A reinforced elgamal scheme proposal against a pohlig-hellman attack

R. Flores-Carapia, V. M. Silva-García, M. D. González-Ramírez, C. Rentería-Márquez

Research output: Contribution to journalArticle

Abstract

Multiplicative group (G, .) where G = Z*p = Zp -{0}, the primitive element calculation, say α, is usually constructed from the following theorem: given a prime p ≥ 2 and α ∈ Zp, which is a primitive or generator element of Z*p if and only if α(p-1)/q ≠ 1 mod p, ∀ prime q such that q | (p - 1). Furthermore, the Dirichlet theorem shows that: given two integers a, b with gcd(a, b) = 1, that is, a, b are relatively prime to each other, then, an infinite number of primes can be constructed in accordance with the formula n(a) + b, where n = 1, 2, . . . . In this work high primality has the following meaning: given a number p ≥ 2, it is said to be prime with an error at most 1/2100. Now, using the two theorems cited above and taking as particular values b = 1 and a = p1p2 where p1, p2, are two high primality numbers; a high primality number p can be built which has the form p = (m)p1p2 + 1. with m = 2n for n = 1, 2, . . . , eliminating those that fulfill p = 0 mod 5. To construct positive integers p-form with high-primality intents that p - 1 can be factorized in a simple way. This research produced 700000 positive p-form integers. The result is that 88.5% of them have a m less or equal to 2000; it is pointed out that p1, p2 are 10200 approximately. Then, one can obtain a primitive α easily, and also a double lock is provided with regard to a Pohlig-Hellman form attack. The first one is the p -1 factorization and the second, solving the discrete logarithm problem when the p - 1 factors are known. © 2013 R. Flores-Carapia et al.
Original languageAmerican English
Pages (from-to)2909-2916
Number of pages2617
JournalApplied Mathematical Sciences
StatePublished - 11 Jun 2013

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Primality
Factorization
Attack
Integer
Dirichlet's theorem
Primitive Element
Discrete Logarithm Problem
Relatively prime
Theorem
Multiplicative
Generator
If and only if
Form

Cite this

@article{dfc04ba33b4747bfbf227f0044e0740c,
title = "A reinforced elgamal scheme proposal against a pohlig-hellman attack",
abstract = "Multiplicative group (G, .) where G = Z*p = Zp -{0}, the primitive element calculation, say α, is usually constructed from the following theorem: given a prime p ≥ 2 and α ∈ Zp, which is a primitive or generator element of Z*p if and only if α(p-1)/q ≠ 1 mod p, ∀ prime q such that q | (p - 1). Furthermore, the Dirichlet theorem shows that: given two integers a, b with gcd(a, b) = 1, that is, a, b are relatively prime to each other, then, an infinite number of primes can be constructed in accordance with the formula n(a) + b, where n = 1, 2, . . . . In this work high primality has the following meaning: given a number p ≥ 2, it is said to be prime with an error at most 1/2100. Now, using the two theorems cited above and taking as particular values b = 1 and a = p1p2 where p1, p2, are two high primality numbers; a high primality number p can be built which has the form p = (m)p1p2 + 1. with m = 2n for n = 1, 2, . . . , eliminating those that fulfill p = 0 mod 5. To construct positive integers p-form with high-primality intents that p - 1 can be factorized in a simple way. This research produced 700000 positive p-form integers. The result is that 88.5{\%} of them have a m less or equal to 2000; it is pointed out that p1, p2 are 10200 approximately. Then, one can obtain a primitive α easily, and also a double lock is provided with regard to a Pohlig-Hellman form attack. The first one is the p -1 factorization and the second, solving the discrete logarithm problem when the p - 1 factors are known. {\circledC} 2013 R. Flores-Carapia et al.",
author = "R. Flores-Carapia and Silva-Garc{\'i}a, {V. M.} and Gonz{\'a}lez-Ram{\'i}rez, {M. D.} and C. Renter{\'i}a-M{\'a}rquez",
year = "2013",
month = "6",
day = "11",
language = "American English",
pages = "2909--2916",
journal = "Applied Mathematical Sciences",
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}

A reinforced elgamal scheme proposal against a pohlig-hellman attack. / Flores-Carapia, R.; Silva-García, V. M.; González-Ramírez, M. D.; Rentería-Márquez, C.

In: Applied Mathematical Sciences, 11.06.2013, p. 2909-2916.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A reinforced elgamal scheme proposal against a pohlig-hellman attack

AU - Flores-Carapia, R.

AU - Silva-García, V. M.

AU - González-Ramírez, M. D.

AU - Rentería-Márquez, C.

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AB - Multiplicative group (G, .) where G = Z*p = Zp -{0}, the primitive element calculation, say α, is usually constructed from the following theorem: given a prime p ≥ 2 and α ∈ Zp, which is a primitive or generator element of Z*p if and only if α(p-1)/q ≠ 1 mod p, ∀ prime q such that q | (p - 1). Furthermore, the Dirichlet theorem shows that: given two integers a, b with gcd(a, b) = 1, that is, a, b are relatively prime to each other, then, an infinite number of primes can be constructed in accordance with the formula n(a) + b, where n = 1, 2, . . . . In this work high primality has the following meaning: given a number p ≥ 2, it is said to be prime with an error at most 1/2100. Now, using the two theorems cited above and taking as particular values b = 1 and a = p1p2 where p1, p2, are two high primality numbers; a high primality number p can be built which has the form p = (m)p1p2 + 1. with m = 2n for n = 1, 2, . . . , eliminating those that fulfill p = 0 mod 5. To construct positive integers p-form with high-primality intents that p - 1 can be factorized in a simple way. This research produced 700000 positive p-form integers. The result is that 88.5% of them have a m less or equal to 2000; it is pointed out that p1, p2 are 10200 approximately. Then, one can obtain a primitive α easily, and also a double lock is provided with regard to a Pohlig-Hellman form attack. The first one is the p -1 factorization and the second, solving the discrete logarithm problem when the p - 1 factors are known. © 2013 R. Flores-Carapia et al.

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