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CVE-2014-6271 was the first vulnerability discovered. A patch can be found here.

From wikipedia explanation:

Function definitions are exported by encoding them within the environment variable list as variables whose values begin with parentheses ("()") followed by a function definition. The new instance of Bash, upon starting, scans its environment variable list for values in this format and converts them back into internal functions.

Bash performs this conversion by creating a fragment of code that defines the function and executing it, but it does not verify that the fragment is merely a function definition. Therefore anyone who can cause Bash to execute with a particular name/value pair in its environment, can also execute arbitrary commands by appending those commands to an exported function definition.

In the source code, we can see the importing of the function variables in variables.c:

/* Initialize the shell variables from the current environment.
   If PRIVMODE is nonzero, don't import functions from ENV or
   parse $SHELLOPTS. */
initialize_shell_variables (env, privmode)
     char **env;
     int privmode;
  for (string_index = 0; string = env[string_index++]; )

      /* If exported function, define it now.  Don't import functions from
     the environment in privileged mode. */
      if (privmode == 0 && read_but_dont_execute == 0 && STREQN ("() {", string, 4))
        parse_and_execute (temp_string, name, SEVAL_NONINT|SEVAL_NOHIST);

We can see a for loop over all the environment variables given to the function, and then an if about whether we are in privileged mode, but that is disabled most times. The "not verify that the fragment is merely a function definition" vulnerability is in the parse_and_execute line. The function description from builtins/evalstring.c:

/* Parse and execute the commands in STRING.  Returns whatever
   execute_command () returns.  This frees STRING.  FLAGS is a
   flags word; look in common.h for the possible values.  Actions
    (flags & SEVAL_NONINT) -> interactive = 0;
    (flags & SEVAL_INTERACT) -> interactive = 1;
    (flags & SEVAL_NOHIST) -> call bash_history_disable ()
    (flags & SEVAL_NOFREE) -> don't free STRING when finished
    (flags & SEVAL_RESETLINE) -> reset line_number to 1
parse_and_execute (string, from_file, flags)
     char *string;
     const char *from_file;
     int flags;

So everything that's passed to the function gets executed as if it would be an ordinary bash command. The flags SEVAL_NONINT and SEVAL_NOHIST are self-explanatory (explanation of interactivity, NOHIST doesn't add the definition to your bash history) don't prevent passing other things than function definitions. The patch introduces flags SEVAL_FUNCDEF and SEVAL_ONECMD that can be passed in the flags field to parse_and_execute:

+ #define SEVAL_FUNCDEF 0x080       /* only allow function definitions */
+ #define SEVAL_ONECMD  0x100       /* only allow a single command */

The patch also adds functionality to parse_and_execute to comply with those new flags, and changes the call to parse_and_execute to pass those flags:

-     parse_and_execute (temp_string, name, SEVAL_NONINT|SEVAL_NOHIST);
+     /* Don't import function names that are invalid identifiers from the
+        environment. */
+     if (legal_identifier (name))
+       parse_and_execute (temp_string, name, SEVAL_NONINT|SEVAL_NOHIST|SEVAL_FUNCDEF|SEVAL_ONECMD);