Torus based cryptography pdf

Cryptography deals with the actual securing of digital data. They interpreted the xtr lv00 and luc sl93 cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented ceilidh, a new torusbased cryptosystem. We introduce the concept of torusbased cryptography, give a new public key system called ceilidh, and compare it to other discrete log based systems including lucas based systems and xtr. We argue that if a chaotic map f is used in cryptography, then it should be implemented as a bijection f m. Ljupco kocarev proposed publickey encryption algorithms based on iteration of onedimensional chebyshev chaotic maps and twodimensional of torus auto orphism chaotic map 1. Applications to cryptography of twisting commutative. Pdf rsa encryption algorithm based on torus automorphisms. It refers to the design of mechanisms based on mathematical algorithms that provide fundamental information security services. Compression in finite fields and torusbased cryptography. Normal elliptic bases and torusbased cryptography by clement dunand and reynald lercier download pdf 391 kb. The main advantage of those schemes is the reduced size of the keys for the same security than the basic schemes. Pdf we propose a publickey encryption algorithm based on torus automorphisms, which is secure, practical, and can be used for both. Summary of torusbased cryptography if there is a birational isomorphism f.

Asymptotically optimal communication for torusbased. The thread followed by these notes is to develop and explain the. A digital image watermarking scheme based on visual. The systems we study, called torus based cryptosystems, were analyzed by karl rubin and alice silverberg in 2003 rs03. Applications of number theory and algebraic geometry to. Normal elliptic bases and torusbased cryptography 5 an explicit computation shows that the w 15,es have a convenient common denominator, namely15. This idea was first introduced by alice silverberg and karl rubin in 2003.

Torusbased cryptography 3 reason is that, reinterpreting the conjectures in 2 in the language of algebraic tori, they say that the coordinate ring of the quotient of t 30 by a certain product of symmetric groups is generated by the. Fpga design for algebraic tori based public key cryptography. It improves on conventional cryptosystems by representing some elements of large finite fields compactly and therefore. When the torus is \rational, then its elements can be compactly represented.

Foreword this is a set of lecture notes on cryptography compiled for 6. Introducing the torus consider a circle in the xyplane with centre r,0 and radius a. How elliptic curve cryptography works technical articles. Jan 21, 2015 effective compression maps for torusbased cryptography andrea montanari 1 designs, codes and cryptography volume 79, pages 1 17 2016 cite this article. Aes rijndael is also block cipher, but different ucomplexitybased cryptography multiplication, exponentiation are oneway fctns examples. This work describes a fast fully homomorphic encryption scheme over the torus tfhe, that revisits, generalizes and improves the fully homomorphic encryption fhe based on gsw and its ring variants. Ninth international conference on finite fields and applications. Feistel networks umany block algorithms are feistel networks examples des, lucifer, freal, khufu, khafre, loki, gost, cast, blowfish, feistel network is a standard form for. On small characteristic algebraic tori in pairing based cryptography volume 9 r. They interpreted the xtr lv00 and luc sl93 cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented ceilidh, a new torus based cryptosystem. Normal elliptic bases and torus based cryptography. However, two recent developments make use of such elds, pairingbased cryptography,57 and torusbased cryptography 19,8,21,27. Owing to these properties, different chaos based maps have been extensively used for the development of numerous block and stream cipher based cryptosystems. Normal elliptic bases and torusbased cryptography core.

On small characteristic algebraic tori in pairingbased cryptography volume 9 r. Namely, any binary gate is evaluated homomorphically in about milliseconds on a single core which improves dm15 by a factor 50, and the mux gate takes about 26 cpums or ms on 2 cores. Torusbased cryptography involves using algebraic tori to construct a group for use in ciphers. An overview of trace based public key cryptography over. Torusbased cryptography crypto wiki fandom powered by. You can think of cryptography as the establishment of a large toolkit containing different techniques in security applications. The volume of a torus using cylindrical and spherical. Recently, an approach based on rational torus repre. Based on similar ideas, rubin and silverberg 42 proposed the notion of torusbased cryptography as an alternative way to obtain compression of elements in the cyclotomic subgroup of a suitable.

We introduce a new concept, namely torusbased cryptography, and give a new torusbased public key cryptosystem that we call ceilidh. Alice and bob first agree to use the same curve and a few other parameters, and then they pick a random point g on the curve. The main ideas come from arithmetic, in particular, algebraic tori. We introduce the concept of torus based cryptography, give a new public key system called ceilidh, and compare it to other discrete log based systems including lucas based systems and xtr. Basic concepts in cryptography fiveminute university. Crypto, asymptotically optimal communication for torusbased cryptography, ps, pdf with marten van. Goldwasser and mihir bellare in the summers of 19962002, 2004, 2005 and 2008. Normal elliptic bases and torus based cryptography 3 the use of a normal elliptic basis can yield a logq speedup. We implement cryptography with chaos following and extending the original program of shannon with 3 selected torus automorphisms, namely the baker map, the horseshoe map and the cat map. Torusbased cryptography crypto wiki fandom powered by wikia. On small characteristic algebraic tori in pairingbased. Cryptography is the art and science of making a cryptosystem that is capable of providing information security. Both of these chapters can be read without having met complexity theory or formal methods before. The systems we study, called torusbased cryptosystems, were analyzed by karl rubin and alice silverberg in 2003 rs03.

It studies ways of securely storing, transmitting, and processing information. Section 4 discusses some of the cryptographic applications of these mappings. More precisely, rubin and silverberg showed that the factor n. Fast fully homomorphic encryption library over the torus. In my view, this hope is misguided, because in its essence cryptography is as much an art as a science. Effective compression maps for torusbased cryptography. Clement dunand, reynald lercier submitted on 1 sep 2009. We introduce cryptography based on algebraic tori, give a new public key system called ceilidh, and compare it to other discrete log based systems including luc and xtr.

Torus based cryptography involves using algebraic tori to construct a group for use in ciphers based on the discrete logarithm problem. Infocr computer science cs cryptography and security cs. While lucasbased systems and xtr are essentially restricted to exponentiation, we are able to perform. Pdf we propose publickey encryption algorithms based on chaotic maps, which are. Infocr computer science cscryptography and security cs. In this gate bootstrapping mode, we show that the scheme fhew of 29. Theory and applications, contemporary mathematics, pages 7153. Much of the approach of the book in relation to public key algorithms is reductionist in nature. Based on similar ideas, rubin and silverberg 42 proposed the notion of torus based cryptography as an alternative way to obtain compression of elements in the cyclotomic subgroup of a suitable. The trace based public key cryptography is one of the quests which ensures higher security over extension field, efficient transmission over subfield and faster computation through linear recursive relation. This idea was first introduced by alice silverberg and karl rubin in 2003 in the form of a public key algorithm by the name of ceilidh. This system has the security of f p2 while transmitting elements of the.

While luc and xtr are essentially restricted to exponentiation, we are able to perform multiplication. The art and science of breaking the cipher text is. The name ceilidh comes from the scots gaelic word ceilidh which means a traditional scottish gathering. Jun 26, 2019 with ellipticcurve cryptography, alice and bob can arrive at a shared secret by moving around an elliptic curve. We introduce the concept of torusbased cryptography, give a new public key system called ceilidh, and compare it to other discrete log based systems including lucasbased systems and xtr. Request pdf asymptotically optimal communication for torusbased cryptography we introduce a compact and ecien t representation of ele ments of the algebraic torus. Cryptography is the mathematical foundation on which one builds secure systems. Torusbased cryptography aims at representing certain field elements in a compact form, while keeping the difficulty of the discrete logarithm problem unchanged.

This is a set of lecture notes on cryptography compiled for 6. Due to these properties the trace based public key cryptosystems pkc have become popular in the last two decades. White paper distributed energy platform based on smart contract the content described in this document is for informational purposes only and torus project team assumes no responsibility for any information, whether explicit or implied. So, torusbased cryptography gives a possibility to work in a subgroup, while maintaining the security of a bigger group. So,x 15 tr 1 1 t r 3 3 t r 5 5 t r 15 15 115,wherether esareconvenientpolynomialsinq, r 1 1, r 3. For this reason, practical evaluation of the hardness of discrete logarithms in such elds is becoming an important issue. Ceilidh is a public key cryptosystem based on the discrete logarithm problem in algebraic torus. Feistel networks umany block algorithms are feistel networks. Though the tori t n in general are only conjectured to be rational, it is known 30 that. Apr 14, 2020 the tfhe library implements a very fast gatebygate bootstrapping, based on cggi16. Normal elliptic bases and torusbased cryptography 3 the use of a normal elliptic basis can yield a logq speedup. The volume of a torus using cylindrical and spherical coordinates.

We introduce the concept of torusbased cryptography, give a new public key system called ceilidh, and compare it to other discrete log based systems. Several examples, including chaotic block cypher and chaotic publickey encryption. Torusbased cryptography the paper introducing the concept in pdf. The information contained in this document is subject to change without notice. This compression reduces transmission size by a factor of. A digital image watermarking scheme based on visual cryptography. We introduce cryptography based on algebraic tori, give a new public. A number of cryptosystems, including the lucasbased,,, gongharn, xtr, and t 2 and ceilidh cryptosystems, and the abelian variety or elliptic curve systems in, can be viewed as being based on the idea that when one does discrete logbased cryptography for either a multiplicative group of a field or an elliptic curve group over. While luc and xtr are essentially restricted to exponentiation, we are able to perform multiplication as well.

A digital watermark which can be the visible logo of the owner is split according to a visual secret sharing scheme 2, 3. We compare ceilidh with other discrete log based systems, and show that it improves on di. Citeseerx document details isaac councill, lee giles, pradeep teregowda. One of the shares is inserted into the image and the other serves as the detection key. Understanding what cryptographic primitives can do, and how they can be composed together, is necessary to build secure systems, but not su cient. In section 1, we describe the concept of visual cryptography. Asymptotically optimal communication for torusbased cryptography. Cryptography and secure communication by richard e. While lucas based systems and xtr are essentially restricted to exponentiation, we are able to perform. The simplest fhe schemes consist in bootstrapped binary gates.