Root optimization of polynomials in the number field sieve

The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.Comment: 16 pages, 18 reference

Root optimization of polynomials in the number field sieve

International audienceThe general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties

Montgomery's method of polynomial selection for the number field sieve

The number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors. For the polynomial selection stage of the algorithm, Montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions. Montgomery's method is analysed in this paper and the existence of suitable geometric progressions is considered

The large sieve, monodromy and zeta functions of curves

We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the ``generic'' irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.Comment: 30 page

Computing the endomorphism ring of an ordinary elliptic curve over a finite field

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log |D_E|, where D_E is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits

Security Analysis of Pairing-based Cryptography

Recent progress in number field sieve (NFS) has shaken the security of Pairing-based Cryptography. For the discrete logarithm problem (DLP) in finite field, we present the first systematic review of the NFS algorithms from three perspectives: the degree $\alpha$, constant $c$, and hidden constant $o(1)$ in the asymptotic complexity $L_Q\left(\alpha,c\right)$ and indicate that further research is required to optimize the hidden constant. Using the special extended tower NFS algorithm, we conduct a thorough security evaluation for all the existing standardized PF curves as well as several commonly utilized curves, which reveals that the BN256 curves recommended by the SM9 and the previous ISO/IEC standard exhibit only 99.92 bits of security, significantly lower than the intended 128-bit level. In addition, we comprehensively analyze the security and efficiency of BN, BLS, and KSS curves for different security levels. Our analysis suggests that the BN curve exhibits superior efficiency for security strength below approximately 105 bit. For a 128-bit security level, BLS12 and BLS24 curves are the optimal choices, while the BLS24 curve offers the best efficiency for security levels of 160bit, 192bit, and 256bit.Comment: 8 figures, 8 tables, 5121 word