You can find the Mac version of this tutorial here:
How to Officially Update and Jailbreak to 1.1.3 iPhone Firmware
I strongly recommend using the Installer method to perform this soft jailbreak. It is way easier and less error prone. It can be found here:
How to Officially Update and Jailbreak to 1.1.3 iPhone Firmware Using Installer
To follow these instructions you need to have a 1.1.2 Jailbroken iPhone. You can find instructions on how to do this here:
Windows: How to Activate and Jailbreak Your OTB 1.1.2 iPhone Using Windows
Mac: How to Activate and Jailbreak Your OTB 1.1.2 iPhone
PREPARE THE PHONE
Press to select Settings from your springboard.
Press to select General from the list of categories.
Press to select Auto-Lock from the General menu.
Press to select Never from the list of Auto-Lock times.
Press the Home button to return to your springboard.
Press to select Installer from your springboard.
Press Donate Later to continue into the Installer app. Note: Come back later and donate as this app is helping you out a ton!
Press to select the Install tab at the bottom of the screen.
Press to select the System category.
Press to select BSD Subsystem from the list of packages.
Press to select the Install button at the top right.
Press the Install button to confirm installation.
NOTE***: Make sure before you continue that you have updated to the latest Installer version if you have not already. The old version of Installer will not operate on 1.1.3.
INSTALL THE JAILBREAK
Download the Jailbreak from: here
Extract the downloaded file (Jailbreak-1.1.3.tar.gz). WinRAR can be used to extract it...
Launch WinSCP by selecting it from your Programs list.
Enter your iPhone's IP Address as your Host name. Enter root as your User name. Enter alpine as your Password. Finally click the Login button.
If you don't know your iPhone's IP Address you can find it here:
How to Find the IP Address of Your iPhone
Navigate to the location of the Jailbreak-1.1.3 folder using the left pane and select all the files in it.
Drag the selected files to the right pane. If prompted to confirm the copy press the Copy button.
Click to select Open Terminal from the Commands Menu.
If asked to open a separate shell session click the Ok button.
In the console window that appears enter the command cd / and click the Execute button.
Now enter the command sh install.sh and click the Execute button.
The script will download the firmware from Apple once it finishes it will just sit there for a few minutes. Don't worry this is normal.
NOTE***: If you ever get this window displayed do not click anything on it. Don't click Abort and don't click the X either. It will disappear in few minutes!
It will then display some encryption information and sit there for a few minutes. Don't worry this is normal.
The script will then display some partition information and sit there for another few minutes. Don't worry this is normal.
The script will then continue to patch your iPhone. It will take several minutes for this to complete and then it will sit there for a longer several minutes again. Once its complete the SSH connection will terminate!
After about a minute your iPhone will reboot into a Jailbroken 1.1.3 iPhone !!!!
NOTES***: If you are experiencing problems with iTunes sync after performing this soft update/jailbreak then follow this tutorial:
How to Fix Sync Problems After Official Soft Jailbreak
THANKS***: asap18, bgm, Bugout, bushing, chris_, davidc__, dinopio, drudge, Fred_, ghost_000, gray, MuscleNerd, NerveGas, netkas, np101137, planetbeing, pr3d4k4t, pr3d4t0r, pumpkin, roxfan, Turbo, Whiterat, Zibri, Zf