The complexity of computing the permanent

New hash functions and their use in authentication and set equality

Much effort has been devoted in the second half of this century to make precise the notion of Randomness. Let us informally recall one of these definitions due to Kolmogorov []. A sequence of bits A =all a2••.•• at is random if the length of the minimal program outputting A is at least k We remark that the above definition is highly non constructive and rules out the possibility of pseudo random number generators. Also. the length of a program, from a Complexity Theory point of view, is a rather unnatural measure. A more operative definition of Randomness should be pursued in the light of modern Complexity Theory.

How to generate cryptographically strong sequences of pseudo random bits

A cryptographic system is described which is secure if and only if computing logarithms over GF(p) is infeasible. Previously published algorithms for computing this function require O(P’/~) complexity in both time and space. An improved algorithm is derived which requires O(log2 p) complexity if p 1 has only small prime factors. Such values of p must be avoided in the cryptosystem. Constructive uses for the new algorithm are also described.

An Improved Algorithm for Computing Logarithms over GF(p) and Its Cryptographic Significance

Current technology is beginning to allow us to manipulate rather than just observe individual quantum phenomena. This opens up the possibility of exploiting quantum effects to perform computations beyond the scope of any classical computer. Recently Peter Shor discovered an efficient algorithm for factoring whole numbers, which uses characteristically quantum effects. The algorithm illustrates the potential power of quantum computation, as there is no known efficient classical method for solving this problem. The authors give an exposition of Shor's algorithm together with an introduction to quantum computation and complexity theory. They discuss experiments that may contribute to its practical implementation. [S0034-6861(96)00303-0]

http://www.phys.uni-sofia.bg/~svetivanov/FourierPapers/1996%20-%20(Th)%20Shor%20-%20Ekert%20-%20RevModPhys.pdf

Quantum Computation and Shor's Factoring Algorithm

: It is shown that the single operation of raising a number to a fixed power modulo a composite modulus is sufficient to implement digital signatures: a way of creating for a (digitized) document a recognizable, unforgeable, document-dependent digitized signature whose authenticity the signer can not later deny. An electronic funds transfer system or electronic mail system clearly could use such a scheme, since the messages must be digitized in order to be transmitted.

/pdf/on-digital-signatures-and-public-key-cryptosystems-3a7bvylh7t.pdf

Len Adleman

Papers

On Digital Signatures and Public-Key Cryptosystems.

Public Key Cryptography