The XTR Public Key System

Lenstra, Arjen K.; Verheul, Eric R.

doi:10.1007/3-540-44598-6_1

Cited by 153 publications

(275 citation statements)

References 20 publications

(21 reference statements)

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“…4536 an XTR exponentiation iteration that can be used for three different purposes. In the first place these improvements result in an XTR double exponentiation method that is on average more than 60% faster than the double exponentiation from [10]. Such exponentiations are used in XTR ElGamal-like signature verifications.…”

Section: Is Presented That Results Inmentioning

confidence: 99%

“…Such exponentiations are used in XTR ElGamal-like signature verifications. Furthermore, they result in two new XTR single exponentiation methods, one that is on average about 60% faster than the method from [10] but that requires a one-time precomputation, and a generic one without precomputation that is on average 35% faster than the old method.…”

Section: Is Presented That Results Inmentioning

confidence: 99%

“…The XTR public key system was introduced at Crypto 2000 [10]. From a security point of view XTR is a traditional subgroup discrete logarithm system, as was proved in [10].…”

Section: Introductionmentioning

confidence: 99%

“…From a security point of view XTR is a traditional subgroup discrete logarithm system, as was proved in [10]. It uses a non-standard way to represent and compute subgroup elements to achieve substantial computational and communication advantages over traditional representations.…”

Section: Introductionmentioning

confidence: 99%

“…This paper describes several important speedups for XTR, while at the same time simplifying its implementation. In the first place the field arithmetic as described in [10] is improved by combining the modular reduction steps. More importantly, a new application of a method from [15] …”

Section: Introductionmentioning

confidence: 99%

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Advances in Cryptology — ASIACRYPT 2001

2001

Lecture Notes in Computer Science

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Abstract. This paper describes several speedups and simplifications for XTR. The most important results are new XTR double and single exponentiation methods where the latter requires a cheap precomputation. Both methods are on average more than 60% faster than the old methods, thus more than doubling the speed of the already fast XTR signature applications. An additional advantage of the new double exponentiation method is that it no longer requires matrices, thereby making XTR easier to implement. Another XTR single exponentiation method is presented that does not require precomputation and that is on average more than 35% faster than the old method. Existing applications of similar methods to LUC and elliptic curve cryptosystems are reviewed.

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Section: Is Presented That Results Inmentioning

confidence: 99%

Section: Is Presented That Results Inmentioning

confidence: 99%

“…The XTR public key system was introduced at Crypto 2000 [10]. From a security point of view XTR is a traditional subgroup discrete logarithm system, as was proved in [10].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Advances in Cryptology — ASIACRYPT 2001

2001

Lecture Notes in Computer Science

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MaTRU‐KE: A key exchange protocol based on MaTRU cryptosystem

Singh

Padhye

2018

Int J Communication

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We propose a quantum -resistant key exchange protocol based on hard problems of lattices using MaTRU cryptosystem as an underlying scheme. A key exchange protocol based on NTRU cryptosystem given by Lie et al is not secure against man-in-the-middle (MITM) attack. To remove this failure and provide a secure protocol, our protocol uses a trusted third party (TTP). Additionally, our protocol is better than NTRU-KE on efficiency and security point of view. In this paper, we propose key exchange protocol with TTP and without TTP, and describe the advantages and disadvantages of both schemes. KEYWORDS closest vector problem, lattice based cryptosystems, MaTRU cryptosystem, NTRU cryptosystem, public key cryptosystems, shortest vectors problem INTRODUCTIONKey exchange protocol is one of the important cryptographic techniques in exchanging secret keys. Without key exchanging, we can not go for secure communication over insecure channel using symmetric key cryptosystem. There may be manual key distributions for symmetric key cryptosystem. In 1976, Diffie and Hellman proposed a key exchange protocol (DH protocol) in their seminal paper "New Directions in Cryptography". 1 The DH-protocol is based on discrete logarithm problem (DLP). Since then, many key exchange protocols are proposed based on different algebraic structures. Most of them are the Diffie-Hellman type schemes. The Diffie-Hellman type means that the schemes are basically the same as the Diffie-Hellman scheme but defined over different groups. Some of the Diffie-Hellman type schemes are the ECDH protocol (elliptic curve-based DH protocol), 2,3 XTR DH protocol (DH protocol based on XTR cryptosystem 4 ), etc. But, P.W. Shor proposed an algorithm 5,6 in which he showed that if quantum computers become reality, then we can solve discrete logarithmic problems and can break all the Diffie-Hellman type key exchange protocols. In this way, we need a quantum-resistant key exchange protocol. To fulfill this requirement, several key exchange schemes 7-12 are proposed by distinguished researchers using learning with error (LWE) problem (introduced by Regev 13 ) and the ring LWE (RLWE) problem, 14 secure against quantum computers (LWE problem is as hard to solve as the worst case lattice problems). Since, hard problems of lattices still remain secure against quantum computers, 15 a lattice-based cryptosystem, NTRU cryptosystem, 16 which is also based on hard problems of lattices (shortest vector problem [SVP] and closest vector problem [CVP]), is considered to be secure in the world of quantum computers, 17 and a strong candidate of post quantum cryptography. Hence, to complete the ciphersuit of NTRU cryptosystem, someone should have to think about the key exchange protocol based on NTRU scheme, which was missing till 2013; and then, by motivating this fact, Lie et al 18 proposed a key exchange protocol based on the NTRU cryptosystem. Lie et al neither used any authentication technique in their key exchange algorithm nor any trusted third party (TTP). Because of this, their ...

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Generating prime degree irreducible polynomials by using irreducible all‐one polynomial over F₂

Makita

Nogami

Sugimura

2005

Electron Comm Jpn Pt III

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SUMMARYIn most of the methods of public key cryptography devised in recent years, a finite field of a large order is used as the field of definition. In contrast, there are many studies in which a higher-degree extension field of characteristic 2 is fast implemented for easier hardware realization. There are also many reports of the generation of the required higher-degree irreducible polynomial, and of the construction of a basis suited to fast implementation, such as an optimal normal basis (ONB). For generating higher-degree irreducible polynomials, there is a method in which a 2m-th degree self-reciprocal irreducible polynomial is generated from an m-th degree irreducible polynomial by a simple polynomial transformation (called the self-reciprocal transformation). This paper considers this transformation and shows that when the set of zeros of the m-th degree irreducible polynomial forms a normal basis, the set of zeros of the generated 2m-th order self-reciprocal irreducible polynomial also forms a normal base. Then it is clearly shown that there is a one-to-one correspondence between the transformed irreducible polynomial and the generated self-reciprocal irreducible polynomial. Consequently, the inverse transformation of the self-reciprocal transformation (selfreciprocal inverse transformation) can be applied to a selfreciprocal irreducible polynomial. It is shown that an m-th degree irreducible polynomial can always be generated from a 2m-th degree self-reciprocal irreducible polynomial by the self-reciprocal inverse transformation. We can use this fact for generating 1/2-degree irreducible polynomials. As an application of 1/2-degree irreducible polynomial generation, this paper proposes a method which generates a prime degree irreducible polynomial with a Type II ONB as its zeros.

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The XTR Public Key System

Cited by 153 publications

References 20 publications

Advances in Cryptology — ASIACRYPT 2001

Advances in Cryptology — ASIACRYPT 2001

MaTRU‐KE: A key exchange protocol based on MaTRU cryptosystem

Generating prime degree irreducible polynomials by using irreducible all‐one polynomial over F₂

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The XTR Public Key System

Cited by 153 publications

References 20 publications

Advances in Cryptology — ASIACRYPT 2001

Advances in Cryptology — ASIACRYPT 2001

MaTRU‐KE: A key exchange protocol based on MaTRU cryptosystem

Generating prime degree irreducible polynomials by using irreducible all‐one polynomial over F2

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Generating prime degree irreducible polynomials by using irreducible all‐one polynomial over F₂