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docs/lectures/cryptography/03_stream_ciphers.md

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+# Modern Stream Ciphers
+
+### Pseudo-randomness
+
+- TRNGs - true Random Number Generator
+  - Not feasible at scale
+- PRNGs - Pseudo Random Number Generator
+- CSPRNGs - Cryptographically Secure Pseudo Random Number Generator
+
+#### LFSRs
+
+- A Linear-feedback Shift Register us a register if buts whose positions shift to the right
+- Usually comprised of flip-flops, the last bit represents the output
+
+(Where the squares at the bottom are flip-flops)
+
+- If initialised to `000`, nothing happens as $0\oplus0 = 0$.
+  - Therefore, we have $2^n-1$ states
+  - Statistical randomness
+- To add more randomness to the setup, we can add another (more) `xor` gate
+  - However, we have fewer states
+
+$$
+s_m \equiv s_{m-1}p_{m-1} + ... + s_1p_1 + s_0p_0\space (mod \space 2)\\
+s_{m+1} \equiv s_{m}p_{m-1} + ... + s_2p_1 + s_1p_0\space (mod \space 2)
+$$
+
+- We usually represent m-bit LFSRs using polynomials of degree m.
+- In general $P(x)=x^m + p_{m-1}x^{m-1} + ... + p_1x + p_0$
+- LFSRs that have primitive polynoimials produce sequences of maximum length
+- There are many and are easily computed
+  - $x^5 + x^2 + 1$ has 31 states
+  - $x^{10} + x^3 + 1$ has 1023
+  - $x^{85}+x^8+x^2+x+1$ has $10^{26}$ states
+
+##### Attacking LFSRs
+
+Suppose an attacker knows $2m-1$ plain text bits
+
+**Step 1** Calculate key bits
+
+$s_i \equiv y_i + x_i \space (mod\space 2), i=0,1,...2_{m-1}$
+
+**Step 2** Reconstruct the LFSR
+
+$s_m \equiv s_{m-1}p_{m-1} + ... + s_1p_1 + s_0p_0$
+
+$s_{m+1} \equiv s_{m}p_{m-1} + ... + s_2p_1 + s_1p_0$
+
+…
+
+$s_{2m+1} \equiv s_{2m-1}p_{m} + ... + s_mp_1 + s_{m-1}p_0$
+
+#### Trivium
+
+- LFSRs are much more cryptographically secure if we combine more than one together in a non-linear way.
+
+Trivium is 3 LFSR in a row
+
+- Feedback between each with non-linear AND gates
+- Initialises the LFSR with an 80-bit key and 80-bit random value
+
+### ChaCha20
+
+- ChaCha is a stream cipher written by Daniel Berstein
+- A modification of a previous cipher, Salsa
+- Very lightweight, using only `add`, `xor` and rotate operations
+- One of two ciphers in `TLS 1.3`
+- Dashes represent bit length
+- Constants are not secret
+- The block number can skip to anywhere
+  - Suppose someone skips ahead on a video stream, the cipher can skip unlike other synchronous stream ciphers
+- Works well on low power devices, due to simplicity of encryption
+- Once the input and the mixed words are added together it is hard to know what the starting thing was
+  - e.g. what two numbers have i added to make 100
+
+ChaCha performs **20** rounds
+
+- Alternates column and diagonal rounds
+  - Each round is 4 quarter rounds
+
+#### Vulnerabilities
+
+- Stream ciphers like ChaCha give us *confidentiality*, but *not integrity*
+- Running a stream cipher by itself is not sufficient