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This comment from Reddit math, in response to a statement about how people can communicate secrets to each other with a third party listening, has a very small, simple example of public key cryptography:

Take your message, treat it as a number and multiply it by a bunch of primes.

Send it to me. I will then multiply by a bunch of primes too.

I send it back to you. You then divide by all of your primes.

Send it back to me. I divide by all of my primes and get the original message.

This comment has 5700+ upvotes by Reddit math, but it strikes me as unusual because there's no modular arithmetic used, only primes.

Does this example actually work, if sufficiently large primes are used?

Community
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mikemaccana
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2 Answers2

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Breaking such a scheme is easy.

Suppose Alice wants to transmit a message $M$ to Bob. First thing, Alice picks an integer $R_a$ and sends the cipher text $C_a = M \times R_a$ to Bob. Bob then picks another integer $R_b$ and transmits $C_b = C_a \times R_b$ back to Alice. Alice calculates $D_a = \frac{C_b}{R_a}$ and sends $D_a$ to Bob. Bob calculates $D_b = \frac{D_a}{R_b}$ and clearly $D_b = M$.

Now, Eve intercepts the messages $C_a, C_b, D_a$ and might easily calculate $R_a = \frac{C_b}{D_a}$ and consequently $M = \frac{C_a}{R_a}$. Eve is clearly not supposed to be able to do this, so this constitutes a break.

Furthermore, it doesn't really matter if the multiplication is done in a finite group or not. In a group, division would be defined as multiplication with the inverse element of the divisor, e.g. $D_a = {C_b} \times {R_a}^{-1}$. Eve would still be able to calculate $M$.

Henrick Hellström
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Except above attack,modular arithmetic reduces the amount of calculation and computation with modular properties(example:5832 mod 33=24) and if you want to have a number similar to n=pq that n is difficult to factor,you should choose two large numbers(p,q) and then calculate n=pq , so your algorithm needs large numbers that it needs big memory and long time for calculation,and it is not good.

in other words modular arithmetic in cryptography has a good advantage in calculation with large number that large numbers are necessary for security of cryptosystems because the attacker can not have a brute force attack