A white-box implementation of a block cipher is a software implementation from which it is difficult for an attacker to extract the cryptographic key. Chow et al. published white-box implementations for AES and DES. These implementations are based on ideas that can be used to derive white-box implementations for other block ciphers as well. In particular, the ideas can be used to derive a white-box implementation for any substitution linear-transformation (SLT) cipher. Although the white-box implementations of AES and DES have been cryptanalyzed, the cryptanalyses published use typical properties of AES and DES. It is therefore an open question whether an SLT cipher exists for which the techniques of Chow et al. result in a secure white-box implementation. In this paper we largely settle this question by presenting an algorithm that is able to extract the key from such an implementation under a mild condition on the diffusion matrix. The condition is, for instance, satisfied by all MDS matrices. Our result can serve as a basis to design block ciphers and to develop white-box techniques that result in secure white-box implementations.

Abstract. At DRM 2002, Chow et al. [4] presented a method for implementing the DES block cipher such that it becomes hard to extract the embedded secret key in a white-box attack context. In such a context, an attacker has full access to the implementation and its execution environment. In order to provide an extra level of security, an implementation shielded with external encodings was introduced by Chow et al. and improved by Link and Neumann [10]. In this paper, we present an algorithm to extract the secret key from such white-box DES implementations. The cryptanalysis is a differential attack on obfuscated rounds, and works regardless of the shielding external encodings that are applied. The cryptanalysis has a average time complexity of 2 14 and a negligible space complexity.

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The multiplication algorithm that we use is based on an algorithm by Bajard and Imbert, extended to work with pseudo-residues (remainders with a larger range than the modulus). The resulting Recursive Residue Number System (RRNS) can be used to implement modular addition, multiplication, and multiply-and-accumulate for very large (2000+ bits) moduli, using only modular operations for small (for example 8-bits) moduli. A hardware implementation of this method allows for massive parallelization.Our method can be applied in cryptographic algorithms such as RSA to realize modular exponentiation with a large (2048bit, or even 4096-bit) modulus. Due to the use of full RNS Montgomery algorithms, the system does not involve any carries, therefore cryptographic attacks that exploit carries cannot be applied.