Abstract In this paper, we give a fast attack against hash function HAVAL-128. HAVAL was presented by Y. L. Zheng et al. at Auscrypto＇92. It can be processed in 3, 4 or 5 passes, and produces 128, 160, 192, or 224-bit fingerprint. We break the HAVAL with 128-bit fingerprint. The conclusion is that, given any 1024-bit message ra, we just make some modifications about m, and the modified message ra can collide with another message m only with probability 1/2^7, where m = m ＋ △m, in which △m is a fixed difference selected in advance. In addition, two collision examples for HAVAL-128 are given in this paper.
In this paper we prove that, with at most O（N^5/12＋ε） exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1（mod 3） can be written as a sum of a prime and two squares of primes.
