*Resources:*

time spent in class: 2.5 hours

Princeton book: Chapters 1.1-1.4

Video on how secure is SHA-256

Hashing functions in python and bash

UBasel course: 1

- Why storing plain
**hashes(passwords)**(without salt) by a website is not acceptable? - Find the largest number N, such that your laptop can find in under 10 minutes a nonce r such that
**sha256(your_first_name || r)**starts with N zeroes. - Read an in-depth article about sha256 hash function.

**Overview of the remaining course topics:
**Cryptography, different consensus protocols, details on Bitcoin/Ethereum, presentations on DeFi primitives, DAOs, NFTs, Scaling, MEV, zk-tech.

# Hash functions

__ Hash function__ is a function that

- has input = any string (in reality there is a large upper bound, bits is common);
- has ouput = some
*fixed*size number (often 256-bit); - is efficiently computable, which means computation for n-bit string takes O(n) time.

You should think of a hash function as a map

**Example:** computing the remainder: H(x)= x (mod 97).

**Applications:** **checksums to detect typos.**

- 3. Checksums are also used in credit cards, vin numbers (US and Canada)

Such simple hash functions are helpful to **detect accidents**, but **not to prevent attacks. **For example, these two IBANs are very similar but have the same checksum, and so an attacker may try to sub one with the other in order to steal money:

CH__ 56__048350123456

__09 vs CH__

**780**__048350123456__

**56**__09__

**877**The above is an example of a *collision. *To prevent attacks we need:

** Cryptographic(ally secure) hash function** is a hash function satisfying:

- Collision resistance
- Pre-image resistance
- Puzzle-friendliness (intuitively ”uniform randomness”)

Let’s take a close look at these properties. (Later on we will also cover the construction of hash functions.)

# Cryptographic hash functions: properties and applications.

### 1. Collision resistance

a function H:X→Y is calledDefinition:collision-resistantif it is infeasible to find x and x’ such that H(x)=H(x’).

**Collisions exist.**
Hash functions do have collisions, since the domain is larger than the target . In fact, trying values as an input will for sure produce a collision. Moreover, if we pick random inputs and compute the hash values, we’ll find a collision with high probability long before examining inputs. In fact, if we randomly choose just inputs, it turns out there is a 99.8% chance that at least two of them are going to collide. This phenomenon in probability is known as as the **birthday paradox** [need only 23 people for >50% of the same birthday].

**Collisions are hard to find via enumeration.**
The catch, is of course, this way of finding a collision is extremely resource-intensive. If a computer computes 10000 hashes per second, we would need years to compute hashes, an average number of hashes needed to find a collision. Thus we can conclude that finding a collision in a life-span of a human (something someone might try to do) is much less probable than an asteroid hitting the earth, and therefore we do not need to worry about it — for practical applications, we can assume hash functions to be collision resistant.

**Collisions may be easy to find by some other, clever method.**
Here is an example where a different, cleverer method to find a collision works. Hash function is not collision-resistant, since and give an obvious collision. [Remember that a string = sequence of bits = binary number] This example showcases that some arbitrary function can very well be not collision-resistant. Turns out, there are no functions that are *proven *to be collision-resistant.

**Cryptographic hash functions are ***empirically*** collision resistant.
**The cryptographic hash functions that we rely on in practice are just functions for which people have tried really, really hard to find collisions and haven’t yet succeeded. In some cases, such as the old MD5 hash function, collisions were eventually found after years of work, leading to the function being deprecated and phased out of practical use.

**Application: recognition of a file that you have seen before.**

### 2. Pre-image resistance

A function H is pre-image resistant if given y=H(x) it is infeasible to guess what x is.

**Problem:** what if we have y=H(”heads”) and also know that x can attain only two values, ”heads” or “tails”. Then a brute-force search algorithm will quickly find the pre-image: try H(”tails)” and try H(”heads”), and pick the one which gives y.

**Solution:** we can add to the hash something very random, a secret string r that can have many many values. Then, knowing y=h(r || ”heads”), one will not be able to guess what x was.

a hash-function h isDefinition:pre-image resistant(orhiding) if: when a secret value r is chosen from a close to uniform distribution on a large set of values, then given y = H(r ‖ x) it is infeasible to guess x, even if x is chosen from a small set. [Note 1: ris not givento the finder] [Note 2: “close to uniform distribution on a large set of values” is used here as a good substitution for the more technical “distribution with high min-entropy”]

**Application 1:** **secure password storage on a website server.**

**Question:** why storing plain **hashes(passwords)** (without salt) by a website is not acceptable?

**Application 2: commitment to a secret message now, with a possibility of revealing the message later.**A commitment scheme consists of two algorithms:

**com := commit(msg, nonce) = H(nonce || msg)**The commit function takes a message and secret random value, called a nonce, as input and returns a commitment.**verify(com, msg, nonce)**The verify function takes a commitment, nonce, and message as input. It returns true if com = commit(msg, nonce) and false otherwise.

We require that the following two security properties hold:

**Hiding:**Given com, it is infeasible to find msg [This follows from the pre-image resistance of H, and ensures that the message is hidden]**Binding:**It is infeasible to find two pairs (msg, nonce) and (msg’, nonce’) such that msg ≠ msg’ and commit(msg, nonce) = commit(msg’, nonce’). [This follows from the collision resistance of H, and ensures that you cannot change your mind later.]

Rmk. Every time you commit to a value, it is important that you choose a new random value nonce. In cryptography, the term nonce is used to refer to a value that can only be used once.

### 3: Puzzle-friendliness (intuitively ”uniform randomness”)

A hash function h is said to beDefinition:puzzle-friendlyif for every possible n‐bit output value y, if k is chosen from a close to uniform distribution on a large set of values, then it is infeasible to find x such that h(k ‖ x) = y in time significantly less than . [Note: kis givento the finder; this is the difference compared to pre-image resistance]

Intuitively, what this means is that if someone wants to target the hash function to come out to some particular output value y, that if there’s part of the input that is chosen in a suitably randomized way, it’s very difficult to find another value that hits exactly that target.

**Application: search puzzle.** It consists of:

- a hash function, H,
- a value, id (which we call the “puzzle‐ID”), chosen from a close to uniform distribution on a large set of values
- and a target set Y

A solution to this puzzle is a value, x, such that H(id ‖ x) ∈ Y.

**How this fits into proof-of-work in Bitcoin.**
There H = sha256, ID = new block, x = nonce that miners try to find, , where the number is the difficulty number automatically adjusted by the Bitcoin nodes’ software once in two weeks so that the average time between blocks is ten minutes. The fact that ID comes from a close to uniform distribution is insured by the fact that it cointains the hash of the previous block.

**The size of Y determines how hard the puzzle is. **
If Y is the set of all n‐bit strings the puzzle is trivial, whereas if Y has only 1 element the puzzle is maximally hard. The fact that the puzzle-ID is randomly chosen from a large value-set ensures that there are no shortcuts. On the contrary, if a particular value of the ID were likely, then someone could cheat, say by pre‐computing a solution to the puzzle with that ID.

**If H is puzzle‐friendly, this implies that there’s no solving strategy for the search-puzzle which is much better than just trying random values of x.**
And therefore, if we want to pose a puzzle that’s difficult to solve, we can do it this way as long as we can generate puzzle‐IDs in a suitably random way. This is crucial for the proof-of-work idea in mining.

# Cryptographic hash functions: construction

There are several components that we need to introduce:

### (1) Block ciphers (cryptography’s workhorse construction)

**Block cipher **consists of two maps satisfying . Each E(k,-) is essentially a permutation on X. E stands for enccryption, D stands for decryption.

**Secure block cipher** is a block cipher where a random permutation from is *probabilistically* *indistinguishable* from a random permutation from . In other words, the subset is scattered *uniformly* inside , as illustrated below.

White:

Black:

**How are secure block ciphers constructed?
**Secure block ciphers are built using **iterations **of a **round function R**:

Key point: if the round function R is chosen appropriately (obv needs to be non-linear) and if n is large enough, this proccess arrives at a secure block cipher. In real life this is done more or less experimentally, two standard algorithms:

**3DES:**X={0,1}^64, K={0,1}^168, 48 iterations**AES:**X={0,1}^128, K={0,1}^128 (or 192 or 256), 10 iterations

### (2) Compression functions

**Compression function** is the name for a *fixed-length-input *hash function, eg . They should satsify the same properties: collision and pre-image resistance, and uniform randomness.

### (3) **Davies-Meyer construction**

Given a block cipher (), one can construct a compression function as follows:

where is bit-wise summation (mod 1) [01=1, 10=1, 00=0, 11=0] (aka “xor” functions, aka exclusive “or”)

__ Thm:__ if a block cipher is secure, then the Davies-Meyer compression function is as collision resistant as possible.

### (4) **Merkle-Damgrad** paradigm

Enables transition [compression function] → [arbitrary-length-input hash function], via the scheme illustrated below (the specific numbers of bits are from SHA-256).

The input is subdivided into substrings, 512 bit each; the last substring is padded at the end with the total length of the initial input, so that the total length of the resulting input is a multiple of 512 bits. IV = Initialization Vector = some number that is chosen a priori.

__ Thm:__ if compression function h is collision resistant, then so is the resulting hash function.

__SHA-256 example__ (hash function used in Bitcoin)

**Block cipher**is chosen: SHACAL-2 (x (256 bit), k (512 bit) ) → x’ (256 bit)**Compression function**is constructed using the**Davies-Meyer**method: h( H (256 bit) , m (512 bit) ) = SHACAL-2 ( H, m ) H**SHA-256 hash function**is constructed using the**Merkle-Damgrad**paradigm.

# Hash pointers and Merkle trees

**Hash pointer** is a data structure that consists of:
(1) a pointer to where data is stored;
(2) hash of that data.

Whereas a regular pointer gives you a way to retrieve the information, a hash pointer also gives you a way to verify that the information hasn’t changed.

**Block chain** data structure is a sequence of blocks with hash pointers to previous blocks.

The key property of a block chain data structure is that it is **temper evident**, which means that by storing the head of a blockchain, any alteration of previous data will result in a change of our hash.

**Merkle tree** is another data structure that uses hash pointers. Each hash is obtained by concatenating the two lower hashes and taking their hash.

It is also temper evident, but has the following advantages over the block chain:

**Proof of membership.**Given n pieces of data (above n=4), only about log(n) hashes are needed in order to prove that some particular piece of data is stored in a Merkle tree.

If a “light client” knows the root hash and wants to be convinced that data3 is stored in the merkle tree, a “full-node” (the one that stores the merkle tree) only needs to send the 2 green hashes to “light-client”. Note that .

**Proof of non-membership.**If pieces of data are sorted (lexicographically, say), then proof of non-membership requires O(log(n)) items to be shown. (just show that two consecutive items are stored, and the item in question would need to be between them)

**Super-important blockchain applications of Merkle trees.** (explained here)

- In Bitcoin, full nodes store all txs inside Merkle trees (takes hundreds of GBs), while light clients (= wallets) store only the hash roots, and can prove membership of transactions in a fast manner by asking full nodes for merkle proofs.
- In Ethereum, not only txs are stored in a Merkle tree, but also the
*Ehereum**state*is stored in a Merkle tree. There are also Merkle trees for txs receipts and for storages of smart contracts (see the diagraь below). __Archival nodes__: in each block they store all txs + all Ethereum state in Merkle trees.__Full nodes (typical node)__: in each block they store all txs in Merkle trees and store only the Merkle root of Ethereum state; can query archival nodes for the state.__Light clients (wallets)__: in each block they store Merkle root of txs and Merkle root of Ethereum state; can query archival nodes for states (usually this is done by a central provider), and full nodes for txs.

Various Ethereum nodes: