Next, we claim that x ∈ Qp is a square if and only if x = py for n ∈ Z, and y ∈ Zp a padic unit. One direction is obvious. The second direction is also fairly obvious–just refer to the previous case, and also note that p is itself NOT a square in Qp (since if it had a square root, then that square root would have valuation 12 ). We can conclude then that [ Qp : Q×2 p ] = 4, and is given by {1, ...

Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n) > π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. If π(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) deno...

With the help of a computer, we obtain the minimum distance of some codes belonging to two families of Z2k -linear codes: the first is the generalized Kerdock codes which aren’t as good as the best linear codes and the second is the Hensel lift of quadratic residue codes. In the latter, we found new codes with same minimum distances as the best linear codes of same length and same cardinality. ...

In this paper, we investigate the relationship between Milnor's K-group and Galois cohomology for the quotient eld of a 2-dimensional complete regular local ring with a nite residue eld. The results given in this paper are considered as a partial answer to the Bloch-Kato conjecture for such a eld.

Three approaches are currently used for devising identitybased encryption schemes. They respectively build on pairings, quadratic residues (QR), and lattices. Among them, the QR-based scheme proposed by Cocks in 2001 is notable in that it works in standard RSA groups: its security relies on the standard quadratic residuosity assumption. But it has also a number of deficiencies, some of them hav...

We present a novel public key cryptosystem in which the public key of a subscriber can be chosen to be a publicly known value, such as his identity. We discuss the security of the proposed scheme, and show that this is related to the difficulty of solving the quadratic residuosity problem

A long standing problem has been to develop “good” binary linear block codes, C, to be used for error-correction. The length of the block is denoted n and the dimension of the code is denoted k. So in this notation C ⊆ GF (2) is a k-dimensional subspace. Another important parameter is the smallest weight of any non-zero codeword, d. This is related to error-correction because C can correct [d−1...

The quick growth of easy communication technologies concluded the last several decades has led in establishment strict standards for functioning productive systems. system execution is improved by reducing computation time with “Residue Number System (RNS)”. It extensively castoff “signal processing” “numeral analysis”, and “cryptoanalysis”, an exact graph-based technique designing perfect conv...