Here is Part 2 of our Low Degree Testing Series. Part 1 is at:

https://kitten-finance.medium.com/low-degree-testing-in-zk-stark-part-1-c0ac6ef0de3c

As an example, assume Bob’s claim is deg(f) < 256.

Bob constructs g(a,b,c,d), such that f(x) = g(x, x⁴, x¹⁶, x⁶⁴).

The task is to prove deg_a(g) < 4, deg_b(g) < 4, deg_c(g) < 4, deg_d(g) < 4.

1. The finite field

Here we assume some knowledge of finite field (the fancy version of modular arithmetic).

Let Fₚ be the finite field with p elements, where p ≡ 1 (mod 64) is a large prime number (much greater than 256).

For any x in Fₚ and any n, we…

You may have heard about zk-SNARK, zk-STARK, etc., which can be used for ETH Layer 2 scaling and more. An excellent explanation is Vitalik’s blog post:

https://vitalik.ca/general/2017/11/09/starks_part_1.html

The Low Degree Testing in zk-STARK can be confusing for most newcomers. It is usually explained using a recursive algorithm.

Here we provide an alternative explanation without recursion. The basic idea is simple:

You can write a number (say 73678235) in radix-4 (becomes 10121003312123), and then you can verify the number is small by bounding each digit of its radix-4 form.

The problem of Low Degree Testing：

• Bob wants to prove to Alice…

This is part 7 of the KittenSwap series, in which we gradually present our next-generation swap+lending+option product.

The lending+option part will be our version of Aave+Hegic which works for all tokens, instead of a few high MC ones.

1. From Put-Call Parity to Lending

Consider the famous put-call parity:

In other words, ASSET = [CALL] + [SELL PUT] + [BORROW CASH].

Assume TOKEN is trading at 1 TOKEN = 0.1 ETH.

If you plan to borrow approximately 3 ETH using 100 TOKEN for two weeks, then you can decompose 100 TOKEN into three parts using our contract:

(i) CALL 100 TOKEN @ 0.03 ETH per TOKEN…

This is part 6 of the KittenSwap series, in which we gradually present our design for the next-generation swap+lending+option solution.

Consider lending. Currently we can only use a number of high MC tokens as collateral on major lending platforms, and low MC tokens are at a disadvantage.

KittenSwap ( https://www.kittenswap.org/ ) can change this. In this article, we discuss a method to build a lending+option pool for any ERC20 token, including those with low liquidity.

1. The Basic Design

Assume a low-liquidity TOKEN is trading at 1 TOKEN = 0.1 ETH on uniswap.

Assume the price-1-month (a parameter) of the TOKEN is 0.03 ETH…

This is part 5 of the KittenSwap series, in which we gradually present our design for the next-generation swap.

KittenSwap is at https://www.kittenswap.org/ and you can trade our first IKO (Initial KittenSwap Offering) token $LIQUID there.

In this article, we discuss a simple method to improve limit orders using AMM information.

1. The Problem with Usual Limit Orders

It is well-known that there are arbitrage opportunities in an OrderBook DEX. As an example:

• Assume the current uniswap price is 1 TOKEN = 0.1 ETH.
• Assume there is a limit order of BUY 1 TOKEN @ 0.08 ETH.
• Assume there is a limit order of SELL 1 TOKEN…

LIQUID (https://www.kittenswap.org/) has a ever-rising price floor, which means LIQUID holders will be able to lock LIQUID for risk-free ETH when our LIQUID lending contract is ready.

This lending contract will be available for all current LIQUID holders. There’s no need to do any migration.

1. Benefits for LIQUID: Less Selling, Easier Buying

For simplicity, assume the current LIQUID price is fixed at 0.2 ETH, and the price floor is fixed at 0.11 ETH.

If you buy 10 LIQUID using 10*0.2=2 ETH, then you will be able to immediately lock it for 10*0.11=1.1 ETH.

This is great for LIQUID holders, because now you can get risk-free extra capital…

This is part 4 of the KittenSwap series, in which we gradually present our design for the next-generation swap.

KittenSwap is at https://www.kittenswap.org/ and you can trade our first IKO (Initial KittenSwap Offering) token $LIQUID there.

LIQUID has a first-in-DeFi feature called AutoBoost. Whenever LIQUID is burned, its price is automatically boosted. You can immediately see the price rise in KittenSwap UI, and you can trade it.

AutoBoost is not a gimmick. It has a solid foundation with formulas, which shows we can shift the bonding curve for better pricing in an AMM.

1. What Happens When You Burn Tokens

It begins with the burning of tokens.

…

This is part 3 of the KittenSwap series, in which we gradually present our design for the next-generation swap.

1. You Don’t Need LPs for AMMs

Here we show it’s possible to have good liquidity without LP.

And then problems like IL (impermanent loss) and “how to provide LP incentives” are gone forever.

Liquidity is essential for a new token, hence there are tokens with LGEs (Liquidity Generation Events), where LPs lock significant amount of ETH to provide permanent liquidity.

However consider this:

1. In the beginning, let the contract claim it has 1000 ETH + 10000 token, even though there is only 0 ETH + 10000 token…