On symmetric and unsymmetric theta functions over a real quadratic field

Theta Functions of Indefinite Quadratic Forms over Real Number Fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

Theta functions of quadratic forms over imaginary quadratic fields

is a modular form of weight n/2 on Γ0(N), where N is the level of Q, i.e. NQ−1 is integral and NQ−1 has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly. Stark [8] gives a different proof by converting θ...

Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields

Secure symmetric ciphers over the real field

Most cryptosystems are defined over finite algebraic structures where arithmetic operations are performed modulo natural numbers. This applies to private key as well as to public key ciphers. No secure cryptosystems defined over the field of real numbers are known. In this work, we demonstrate the feasibility of constructing secure symmetric key ciphers defined over the field of real numbers. W...

Defining relations of a group $Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group $Gamma$ generated by the linear-fractional‎ ‎transformations $r:zrightarrow frac{z-1}{z}$ and $s:zrightarrow frac{-1}{2(z+1)}$ on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that $r^{3}=s^{4}=1$ are defining relations for $Gamma$‎. ‎Furt...