## Primality proving via one round in ECPP and one iteration in AKS (2003)

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Venue: | Advances in Cryptology – CRYPTO 2003 |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Cheng03primalityproving,

author = {Qi Cheng},

title = {Primality proving via one round in ECPP and one iteration in AKS},

booktitle = {Advances in Cryptology – CRYPTO 2003},

year = {2003},

pages = {338--348},

publisher = {Springer Verlag}

}

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### Abstract

On August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial time primality testing algorithm. For an input n, the AKS algorithm runs in heuristic time Õ(log6 n). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving algorithm (ECPP), runs in random heuristic time Õ(log6 n) ( Õ(log5 n) if the fast multiplication is used), and generates certificates which can be easily verified. More recently, Berrizbeitia gave a variant of the AKS algorithm, in which some primes cost much less time to prove than a general prime does. Building on these celebrated results, this paper explores the possibility of designing a more efficient algorithm. A random primality proving algorithm with heuristic time complexity Õ(log4 n) is presented. It generates a certificate of primality which is O(log n) bits long and can be verified in deterministic time Õ(log 4 n). The reduction in time complexity is achieved by first generalizing Berrizbeitia’s algorithm to one which has higher density of easily-proved primes. For a general prime, one round of ECPP is deployed to reduce its primality proof to the proof of a random easily-proved prime. 1

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Citation Context ...ber of non-zero-divisors between 1 and m is φ(m) = m ∏ 1 p (1 − prime,b1≤p≤b2 p ), where φ is the Euler phi-function. First we estimate the quantity: βb1,b2 = ∏ p prime,b1≤i≤b2 (1 − 1 p ) It is known =-=[11]-=- that ∏ 1 e−γ 1 p<x,p prime (1 − p ) = ln x (1 + O( ln x )), where γ is the Euler constant. There must exist two absolute constants c1, c2, such that e−γ c1 (1 + ) ≤ ln x ln x ∏ Set C = c where c repr... |

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Citation Context ... certificate of compositeness is sometimes called compositeness proving algorithm. Very efficient random compositeness proving algorithms have long been known. Curiously, primality proving algorithms =-=[5, 1]-=- lag far behind of compositeness proving algorithms in term of efficiency and simplicity. Recently, Berrizbeitia [7] proposed a brilliant modification to the AKS original algorithm. He used the polyno... |

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(Show Context)
Citation Context ...of primes is in the complexity class P. For a given integer n, the AKS algorithm runs in time no longer than Õ(log12 n), while the best deterministic algorithm before it has subexponential complexity =-=[2]-=-. Under a reasonable conjecture, The AKS algorithm should give out answer in time Õ(log6 n). Notation: In this paper, we use “ln” for logarithm base e and “log” for logarithm base 2. We write rα ||n, ... |

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Citation Context ...val are usually very hard. 1.2 Algorithm for the general primes For general primes, we apply the idea in the Elliptic Curve Primality Proving algorithm (ECPP). ECPP was proposed by Goldwasser, Kilian =-=[8]-=- and Atkin [4] and implemented by Atkin and Morain [5]. In practice, ECPP performs much better than the current version of AKS. It has been used to prove primality of numbers up to thousands of decima... |

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Citation Context ... certificate of compositeness is sometimes called compositeness proving algorithm. Very efficient random compositeness proving algorithms have long been known. Curiously, primality proving algorithms =-=[5, 1]-=- lag far behind of compositeness proving algorithms in term of efficiency and simplicity. Recently, Berrizbeitia [7] proposed a brilliant modification to the AKS original algorithm. He used the polyno... |

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(Show Context)
Citation Context ...n [4] and implemented by Atkin and Morain [5]. In practice, ECPP performs much better than the current version of AKS. It has been used to prove primality of numbers up to thousands of decimal digits =-=[10]-=-. In ECPP, if we want to prove that an integer n is a prime, we reduce the problem to the proof of primality of a smaller number (less than n/2). To achieve this, we try to find an elliptic curve with... |

5 |
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(Show Context)
Citation Context ...proving algorithms have long been known. Curiously, primality proving algorithms [5, 1] lag far behind of compositeness proving algorithms in term of efficiency and simplicity. Recently, Berrizbeitia =-=[7]-=- proposed a brilliant modification to the AKS original algorithm. He used the polynomial x2s − a instead of xr − 1 in equation (1), where 2s ≈ log 2 n. Among others, he was able to prove the following... |

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2 |
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(Show Context)
Citation Context ...he same probability as a random integer does. ECPP needs O(log n) rounds of reductions to eventually reduce the problem to a primality proof of a very small prime, say, less than 1000. As observed in =-=[9]-=-, one round of reduction takes heuristic time Õ(log5 n), or Õ(log4 n) if we use the fast multiplication. To get the time complexity, it is assumed that the number of primes between n − 2 √ n + 1 and n... |

1 |
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(Show Context)
Citation Context ...y very hard. 1.2 Algorithm for the general primes For general primes, we apply the idea in the Elliptic Curve Primality Proving algorithm (ECPP). ECPP was proposed by Goldwasser, Kilian [8] and Atkin =-=[4]-=- and implemented by Atkin and Morain [5]. In practice, ECPP performs much better than the current version of AKS. It has been used to prove primality of numbers up to thousands of decimal digits [10].... |