Number Theory Learning Group

1. Why we do it

As web3 buidlers, we(the learning group) have strong interests in research or application field of zero-knowledge, such as verifiable computation, private computation as well as anonymous transaction and etc.

For senior or skillful developers, it seems not difficult to implement zk dapp based on circuit development. But there is long way to go understanding zero knowledge deeply and broadly which is an irreplaceable fascinating part of the path learning or applying zk.

Compared to hand on zk dapp development, diving into exploring zk deeply and broadly is hard and learning some underlying abstract theorem or algorithm requires motivation, persistence and in-depth studying. Thus, this learning group is organized for supporting each other in the path of learning number theory which is necessary for understanding zk theoretically.

2. Plan Description

In 10 weeks starting from 11-01, 2022. In every week, group members learning certain part of number theory and weekly discussions in fixed time period are hosted by each member one by one.

For inlcusion in membership associations, we keep number of involvers between 5-10. There is group leader who responsible for making learning material before weekly discussion.

3. Stake to Learn(S2L)

Apart from self-discipline driven automously and personally, S2L is introduced as external and financial motivation.

Rules of S2L:

Gnosis Safe Multi-sig wallet is created for staked ethers which is from and managered by all members' etherem addresses. Amount staked is 0.2 ether per member. (least confirmation percentage is 60%)

Every absence of discussion hosted by others results in 0.2 * 0.75 * 1 / ( 10 - 10 / n ) ethers penalty charged; Every absence of discussion which should be hosted by defaulter results in 0.2 * 0.25 * ( n / 10 ) ethers penalty charged; charged ethers are distrubuted to others who abide by the rules. (n in formula is member number)

Standard for attendance in discussion:
- Enter meeting rooms within 15 minutes after it begins and not leave within 30 minutes after it begins.
- As host, the discussion length is hosted over 20 minutes.
Remission of penalty：
- If the host initiates a rescheduled vote on dc channel over 24 hours before discussion, then 50% penalty for that time can be reduced. If any host's total reschedulement over 2 times(including 2 times), then all remissions are cancelled.
- If the one not the host asks for absence over 24 hours before discussion, then 30% penalty for that time can be reduced. If any member's total reschedulement over 3 times(including 3 times, not included when he/she is host), then all remissions are cancelled.
Group Leader are not incentivised if penalty ethers > staked ethers * 20%, otherwise can get 1/2 * penalty ethers * 100 * (penalty ethers / staked ethers - 20%)^2.

4. Learning Path

W1～W3 Plan：

Reference: https://crypto.stanford.edu/pbc/notes/numbertheory/

Week1: Modular Arithmetic, Euclid’s Algorithm, Division, The Chinese Remainder Theorem
Week2: Units, Exponentiation, Order of a Unit, The Miller-Rabin Test
Week3: Generators, Cyclic Groups, Quadratic Residues

W4～W7 Plan:

Reference: https://crypto.stanford.edu/pbc/notes/elliptic/

Week4: Elliptic Curves, Weierstrass Form, Group of Points, Explicit Formulas
Week5: Rational Functions, Zeroes & Poles, Rational Maps, Torsion Points
Week6: Weil Pairing, Weil Pairing II, Counting Points, Hyperelliptic Curves
Week7: Tate Pairing, MOV Attack, Trace 0 Points

W8～W10 Plan:

Reference: https://crypto.stanford.edu/pbc/notes/ep/

Week8: Programming ECC， Tonelli’s Algorithm， Elliptic Curves，Complex
Week9: Multiplication, MNT Curves, Hilbert Class, Pell Equation
Week10: Roots of Polynomials, Bilinear Pairing, Tate Pairing, B-K Theorem

AmagiDDmxh / Season2