\, \mbox{mod}\, n$

If $c'$ is PKCS conforming we know that the two most significant bytes of $ms = c'^{d} \mbox{mod}\,n$ were equal to 0x00 0x02. If we define

$B=2^{8(k-2)}$

where $k$ is the length of $n$ in bytes, then we know

$2B\leq ms\,\mbox{mod}\,n<3B$

Now we have a range $M_{0}$ in which we know $ms$ to lie:

$M_{0} = {[2B, 3B-1]}$

The attack now proceeds by iteratively generating more valid forged ciphertexts for integers $s_{i}$ and with the gained knowledge reduce the range of $M_{i}$

I am having trouble understanding Step 3 in the paper which deals with narrowing the set of solutions:

$M_{i} \leftarrow \cup_{a, b, r} \lbrace [\mbox{max}(a, [\frac{2B+rn}{s_{i}}]), \mbox{min}(b, [\frac{3B-1+rn}{s_{i}}])]\rbrace$

for all $[a,b] \in M_{i-1}$ and $\frac{as_{i}-3B+1}{n}\leq r< \frac{bs_{i}-2B}{n}$

How would this be implemented in code? It looks to me that we would need to compute each range for every $r$ within the specified bounds e.g.:

for (r=r_min; r<r_max; ++r)
max(a, x);
min(b, y);


However this seems to be intractable to me and I think I am making an error in trying to convert the math to code. Anyone maybe see where I am going wrong?

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Yes, you have to compute the interval for each possible $r$ (by looping over $r$). The reason is that for big $s_i$ the current interval $[a, b]$ can be stretched by multiplying with $s_i$ so long that it intersects $[2B+rn, 3Brn-1]$ for several $r$. (In the line starting with $M_i \leftarrow$ you should have $s_i$ in the denominator). –  j.p. Oct 19 '13 at 13:12
This might help crypto.stackexchange.com/questions/12688/… –  Bush Oct 1 at 10:33