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1

Asymmetric cryptography and trapdoor one-way functions

Krzyworzeka N.

Automatyka / Automatics

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2016

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Vol. 20, no. 2

39--51

EN

The asymmetric-key (public-key) encryption scheme is considered to be the most important discovery in the history of cryptography. It is based on the use of two complementary keys generated according to a chosen trapdoor one-way function (TOWF). Since its first implementation, asymmetric encryption has revolutionized our way of communicating as well as the safety of information transfer, and it is now widely used around the world for various purposes, especially in the field of online transaction security. The safety of the asymmetric-key scheme relies on the assumption that any known cryptographic attack using an efficient problem-solving algorithm will not be able to succeed in applying the inverse (decryption) function onto the cryptogram in a polynomial time without additional knowledge (secret information). The most-challenging aspect of creating a new asymmetric cryptographic algorithm is selecting a one-way function for encryption purposes and finding a trapdoor in its inverse. In this paper, the concept of public-key cryptography will be explained using the RSA algorithm as an example. In addition, the review of the most-important functions that are considered to be trapdoor one- -way functions will be conducted.

PL

Odkrycie szyfrowania asymetrycznego jest uważane za największy przełom w dziedzinie kryptografii. W założeniu kryptografia asymetryczna polega na generowaniu dwóch komplementarnych kluczy według wybranej jednokierunkowej funkcji zapadkowej (trapdoor one-way function). Odkrycie tego sposobu szyfrowania całkowicie zrewolucjonizowało metody bezpiecznego przesyłu informacji i jest obecnie wykorzystywane w wielu dziedzinach, szczególnie przy uwierzytelnianiu danych oraz w transakcjach online. Skuteczność omawianej metody pozwala założyć, że żaden algorytm deszyfrujący działający w realnym czasie nie będzie w stanie bez dodatkowych informacji (tzw. zapadki) efektywnie odgadnąć funkcji deszyfrującej. Największym utrudnieniem przy tworzeniu nowego algorytmu asymetrycznego jest odkrycie nowej, jednokierunkowej funkcji zapadkowej. W poniższym artykule zostanie dokładnie opisane działanie najbardziej znanego algorytmu asymetrycznego – RSA. Przeprowadzono również przegląd najważniejszych funkcji jednokierunkowych.

2

Public-Key Cryptography Based on a Cubic Extension of the Lucas Functions

Roettger E., Williams H. C.

Fundamenta Informaticae

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2012

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Vol. 114, nr 3/4

325-344

EN

One of the goals of public-key cryptography is to securely exchange a key by use of a public channel without the users previously communicating with one another. In 1976W. Diffie and M. Hellman had an idea how to do this by exploiting mathematically difficult one-way problems. Diffie-Hellman key exchange is based on the believed difficulty of the discrete log problem. This paper presents a new key exchange protocol based on functions that were developed to generalize the Lucas functions. Relevant results about this generalization of the Lucas functions are provided that provide the machinery for the Diffie-Hellman-like key exchange presented here. Lastly, there is a brief discussion about the efficiency of our system versus Diffie-Hellman key exchange and LUCDIF.

3

Forward-Secure Identity-Based Public-Key Encryption without Random Oracles

Yu J., Cheng X., Hao R., Kong F., Fan J.

Fundamenta Informaticae

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2011

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Vol. 111, nr 2

241-256

EN

In traditional identity-based encryption schemes, security will be entirely lost once secret keys are exposed. However, with more and more use of mobile and unprotected devices, key exposure seems unavoidable. To deal with this problem, we newly propose a forward-secure identitybased public-key encryption scheme. In this primitive, the exposure of the secret key in one period doesn't affect the security of the ciphertext generated in previous periods. Any parameter in our scheme has at most log-squared complexity in terms of the total number of time periods. We also give the semantic security notions of forward-secure identity-based public-key encryption. The proposed scheme is proven semantically secure in the standard model. As far as we are concerned, it is the first forward-secure identity-based public-key encryption scheme without random oracles.

4

A new approach to the elgamal encryption scheme

Kościelny Cz.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 2

265-267

EN

The ElGamal encryption scheme can be used for both digital signatures and encryption, and its security results from the difficulty of calculating discrete logarithms in a finite field. This algorithm usually works in a multiplicative group of GF(p) and in this case the progress in the discrete logarithm problem forces the users of such a basic ElGamal public key cryptosystem to permanently increase a prime modulus p in order to ensure the desired security. But the task of finding a multiplicative group of GF(p) is unfeasible for an ordinary user. It is possible to overcome this inconvenience by forming an ElGamal encryption scheme which works in a multiplicative group of GF(pm). Therefore, it is shown in the paper how to implement this cryptosystem for work in the multiplicative group of GF(pm), in its subgroup, and in an algebraic system named the spurious multiplicative group of GF(pm).

5

Asymmetric cryptography and practical security

Pointcheval D.

Journal of Telecommunications and Information Technology

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2002

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nr 4

41-56

EN

Since the appearance of public-key cryptography in Diffie-Hellman seminal paper, many schemes have been proposed, but many have been broken. Indeed, for many people, the simple fact that a cryptographic algorithm withstands cryptanalytic attacks for several years is considered as a kind of validation. But some schemes took a long time before being widely studied, and maybe thereafter being broken. A much more convincing line of research has tried to provide "provable" security for cryptographic protocols, in a complexity theory sense: if one can break the cryptographic protocol, one can efficiently solve the underlying problem. Unfortunately, very few practical schemes can be proven in this so-called "standard model" because such a security level rarely meets with efficiency. A convenient but recent way to achieve some kind of validation of efficient schemes has been to identify some concrete cryptographic objects with ideal random ones: hash functions are considered as behaving like random functions, in the so-called "random oracle model", block ciphers are assumed to provide perfectly independent and random permutations for each key in the "ideal cipher model", and groups are used as black-box groups in the "generic model". In this paper, we focus on practical asymmetric protocols together with their "reductionist" security proofs. We cover the two main goals that public-key cryptography is devoted to solve: authentication with digital signatures, and confidentiality with public-key encryption schemes.