On Some Computations on Sparse Polynomials

In arithmetic circuit complexity the standard operations are +,x. Yet, in some scenarios exponentiation gates are considered as well. In this paper we study the question of efficiently evaluating a polynomial given an oracle access to its power. Among applications, we show that: * A reconstruction algorithm for a circuit class c can be extended to handle f^e for f in C. * There exists an efficient deterministic algorithm for factoring sparse multiquadratic polynomials. * There is a deterministic algorithm for testing a factorization of sparse polynomials, with constant individual degrees, into sparse irreducible factors. That is, testing if f = g_1 x ... x g_m when f has constant individual degrees and g_i-s are irreducible. * There is a deterministic reconstruction algorithm for multilinear depth-4 circuits with two multiplication gates. * There exists an efficient deterministic algorithm for testing whether two powers of sparse polynomials are equal. That is, f^d = g^e when f and g are sparse

Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

Informal proof, formal proof, formalism

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened

The Interpolating Random Spline Cryptosystem and the Chaotic-Map Public-Key Cryptosystem

The feasibility of implementing the interpolating cubic spline function as encryption and decryption transformations is presented. The encryption method can be viewed as computing a transposed polynomial. The main characteristic of the spline cryptosystem is that the domain and range of encryption are defined over real numbers, instead of the traditional integer numbers. Moreover, the spline cryptosystem can be implemented in terms of inexpensive multiplications and additions. Using spline functions, a series of discontiguous spline segments can execute the modular arithmetic of the RSA system. The similarity of the RSA and spline functions within the integer domain is demonstrated. Furthermore, we observe that such a reformulation of RSA cryptosystem can be characterized as polynomials with random offsets between ciphertext values and plaintext values. This contrasts with the spline cryptosystems, so that a random spline system has been developed. The random spline cryptosystem is an advanced structure of spline cryptosystem. Its mathematical indeterminacy on computing keys with interpolants no more than 4 and numerical sensitivity to the random offset t( increases its utility. This article also presents a chaotic public-key cryptosystem employing a one-dimensional difference equation as well as a quadratic difference equation. This system makes use of the El Gamal’s scheme to accomplish the encryption process. We note that breaking this system requires the identical work factor that is needed in solving discrete logarithm with the same size of moduli