Bitcoin Public and Private Keys - dummies

Bitcoin addresses and private keys are not all that random. Please take caution.

I replied to another comment just now and thought others would be interested. I no longer let wallet programs generate address for me. They are simply not secure.
Take Electrum, for example. Try to export your private keys, transform each key to its 64-char hex value, convert that to an integer, and then start a simple lookup of all 10 million values before and after your own hex value. You will find other people's used addresses. Do it to all your private keys and let it run. After a day you will start seeing the collisions.
Address: 15DTFio6WKG5HRHfRQNXXVhFroDbWwxJgp
Private key: 5JSTpckYPqsPRxrkV3zZzVKMJFNGfVu9JWX721QXTMb5Wr82mSf
Hex: 5224af20889ac590f79ca68588689d873ef60b206b33da6025ace0c90705c0b8
Int val of hex: 37154468760874988449110293609706090360941268111629659303434866284112583704760
Add and substract 100000000 to that last int, generate the private key and then address for each of those and test against a dump of all address with balances from a bitcoind.
Wallets such as Electrum don't generate their private key from 1 115792089237316195423570985008687907853269984665640564039457584007913129639935. They stay within a much smaller bound somewhere in between.
Pretty quickly you end with a dump like
176w5WyxEFVyzew5qSjoThoAt1ERwaqpKz 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmhS9GJNrt7 1GVBWqa8yU7s9JtNacVy72f6PQVdRbhaTt 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmkFt4TbqQt 1JiDAzNpoMX1zKBBGm5LGqp9W9cTNJbrEc 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmjg8AMcAA8 198Ld4Xd3Fvd9mpANheYQHhNrv19QNV7aA 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmjSkK5fVxX 16xaFP1PkrbE8a1phY4Tq1YTREH7Fbtq9y 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmjxNp2PDzY 14R4cWRLsKoKmVjqc57nCEAbACvtydExwe 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmiqT4FPnEm 17KdvJXmynTUsSRaf9FvGkVRCQyehrxgPh 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmiSXhHXrJ9 1FFgSq6nwTMRkRe33RpXa6mBhgJzr3fwY7 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmimdfA1Arf 14pGZ4BEewBkuTnwBeAX1Q54Voa9xV8y8B 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmhx4vtSWgj 14AS4r8eWtRHdHgLUb3x8FEwyvKyMKAaEu 5KHSTeKtpXcaUJyz3iVAif76AHKZmzE4gqKkgnfBUmkUWvFH5k2
and your address will be there too, along with others.
That's why I generate my own addresses by hand from made up hex values that software wallets are unlikely to generate.
I know the conventional wisdom around here is that there are more bitcoin addresses than atoms on our planet, but in reality that is simply not the case. We will start seeing collision faster than everyone assumes. Your addresses and private keys are not truly random and have never been that.
Benford's Law applies here
I don't care either way. You are free to continue to use your addresses and keys as before. I'm just letting you know - they are not as safe as you are being led to believe.
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