If an encryption algorithm is tested, examined, and found that the mathematics behind it are solid, and secure, then why do so many encryption protocols get broken?
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The fact is that flaws do exist. Mathematicians will tell you the beauty of numbers is that they are predictable. However, even in a world of predictability, we have not perfected the science of encryption.
I find most of the time it is not the math behind the encryption that is flawed. It is the implementation or the key distribution methods.
In order to build strong encryption values you need random keys. How do we get random numbers? Most of the time we use mathematical algorithms that generate randomness for us. The flaw with this entire process is that the algorithms we use to generate randomness are not random themselves.
There are implementations that strive to generate entropy, and do a great job at it. However, at the end of the day there will be some static part to the formula, and that is the part that is vulnerable to attacks.
Asymmetric keys are based on factoring large, random prime numbers. If I know the value of those prime numbers I will may able to crack the encryption.
The biggest problem is that is doesn’t matter how BIG your key size or encryption protocols are. What matters is how you secure your key distribution.