Randomness Complexity of Private Circuits for Multiplication

Conference paper

DOI: 10.1007/978-3-662-49896-5_22

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9666)
Cite this paper as:
Belaïd S., Benhamouda F., Passelègue A., Prouff E., Thillard A., Vergnaud D. (2016) Randomness Complexity of Private Circuits for Multiplication. In: Fischlin M., Coron JS. (eds) Advances in Cryptology – EUROCRYPT 2016. EUROCRYPT 2016. Lecture Notes in Computer Science, vol 9666. Springer, Berlin, Heidelberg


Many cryptographic algorithms are vulnerable to side channel analysis and several leakage models have been introduced to better understand these flaws. In 2003, Ishai, Sahai and Wagner introduced the d-probing security model, in which an attacker can observe at most d intermediate values during a processing. They also proposed an algorithm that securely performs the multiplication of 2 bits in this model, using only \(d(d+1)/2\) random bits to protect the computation. We study the randomness complexity of multiplication algorithms secure in the d-probing model. We propose several contributions: we provide new theoretical characterizations and constructions, new practical constructions and a new efficient algorithmic tool to analyze the security of such schemes.

We start with a theoretical treatment of the subject: we propose an algebraic model for multiplication algorithms and exhibit an algebraic characterization of the security in the d-probing model. Using this characterization, we prove a linear (in d) lower bound and a quasi-linear (non-constructive) upper bound for this randomness cost. Then, we construct a new generic algorithm to perform secure multiplication in the d-probing model that only uses \(d + d^2/4\) random bits.

From a practical point of view, we consider the important cases \(d \le 4\) that are actually used in current real-life implementations and we build algorithms with a randomness complexity matching our theoretical lower bound for these small-order cases. Finally, still using our algebraic characterization, we provide a new dedicated verification tool, based on information set decoding, which aims at finding attacks on algorithms for fixed order d at a very low computational cost.


Side-channel analysis Probing model Randomness complexity Constructions Lower bounds Probabilistic method Information set decoding Algorithmic tool 

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.ENS, CNRS, INRIA, and PSLParisFrance
  2. 2.Thales Communications and SecurityGennevilliersFrance
  3. 3.ANSSIParisFrance
  4. 4.UPMC, POLSYS, LIP6ParisFrance

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