On the Affine Equivalence and Nonlinearity Preserving Bijective Mappings

It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, $\mathcal{F}_n$, over \bbbf_2^n, is naturally regarded as the $2^n$ dimensional vector space, \bbbf_2^{2^n}. Thus, while analyzing the transformations acting on $\mathcal{F}_n$, $S_{2^{2^n}}$, the group of all bijective mappings, defined from \bbbf_2^{2^n} onto itself should be considered. As it is shown in \cite{ser,ser:dog,ser:dog:2}, there exist non-affine bijective transformations that preserve nonlinearity. In this paper, first, we prove that the group of affine equivalence relations is isomorphic to the automorphism group of Sylvester Hadamard matrices. Then, we show that new nonlinearity preserving non-affine bijective mappings also exist. Moreover, we propose that the automorphism group of nonlinearity classes, should be studied as a subgroup of $S_{2^{2^n}}$, since it contains transformations which are not affine equivalence relations

Fully Verifiable Secure Delegation of Pairing Computation: Cryptanalysis and An Efficient Construction

We address the problem of secure and verifiable delegation of general pairing computation. We first analyze some recently proposed pairing delegation schemes and present several attacks on their security and/or verifiability properties. In particular, we show that none of these achieve the claimed security and verifiability properties simultaneously. We then provide a fully verifiable secure delegation scheme ${\sf VerPair}$ under one-malicious version of a two-untrusted-program model (OMTUP). ${\sf VerPair}$ not only significantly improves the efficiency of all the previous schemes, such as fully verifiable schemes of Chevallier-Mames et al. and Canard et al. by eliminating the impractical exponentiation- and scalar-multiplication-consuming steps, but also offers for the first time the desired full verifiability property unlike other practical schemes. Furthermore, we give a more efficient and less memory consuming invocation of the subroutine ${\sf Rand}$ for ${\sf VerPair}$ by eliminating the requirement of offline computations of modular exponentiations and scalar-multiplications. In particular, ${\sf Rand}$ includes a fully verifiable partial delegation under the OMTUP assumption. The partial delegation of ${\sf Rand}$ distinguishes ${\sf VerPair}$ as a useful lightweight delegation scheme when the delegator is resource-constrained (e.g. RFID tags, smart cards or sensor nodes)

An Efficient ID-Based Message Recoverable Privacy-Preserving Auditing Scheme

One of the most important benefits of public cloud storage is outsourcing of management and maintenance with easy accessibility and retrievability over the internet. However, outsourcing data on the cloud brings new challenges such as integrity verification and privacy of data. More concretely, once the users outsource their data on the cloud they have no longer physical control over the data and this leads to the integrity protection issue. Hence, it is crucial to guarantee proof of data storage and integrity of the outsourced data. Several pairing-based au- diting solutions have been proposed utilizing the Boneh-Lynn-Shacham (BLS) short signatures. They basically provide a desirable and efficient property of non-repudiation protocols. In this work, we propose the first ID-based privacy-preserving public auditing scheme with message recov- erable signatures. Because of message recoverable auditing scheme, the message itself is implicitly included during the verification step that was not possible in previously proposed auditing schemes. Furthermore, we point out that the algorithm suites of existing schemes is either insecure or very inefficient due to the choice of the underlying bilinear map and its baseline parameter selections. We show that our scheme is more ef- ficient than the recently proposed auditing schemes based on BLS like short signatures

Nonlineeriteyi koruyan ard dönüşümler.

Boolean functions are accepted to be cryptographically strong if they satisfy some common pre-determined criteria. It is expected that any design criteria should remain invariant under a large group of transformations due to the theory of similarity of secrecy systems proposed by Shannon. One of the most important design criteria for cryptographically strong Boolean functions is the nonlinearity criterion. Meier and Staffelbach studied nonlinearity preserving transformations, by considering the invertible transformations acting on the arguments of Boolean functions, namely the pre-transformations. In this thesis, first, the results obtained by Meier and Staffelbach are presented. Then, the invertible transformations acting on the truth tables of Boolean functions, namely the post-transformations, are studied in order to determine whether they keep the nonlinearity criterion invariant. The equivalent counterparts of Meier and Staffelbach̕s results are obtained in terms of the post-transformations. In addition, the existence of nonlinearity preserving post-transformations, which are not equivalent to pre-transformations, is proved. The necessary and sufficient conditions for an affine post-transformation to preserve nonlinearity are proposed and proved. Moreover, the sufficient conditions for an non-affine post-transformation to keep nonlinearity invariant are proposed. Furthermore, it is proved that the smart hill climbing method, which is introduced to improve nonlinearity of Boolean functions by Millan et. al., is equivalent to applying a post-transformation to a single Boolean function. Finally, the necessary and sufficient condition for an affine pre-transformation to preserve the strict avalanche criterion is proposed and proved.M.S. - Master of Scienc

Boole fonksiyonları üzerine tanımlı nonlineerite ve hamming ağırlığını koruyan tersinir dönüşümler.

Boolean functions are widely studied in cryptography due to their key role and ap- plications in various cryptographic schemes. Particularly in order to make symmetric crypto-systems resistant against cryptanalytic attacks, Boolean functions are associ- ated some cryptographic design criteria. As a result of Shannon’s similarity of secrecy systems theory, cryptographic design criteria should be at least preserved under the action of basic transformations. Among these design criteria, Meier and Staffelbach analyzed behavior of the nonlinearity criteria under the action of bijective mappings defined on input values of the functions. Later, Preneel proved that nonlinearity still remains invariant under the action of affine equivalence mappings. Motivated by the previous studies, in his master thesis, the author showed the existence of new nonlin- earity preserving bijective mappings. In this thesis, we first give definition of the maximal group that can act on Boolean functions. This maximal group is the symmetric group of the vector space that cor- responds to the set comprised of the truth table of the Boolean functions. We give a representation, based on the coordinate functions’ algebraic normal form, for the ele- ments of this symmetric group and then we list its subgroups that we mainly focus on. Regarding these subgroups, our aim is to enumerate or classify these bijective map- pings with respect to preserving a cryptographic design criterion. After the necessary definitions and notions, we mainly investigate the nonlinearity preserving bijective mappings. Then we apply the procedures constructed on nonlinearity preservability to another cryptographic design criterion, namely the Hamming weight. From a the- oretical viewpoint, our basic result is that we show the existence of new families of bijective mappings that leaves nonlinearity (respectively, Hamming weight) invariant. Under the action of linear and affine bijective mappings we give the necessary and sufficient conditions to keep nonlinearity invariant. We explicitly construct an iso- morphism between the affine equivalency mappings subgroup and the automorphism group of the Sylvester Hadamard matrices and give the order of this automorphism group. Next we construct a family of non-affine nonlinearity preserving bijective map- pings explicitly. However, we also show that all of these explicitly constructed non- linearity preserving bijective mappings produce the same orbit structure as the affine equivalency mappings. On the other hand, we give the exact number of nonlinearity preserving bijective mappings for the functions with n ≤ 6 variables. Then, based on these cardinalities, we prove the existence of new non-affine nonlinearity preserving mappings, without constructing explicitly. We demonstrate some examples for these non-affine mappings. Following the results obtained for nonlinearity preserving bijective mappings, we ex- tend our study to the Hamming weight preserving bijective mappings. First we com- pletely solve the enumeration problem of Hamming weight preserving bijective mappings, and give the exact number of the Hamming weight preserving bijective map- pings for all Boolean functions. Afterwards, we study the classification problem and give partial results. Lechner proved that the Hamming weight property is preserved un- der the action of symmetric group of input vector space. We further prove that among the affine bijective mappings only these mappings preserve the Hamming weight. Finally, again based on the enumeration of the Hamming weight preserving bijective mappings we proved the existence of Hamming weight preserving non-affine bijective mappings.Ph.D. - Doctoral Progra

Affine Equivalency and Nonlinearity Preserving Bijective Mappings over F-2

We first give a proof of an isomorphism between the group of affine equivalent maps and the automorphism group of Sylvester Hadamard matrices. Secondly, we prove the existence of new nonlinearity preserving bijective mappings without explicit construction. Continuing the study of the group of nonlinearity preserving bijective mappings acting on n-variable Boolean functions, we further give the exact number of those mappings for n <= 6. Moreover, we observe that it is more beneficial to study the automorphism group of bijective mappings as a subgroup of the symmetric group of the 2(n) dimensional F-2-vector space due to the existence of non-affine mapping classes

More Efficient Secure Outsourcing Methods for Bilinear Maps

Bilinear maps are popular cryptographic primitives which have been commonly used in various modern cryptographic protocols. However, the cost of computation for bilinear maps is expensive because of their realization using variants of Weil and Tate pairings of elliptic curves. Due to increasing availability of cloud computing services, devices with limited computational resources can outsource this heavy computation to more powerful external servers. Currently, the checkability probability of the most efficient outsourcing algorithm is $1/2$ and the overall computation requires $4$ point addition in the preimage and $3$ multiplications in the image of the bilinear map under the one-malicious version of a two-untrusted-program model. In this paper, we propose two efficient new algorithms which decrease not only the memory requirement but also the overall communication overhead