An extension of the noncommutative Bergman’s ring with a large number of noninvertible elements

Climent, Joan‐Josep; Navarro, Pedro R.; Tortosa, Leandro

doi:10.1007/s00200-014-0231-6

Cited by 9 publications

(12 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, m. So, the mull matrix and the identity matrix are the additive and multiplicative identities of E (m) p . As we mentioned earlier in Section 1, the number of noninvertible elements in E (m) p is very large when m is large compared with p. For example, for p = 7 and m = 2, 4, 8, 16, 32, the number of noninvertible elements is about 26.53 %, 46.02 %, 70.86 %, 91.51 % and 99.28 % respectively (see [11]). …”

Section: The Ring E (M) Pmentioning

confidence: 76%

See 1 more Smart Citation

Key agreement protocols for distributed secure multicast over the ringE^(m)_p

Climent

Ramos

Navarro

et al. 2013

WIT Transactions on Information and Communication Technologies

View full text Add to dashboard Cite

Protocols for authenticated key exchange allow parties within an insecure network to establish a common session key which can then be used to secure their future communication. In this paper we introduce a protocol for distributed key agreement over a noncommutative ring with a large number of noninvertible elements. This protocol uses polynomials with coefficients in the center of the ring. We also present the necessary steps for recalculating the shared secret key when a new user joins the system, or when a user leaves the system. Secure communications, key exchange, noncommutative ring, multicast protocol

show abstract

Section: The Ring E (M) Pmentioning

confidence: 76%

“…To avoid this weakness Climent, Navarro and Tortosa [11] introduce an extension E (m) p of E p that maintains the main properties of E p . In particular, it can not be embedded in a ring of matrices over a commutative ring.…”

Section: Introductionmentioning

confidence: 99%

Key agreement protocols for distributed secure multicast over the ringE^(m)_p

Climent

Ramos

Navarro

et al. 2013

WIT Transactions on Information and Communication Technologies

View full text Add to dashboard Cite

show abstract

“…If Eve is able to find M ∈ Z(E with h 0 ≡ p 0 (cf. [8]). Thus Eve writes MH −1 H and gets that A 1 XA 2 = MX = MH −1 XH, solving the DP and therefore she solves the DHDP.…”

Section: Let Us Consider the Ring Ementioning

confidence: 99%

Public key protocols over the ring $E_{p}^{(m)}$

Climent¹,

López-Ramos²

2016

AMC

Self Cite

View full text Add to dashboard Cite

In this paper we use the nonrepresentable ring E (m) p to introduce public key cryptosystems in noncommutative settings and based on the Semigroup Action Problem and the Decomposition Problem respectively. IntroductionMost public-key cryptosystems are based on certain specific problems of number theory.One of these problems is the Integer Factorization Problem (IFP) over the ring Z n , being n the product of two large prime numbers; the well known cryptosystem RSA [22] is based in this problem. The second classical problem is the Discrete Logarithm Problem (DLP) over a finite field Z p , being p a large prime; the Diffie-Hellman key exchange protocol [9] and ElGamal protocol [10] are based on this problem. In general, we can say that their robustness depends on the computational difficulty of solving certain mathematical problems over finite commutative algebraic structures. Some efficient attacks based on the commutative property of these structures are well known: Quadratic Sieve, General Number Field Sieve, Pollard's rho algorithm, Index-Calculus, etc. (see, for example, [19,28], and the references within these books).This fact, together with the increase of the computational power of modern computers, has made these techniques become more and more insecure. As a result there exists an active field of research known as noncommutative algebraic cryptography (see, for example, [2,12,13,23,26]) aiming to develop and analyse new cryptosystems and key *

show abstract

“…This ring is a generalization of the ring E p , Climent, Navarro and Tartosa introduced in [15]. The ring E (m) p admits only few invertible elements [16,Corollary 1], for which it avoids most of the attacks (see [17]). In addition, another nice property of such rings is that they do not admit embeddings into matrix rings over a field (see [18]), which is often the main problem of cryptographic schemes over matrix rings (see for example [19] supported also by the results in [12]).…”

Section: Introductionmentioning

confidence: 99%