CVE-2024-1544

Generating the ECDSA nonce k samples a random number r and then <br /> truncates this randomness with a modular reduction mod n where n is the <br /> order of the elliptic curve. Meaning k = r mod n. The division used <br /> during the reduction estimates a factor q_e by dividing the upper two <br /> digits (a digit having e.g. a size of 8 byte) of r by the upper digit of <br /> n and then decrements q_e in a loop until it has the correct size. <br /> Observing the number of times q_e is decremented through a control-flow <br /> revealing side-channel reveals a bias in the most significant bits of <br /> k. Depending on the curve this is either a negligible bias or a <br /> significant bias large enough to reconstruct k with lattice reduction <br /> methods. For SECP160R1, e.g., we find a bias of 15 bits.