section    .text
   global _start         ;must be declared for using gcc
    
_start:                     ;tell linker entry point

   mov    eax, 45         ;sys_brk
   xor    ebx, ebx
   int    80h

   add    eax, 16384     ;number of bytes to be reserved
   mov    ebx, eax
   mov    eax, 45         ;sys_brk
   int    80h
    
   cmp    eax, 0
   jl    exit    ;exit, if error 
   mov    edi, eax     ;EDI = highest available address
   sub    edi, 4         ;pointing to the last DWORD  
   mov    ecx, 4096     ;number of DWORDs allocated
   xor    eax, eax     ;clear eax
   std             ;backward
   rep    stosd            ;repete for entire allocated area
   cld             ;put DF flag to normal state
    
   mov    eax, 4
   mov    ebx, 1
   mov    ecx, msg
   mov    edx, len
   int    80h         ;print a message

exit:
   mov    eax, 1
   xor    ebx, ebx
   int    80h
    
section    .data
msg        db    "Allocated 16 kb of memory!", 10
len     equ    $ - msg

Allocated 16 kb of memory!