Approximation of the discrete logarithm in finite fields of even characteristic by real polynomials

summary:We obtain lower bounds on degree and additive complexity of real polynomials approximating the discrete logarithm in finite fields of even characteristic. These bounds complement earlier results for finite fields of odd characteristic

On Small Degree Extension Fields in Cryptology

This thesis studies the implications of using public key cryptographic primitives that are based in, or map to, the multiplicative group of finite fields with small extension degree. A central observation is that the multiplicative group of extension fields essentially decomposes as a product of algebraic tori, whose properties allow for improved communication efficiency. Part I of this thesis is concerned with the constructive implications of this idea. Firstly, algorithms are developed for the efficient implementation of torus-based cryptosystems and their performance compared with previous work. It is then shown how to apply these methods to operations required in low characteristic pairing-based cryptography. Finally, practical schemes for high-dimensional tori are discussed. Highly optimised implementations and benchmark timings are provided for each of these systems. Part II addresses the security of the schemes presented in Part I, i.e., the hardness of the discrete logarithm problem. Firstly, an heuristic analysis of the effectiveness of the Function Field Sieve in small characteristic is given. Next presented is an implementation of this algorithm for characteristic three fields used in pairing-based cryptography. Finally, a new index calculus algorithm for solving the discrete logarithm problem on algebraic tori is described and analysed

Galois invariant smoothness basis

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field \bK, one looks for a smoothness basis for \bK^* that is left invariant by automorphisms of \bK. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve)

Distribution and Polynomial Interpolation of the Dodis-Yampolskiy Pseudo-Random Function

International audienceWe give some theoretical support to the security of the cryptographic pseudo-random function proposed by Dodis and Yampolskiy in 2005. We study the distribution of the function values over general finite fields and over elliptic curves defined over prime finite fields. We also prove lower bounds on the degree of polynomials interpolating the values of these functions in these two settings

Hardness of Computing Individual Bits for One-way Functions on Elliptic Curves

We prove that if one can predict any of the bits of the input to an elliptic curve based one-way function over a finite field, then we can invert the function. In particular, our result implies that if one can predict any of the bits of the input to a classical pairing-based one-way function with non-negligible advantage over a random guess then one can efficiently invert this function and thus, solve the Fixed Argument Pairing Inversion problem (FAPI-1/FAPI-2). The latter has implications on the security of various pairing-based schemes such as the identity-based encryption scheme of Boneh–Franklin, Hess’ identity-based signature scheme, as well as Joux’s three-party one-round key agreement protocol. Moreover, if one can solve FAPI-1 and FAPI-2 in polynomial time then one can solve the Computational Diffie--Hellman problem (CDH) in polynomial time. Our result implies that all the bits of the functions defined above are hard-to-compute assuming these functions are one-way. The argument is based on a list-decoding technique via discrete Fourier transforms due to Akavia--Goldwasser–Safra as well as an idea due to Boneh–Shparlinski

Cryptographic Pairings: Efficiency and DLP security

This thesis studies two important aspects of the use of pairings in cryptography, efficient algorithms and security. Pairings are very useful tools in cryptography, originally used for the cryptanalysis of elliptic curve cryptography, they are now used in key exchange protocols, signature schemes and Identity-based cryptography. This thesis comprises of two parts: Security and Efficient Algorithms. In Part I: Security, the security of pairing-based protocols is considered, with a thorough examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the relationship between the two instances of the DLP will be presented along with a discussion about the appropriate selection of parameters to ensure particular security level. In Part II: Efficient Algorithms, some of the computational issues which arise when using pairings in cryptography are addressed. Pairings can be computationally expensive, so the Pairing-Based Cryptography (PBC) research community is constantly striving to find computational improvements for all aspects of protocols using pairings. The improvements given in this section contribute towards more efficient methods for the computation of pairings, and increase the efficiency of operations necessary in some pairing-based protocol

Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations

Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any finite abelian group. Their algorithm stimulated a lot of interest in the cryptography community, especially in the context of `bit security'. This manuscript attempts to be a friendly and comprehensive guide to the tools and results in this field. The intended readership is cryptographers who have heard about these tools and seek an understanding of their mechanics and their usefulness and limitations. A compact overview of the algorithm is presented with emphasis on the ideas behind it. We show how these ideas can be extended to a `modulus-switching' variant of the algorithm. We survey some applications of this algorithm, and explain that several results should be taken in the right context. In particular, we point out that some of the most important bit security problems are still open. Our original contributions include: a discussion of the limitations on the usefulness of these tools; an answer to an open question about the modular inversion hidden number problem