In this paper we study systematically the ℓ-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between ℓ-adic Eisenstein classes and elliptic Soulé elements. This allows to give a new proof of the formula for the residue of the ℓ-adic Eisenstein classes at the cusps and the formula for the cup-product construction in [HK99], which relies only on the explicit description of elliptic units. This computation is the main input in the proof of Bloch-Kato's compatibility conjecture 6.2. needed in the proof of Tamagawa number conjecture for the Riemann zeta function.

In this paper we propose an efficient multivariate public key cryptosystem. Public key of our cryptosystem contains polynomials of total degree three in plaintext and ciphertext variables, two in plaintext variables and one in ciphertext variables. However, it is possible to reduce the public key size by writing it as two sets of quadratic multivariate polynomials. The complexity of encryption in our public key cryptosystem is O(n 3 ), where n is bit size, which is equivalent to other multivariate public key cryptosystems. For decryption we need only four exponentiations in the binary field. Our Public key algorithm is bijective and can be used for encryption as well as for signatures.

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Abstract. Let (R, m) be a Noetherian local ring of dimension d > 0 and depth R ≥ d − 1. Let Q be a parameter ideal of R. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient ei(Q) under certain assumptions on the depth of associated graded ringFurther, we obtain a necessary condition for the vanishing of the last coefficient e d (Q). As a consequence, we characterize the vanishing of e2(Q