Number Theory

Encryption with Number Theory

Public-Key Encryption


What sort of mathematical breakthrough would endanger the security of the RSA encryption algorithm?

In an encoding scheme, each letter is replaced as follows: A=1, B=2, C=3,  , Z=26.A = 1, \ B = 2, \ C = 3,\ \ldots\ , \ Z = 26. For example, CODECODE would be encoded as 31545,31545, since C=3,C = 3, O=15,O = 15, D=4,D = 4, and E=5.E = 5.

What is the main problem with such a system?

To crack a code, Alice and Bob need to find the product of their secret large primes, aa and b,b, so they each send these primes to each other.

Alice receives b b and performs ba b \cdot a , while Bob receives a a and performs ab a \cdot b .

What's the main issue with this encryption system?

Suppose Alice and Bob want to establish an encryption key. They start by choosing a large prime p p , which is shared in public.

Alice secretly picks a large prime a a , and Bob secretly picks a large prime b b . Then both Alice and Bob multiply their primes by p p , so Alice now has ap a \cdot p and Bob has bp b \cdot p .

They send ap a \cdot p and bp b \cdot p to each other. They then multiply what they receive by what their own secret prime. That is, Alice receives bp b \cdot p and performs bpa b \cdot p \cdot a , and Bob receives ap a \cdot p and performs apb a \cdot p \cdot b .

Alice and Bob now have the same number that they can use as their secret key.

Is this encryption system secure?

Suppose a cryptography system included the step where, given m, m , e, e , and n, n, it is necessary to calculate c c in mec(modn).m^e \equiv c \pmod{n}. (That is, first me m^e is found, then the value is divided by n n with the remainder c c . )

e, e, n, n, and c c are going to be shared in public, and it is important that m m remains a secret.

Which of these circumstances (if known by someone trying to break the code) would easily compromise m? m?


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