Known–Plaintext–Only Attack on RSA–CRT with Montgomery Multiplication

  • Martin Hlaváč
Conference paper

DOI: 10.1007/978-3-642-04138-9_10

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5747)
Cite this paper as:
Hlaváč M. (2009) Known–Plaintext–Only Attack on RSA–CRT with Montgomery Multiplication. In: Clavier C., Gaj K. (eds) Cryptographic Hardware and Embedded Systems - CHES 2009. Lecture Notes in Computer Science, vol 5747. Springer, Berlin, Heidelberg


The paper describes a new attack on RSA–CRT employing Montgomery exponentiation. Given the amount of so-called final subtractions during the exponentiation of a known message (not chosen, just known), it creates an instance of the well known Hidden Number Problem (HNP, [2]). Solving the problem reveals the factorization of RSA modulus, i.e. breaks the scheme.

The main advantage of the approach compared to other attacks [14,17] is the lack of the chosen plaintext condition. The existing attacks, for instance, cannot harm so-called Active Authentication (AA) mechanism of the recently deployed electronic passports. Here, the challenge, i.e. the plaintext, is jointly chosen by both parties, the passport and the terminal, thus it can not be conveniently chosen by the attacker. The attack described here deals well with such a situation and it is able to solve the HNP instance with 150 measurements filtered from app. 7000. Once the secret key used by the passport during AA is available to the attacker, he can create a fully functional copy of the RFID chip in the passport he observes.

A possible way to obtain the side information needed for the attack within the electromagnetic traces is sketched in the paper. Having no access to high precision measurement equipment, its existence has not been experimentally verified, yet. The attack, however, should be taken into account by the laboratories testing the resilience of (not only) electronic passports to the side channel attacks.


RSA Chinese Remainder Theorem Montgomery exponentiation Hidden Number Problem side channel attack electronic passport 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Martin Hlaváč
    • 1
  1. 1.Department of AlgebraCharles University in PraguePrague 8Czech Republic

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