Abstract. In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gröbner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree.Let k be a finite field of cardinality q n and let k ′ be its subfield of cardinality q. Let F ⊂ k[X 0 , . . . , X m−1 ] be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of F , which depends on q, m, the last fall degree of F , the degree of F and the number of solutions of F , but not on n. This shows that such Weil descent systems can be solved efficiently if n grows. In particular, we apply these results for multi-HFE and essentially show that multi-HFE is insecure.Finally, we discuss that the degree of regularity (or last fall degree) of Weil descent systems coming from summation polynomials to solve the elliptic curve discrete logarithm problem might depend on n, since such systems without field equations are not zero-dimensional.
AbstractWe initiate the study of a new class of polynomials which we call quasi-subfield polynomials. First, we show that this class of polynomials could lead to more efficient attacks for the elliptic curve discrete logarithm problem via the index calculus approach. Specifically, we use these polynomials to construct factor bases for the index calculus approach and we provide explicit complexity bounds. Next, we investigate the existence of quasi-subfield polynomials.
We present here a construction of horseshoes for any C 1+α mapping f preserving an ergodic hyperbolic measure µ with hµ(f ) > 0 and then deduce that the exponential growth rate of the number of periodic points for any C 1+α mapping f is greater than or equal to hµ(f ). We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.
In this paper we consider centroaffine immersions of codimension two. We will calculate the first and also second variational formulas of centroaffine volume integral with moving frames and then give some examples of minimal centroaffine immersions of codimension two.
Mathematics Subject Classification (2000). Primary 53A15; Secondary 53C42, 58E30.
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