Implementation of a Secure Internet Voting Protocol

Voting is one of the most important activities in a democratic society. In a traditional voting environment voting process sometimes becomes quite inconvenient due to the reluctance of certain voters to visit a polling booth to cast votes besides involving huge social and human resources. The development of computer networks and elaboration of cryptographic techniques facilitate the implementation of electronic voting. In this work we propose a secure electronic voting protocol that is suitable for large scale voting over the Internet. The protocol allows a voter to cast his or her ballot anonymously, by exchanging untraceable yet authentic messages. The e-voting protocol is based on blind signatures and has the properties of anonymity, mobility, efficiency, robustness, authentication, uniqueness, and universal verifiability and coercion-resistant. The proposed protocol encompasses three distinct phases - that of registration phase, voting phase and counting phase involving five parties, the voter, certification centre, authentication server, voting server and a tallying server

On the Multiple Fault Attack on RSA Signatures with LSBs of Messages Unknown

In CHES 2009, Coron, Joux, Kizhvatov, Naccache and Paillier(CJKNP) introduced a fault attack on RSA signatures with partially unknown messages. They factored RSA modulus $N$ using a single faulty signature and increased the bound of unknown messages by multiple fault attack, however, the complexity multiple fault attack is exponential in the number of faulty signatures. At RSA 2010, it was improved which run in polynomial time in number of faults. Both previous multiple fault attacks deal with the general case that the unknown part of message is in the middle. This paper handles a special situation that some least significant bits of messages are unknown. First, we describe a sample attack by utilizing the technique of solving simultaneous diophantine approximation problem, and the bound of unknown message is $N^{\frac1{2}-\frac1{2\ell}}$ where $\ell$ is the number of faulty signatures. Our attacks are heuristic but very efficient in practice. Furthermore, the new bound can be extended up to $N^{\frac1{2}^{1+\frac1{\ell}}}$ by the Cohn-Heninger technique. Comparison between previous attacks and new attacks with LSBs of message unknown will be given by simulation test

Fault Attacks Against EMV Signatures

At CHES 2009, Coron, Joux, Kizhvatov, Naccache and Paillier (CJKNP) exhibited a fault attack against RSA signatures with partially known messages. This attack allows factoring the public modulus N. While the size of the unknown message part (UMP) increases with the number of faulty signatures available, the complexity of CJKNP\u27s attack increases exponentially with the number of faulty signatures. This paper describes a simpler attack, whose complexity is polynomial in the number of faults; consequently, the new attack can handle much larger UMPs. The new technique can factor N in a fraction of a second using ten faulty EMV signatures -- a target beyond CJKNP\u27s reach. We show how to apply the attack even when N is unknown, a frequent situation in real-life attacks

When e-th Roots Become Easier Than Factoring

We show that computing $e$-th roots modulo $n$ is easier than factoring $n$ with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form $x_i + c$. Here $c$ is fixed and $x_i$ denotes small integers of the attacker\u27s choosing. Several variants of the attack are presented, with varying assumptions on the oracle, and goals ranging from selective to universal forgeries. The computational complexity of the attack is $L_n(\frac{1}{3}, \sqrt[3]{\frac{32}{9}})$ in most significant situations, which matches the {\sl special} number field sieve\u27s ({\sc snfs}) complexity. This sheds additional light on {\sc rsa}\u27s malleability in general and on {\sc rsa}\u27s resistance to affine forgeries in particular -- a problem known to be polynomial for $x_i > \sqrt[3]{n}$, but for which no algorithm faster than factoring was known before this work

Partial key exposure attacks on multi-power RSA

Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.In this thesis, our main focus is a type of cryptanalysis of a variant of RSA, namely multi-power RSA. In multi-power RSA, the modulus is chosen as N = prq, where r ≥ 2. Building on Coppersmith’s method of finding small roots of polynomials, Boneh and Durfee show a very crucial result (a small private exponent attack) for standard RSA. According to this study, N = pq can be factored in polynomial time in log N when d < N 0.292 . In 2014, Sarkar improve the existing small private exponent attacks on multi-power RSA for r ≤ 5. He shows that one can factor N in polynomial time in log N if d < N 0.395 for r = 2 . Extending the ideas in Sarkar’s work, we develop a new partial key exposure attack on multi-power RSA. Prior knowledge of least significant bits (LSBs) of the private exponent d is required to realize this attack. Our result is a generalization of Sarkar’s result, and his result can be seen as a corollary of our result. Our attack has the following properties: the required known part of LSBs becomes smaller in the size of the public exponent e and it works for all exponents e (resp. d) when the exponent d (resp. e) has full-size bit length. For practical validation of our attack, we demonstrate several computer algebra experiments. In the experiments, we use the LLL algorithm and Gröbner basis computation. We achieve to obtain better experimental results than our theoretical result indicates for some cases.Declaration of Authorship ii Abstract iii Öz iv Acknowledgments v List of Figures viii List of Tables ix Abbreviations x 1 Introduction 1 1.1 A Short History of the Partial Key Exposure Attacks . . . . . . . . . . . . 4 1.2 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 The RSA Cryptosystem 8 2.1 RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 RSA Key Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Multi-power RSA (Takagi’s Variant) . . . . . . . . . . . . . . . . . . . . . 10 2.4 Cryptanalysis of RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.1 Factoring N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 Implementation Attacks . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2.1 Side-Channel Analysis . . . . . . . . . . . . . . . . . . . . 12 2.4.2.2 Bleichenbacher’s Attack . . . . . . . . . . . . . . . . . . . 13 2.4.3 Message Recovery Attacks . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.3.1 Håstad’s Attack . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.3.2 Franklin-Reiter Attack . . . . . . . . . . . . . . . . . . . . 15 2.4.3.3 Coppersmith’s Short Pad Attack . . . . . . . . . . . . . . 15 2.4.4 Attacks Using Extra Knowledge on RSA Parameters . . . . . . . . 15 2.4.4.1 Wiener’s Attack . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.4.2 Boneh-Durfee Attack . . . . . . . . . . . . . . . . . . . . 17 3 Preliminaries 18 3.1 Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Finding Small Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Finding Small Modular Roots . . . . . . . . . . . . . . . . . . . . . 21 3.2.2 Complexity of the Attacks . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.2.1 Polynomial Reduction . . . . . . . . . . . . . . . . . . . . 25 3.2.2.2 Root Extraction . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3 Boneh-Durfee Attack . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Partial Key Exposure Attacks on Multi-Power RSA 28 4.1 Known Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.1 Attacks when ed ≡ 1 mod ( p−1)( q−1) . . . . . . . . . . . . . . . 29 4.1.2 Attacks when ed ≡ 1 mod ( pr −pr−1)( q−1) . . . . . . . . . . . . . 29 4.2 A New Attack with Known LSBs . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Conclusion and Discussions 39 Bibliograph

Proper Usage of the Group Signature Scheme in ISO/IEC 20008-2

In ISO/IEC 20008-2, several anonymous digital signature schemes are specified. Among these, the scheme denoted as Mechanism 6, is the only plain group signature scheme that does not aim at providing additional functionalities. The Intel Enhanced Privacy Identification (EPID) scheme, which has many applications in connection with Intel Software Guard Extensions (Intel SGX), is in practice derived from Mechanism 6. In this paper, we firstly show that Mechanism 6 does not satisfy anonymity in the standard security model, i.e., the Bellare-Shi-Zhang model [CT-RSA 2005]. We then provide a detailed analysis of the security properties offered by Mechanism 6 and characterize the conditions under which its anonymity is preserved. Consequently, it is seen that Mechanism 6 is secure under the condition that the issuer, who generates user signing keys, does not join the attack. We also derive a simple patch for Mechanism 6 from the analysis

On the security of RSA padding

This paper presents a new signature forgery strategy