# Keeping secrets is fundamental for democracies 1. RSA algorithm is the workhorse technique for public cryptography. How it works can be understood with elementary number theory. Understanding why it works is a whole another topic.

2. Consider this. Sending a secret message in a sealed envelope is insecure because the mailman can unseal the envelope.

3. So, your banker builds a public mailbox available to the entire world. Anybody can drop envelopes into it. But, only the banker has a private key to open this public mailbox. 4. This analogy explains how RSA algorithm works. The Bank publishes a public key to encrypt messages and guards the private key to decrypt those messages.

5. Before we talk about RSA mechanics - lets clarify what Modulo Operation does. It returns the remainder resulting from any $$x/y$$. For example, =MOD(17,5) returns $$2$$ in Excel. Its math notation is $$17\bmod 5 = 2$$ 6. RSA algorithm needs three elements to encrypt a message $$m$$ to its cipher text $$c$$ and decrypt it back to $$m$$:
• A public modulus, $$n$$
• A public encryption key, $$e$$
• A private decryption key, $$d$$
7. With these elements,
• $$m$$ is encrypted to $$c$$ with function $$m^e \bmod n$$.
• $$c$$ is decrypted back to $$m$$ with function $$c^e \bmod n$$
• You can see the two functions combined gives us: $$(m^e \bmod n)^ d \bmod n = m$$
8. For example, if $$n$$ = $$33$$, $$e$$ = $$7$$, $$d$$ = $$3$$ and $$m$$ = $$31$$.
$$m$$ is encrypted to $$c$$ with : $$m^e \bmod n \\ \equiv {31^7} \bmod {33} \\ \equiv remainder \ of \ {31^7}/{33} \\ = 4$$

9. $$c$$ is decrypted back to $$m$$ with : $$c^d \bmod n \\ \equiv {4^3} \bmod {33} \\ \equiv remainder \ of \ {4^3} / {33} \\ = 31$$ 10. The algorithm holds true only in the bounds of 4 constraints.
• $$n$$ must be $$p \times q$$ where

both $$p$$ and $$q$$ are prime numbers.

• $$e$$ must such that it is

a co-prime of $$(p-1) \times (q-1)$$ and $$>1$$.

• $$d$$ must be such that

$$( d \times e) \bmod ((p-1) \times (q-1)) \equiv 1$$.

• $$m$$ must be less than $$n$$
11. These constraints hold for $$n$$ = $$33$$, $$e$$ = $$7$$, $$d$$ = $$3$$ and $$m = 31$$
• $$n$$ is $$3 \times 11$$

both $$3$$ and $$11$$ are prime numbers;

• $$7$$ is a co-prime of $$20$$.

$$20$$ is $$(p-1) \times (q-1)$$;

• $$21 \bmod 20$$ is $$1$$

$$21$$ is $$( d \times e) \equiv (3 \times 7)$$
and $$20$$ is $$(p-1) \times (q-1)$$;

• $$m \equiv 31$$ is less than $$n \equiv 33$$