A security issue in the ModularSquareRoot function leads to a DOS attack

[roadicing]

Hi, recently I found a security issue in the ModularSquareRoot function of Crypto++ library that would cause an infinite loop, since this function is being used in ECP::DecodePoint, an attacker could potentially craft a malformed DER public key file, and any user or server attempting to read this public key file in processes such as ECDSA may be susceptible to a DOS attack.

Issue

The issue lies in the second while loop in this function, the loop starts with n = 2 and increments n by one each time until we find an n such that Jacobi(n, p) = -1. However, this overlooks the case when p is in the form of m^2 and m is an odd number.

In this case, jacobi(n, p) = jacobi(n, m^2) can be converted into the product of a series of squares of Jacobi symbols using the properties of Jacobi symbols, this means that its value can only be 1 or 0 but not -1, therefore, no matter how n continues to increase, it will never satisfy Jacobi(n, p) = -1 to break out of the loop, thus leading to an infinite loop.

Furthermore, since this function is being used in ECP::DecodePoint, we can construct a malformed DER public key file that includes an elliptic curve parameter p as:

72358384006116823815439217615866351214375729203207450702838342058601772551609

which is the square of:

268995137513890432434389773128616504853

and repackage it into a DER file, then any attempt to read and parse this DER file in Crypto++ will trigger an infinite loop immediately.

In addition, since the ModularSquareRoot function is in the branch that handles 03 compression encoding, we need to ensure that the points in the constructed DER file are in compressed form (start with 03).

Fix

To fix this issue, the simplest method is to check whether p is a prime number at the beginning of the ModularSquareRoot function. If it is not a prime number, then directly reject any further calculations (after all, even if it is not rejected, the result calculated in this way will be wrong).