From df47c73acdfa8a709708321a7db37fd8bb01c3b3 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 25 Aug 2019 14:29:59 +0200 Subject: [PATCH] first commit, add slides: zksnarks, shamirsecretsharing, rsa --- README.md | 7 + rsa-and-homomorphicmultiplication.pdf | Bin 0 -> 532961 bytes shamirsecretsharing.pdf | Bin 0 -> 544336 bytes .../rsa-and-homomorphicmultiplication.md | 358 +++++++ .../shamirsecretsharing.md | 176 ++++ .../demo/go.mod | 8 + .../demo/go.sum | 15 + .../demo/main.go | 109 +++ .../imgs/arnaucube.png | Bin 0 -> 194441 bytes .../imgs/cat00.jpeg | Bin 0 -> 73924 bytes .../imgs/cat01.jpeg | Bin 0 -> 62920 bytes .../imgs/cat02.jpeg | Bin 0 -> 74843 bytes .../imgs/cat03.jpeg | Bin 0 -> 23262 bytes .../imgs/cat04.jpeg | Bin 0 -> 48350 bytes .../imgs/cat05.jpeg | Bin 0 -> 9246 bytes .../imgs/iden3.png | Bin 0 -> 4663 bytes .../imgs/qap-screenshot.png | Bin 0 -> 275617 bytes .../imgs/zksnark-concept-flow.png | Bin 0 -> 16120 bytes .../imgs/zksnark-concept-flow.xml | 1 + .../imgs/zksnark-prover.png | Bin 0 -> 13234 bytes .../imgs/zksnark-prover.xml | 1 + .../imgs/zksnark-verifier.png | Bin 0 -> 9745 bytes .../imgs/zksnark-verifier.xml | 1 + ...ks-from-scratch-a-technical-explanation.md | 877 ++++++++++++++++++ ...s-from-scratch-a-technical-explanation.pdf | Bin 0 -> 1194298 bytes 25 files changed, 1553 insertions(+) create mode 100644 README.md create mode 100644 rsa-and-homomorphicmultiplication.pdf create mode 100644 shamirsecretsharing.pdf create mode 100644 src/rsa-and-homomorphicmultiplication/rsa-and-homomorphicmultiplication.md create mode 100644 src/shamirsecretsharing/shamirsecretsharing.md create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/demo/go.mod create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/demo/go.sum create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/demo/main.go create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/arnaucube.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat00.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat01.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat02.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat03.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat04.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/cat05.jpeg create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/iden3.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/qap-screenshot.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-concept-flow.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-concept-flow.xml create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-prover.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-prover.xml create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-verifier.png create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/imgs/zksnark-verifier.xml create mode 100644 src/zksnarks-from-scratch-a-technical-explanation/zksnarks-from-scratch-a-technical-explanation.md create mode 100644 zksnarks-from-scratch-a-technical-explanation.pdf diff --git a/README.md b/README.md new file mode 100644 index 0000000..7f27970 --- /dev/null +++ b/README.md @@ -0,0 +1,7 @@ +# slides + +- [RSA and Homomorphic Multiplication](https://github.com/arnaucube/slides/blob/master/rsa-and-homomorphicmultiplication.pdf) +- [Shamir's Secret Sharing](https://github.com/arnaucube/slides/blob/master/rsa-and-homomorphicmultiplication.pdf) +- [zkSNARKs from scratch, a technical explanation](https://github.com/arnaucube/slides/blob/master/zksnarks-from-scratch-a-technical-explanation.pdf) + +Any error, 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z$mgeO2-1q-YG`)|mo+l|x}TTQG->>SIq=dW2kNNi#p|~$MEgtqjQVrG;$P}gh@Z+mY(TQ7e^F`vH=0pDi~axOv_EGFoC&s2owtjz}y22kf-JBj59X&rX9wgpW9BFt8w?r~^m9kKTwPfH(hu zpa$^I%v@~TEG-=E?HwJkKRId`8w*PZtR+^++(W>@LI7(f1h9OBezMa3LVG(R*ZmVN z9atE0WG4i80eEy^(IaaC!q7jW)0sItn7LZGnp+A2_WoB{1O^loMGyde2?iu6L;eHb zS=qQ)yPE$rCU@!=Tv=0$Lu89AxP&HfFVg^2?Ij` zCM#)sP$-yW9>9PVeWU>T%Qzsw+9$OI0!EVViNGocK+?Lv7Dnn9K$n0tMnGNB-~0y= z0U|=$9`VqQ|Bqz)j*dnlZAV8JwxbQ3O&efJI{>L*0YiWO>xgwRb9VW8hysm3VF)r# JPB{(v{{h_pd+GoH literal 0 HcmV?d00001 diff --git a/src/rsa-and-homomorphicmultiplication/rsa-and-homomorphicmultiplication.md b/src/rsa-and-homomorphicmultiplication/rsa-and-homomorphicmultiplication.md new file mode 100644 index 0000000..281bc0b --- /dev/null +++ b/src/rsa-and-homomorphicmultiplication/rsa-and-homomorphicmultiplication.md @@ -0,0 +1,358 @@ +# RSA and Homomorphic Multiplication + + + + +- https://arnaucube.com +- https://github.com/arnaucube +- https://twitter.com/arnaucube +


+

+ +
+ 2018-11-30 +
+ +--- + +- Intro +- Public key cryptography +- Basics of modular arithmetic +- Brief history of RSA +- Keys generation +- Prime numbers +- Encryption +- Decryption +- What's going on in encryption and decryption? +- Signature +- Verification of the signature +- Homomorphic Multiplication with RSA +- Resources + +--- +# Intro +- I'm not an expert on the field, neither a mathematician. Just an engineer with interest for cryptography +- Short talk (15 min), with the objective to make a practical introduction to the RSA cryptosystem +- Is not a talk about mathematical demostrations, is a talk with the objective to get the basic notions to be able to do a practical implementation of the algorithm +- After the talk, we will do a practical workshop to implement the concepts. We can offer support for Go, Rust, Python and Nodejs (you can choose any other language, but we will not be able to help) + +--- +# Public key cryptography +![pubkencr](https://upload.wikimedia.org/wikipedia/commons/f/f9/Public_key_encryption.svg "pubkencr") + +--- +Some examples: +- RSA +- Paillier +- ECC (Corba el·líptica) + +--- +# Basics of modular arithmetic +- Modulus, `mod`, `%` +- Remainder after division of two numbers + +![clocks](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/220px-Clock_group.svg.png "clocks") + +``` +5 mod 12 = 5 +14 mod 12 = 2 +83 mod 10 = 3 +``` + +``` +5 + 3 mod 6 = 8 mod 6 = 2 +``` + +--- + +# Brief history of RSA +- RSA (Rivest–Shamir–Adleman): Ron Rivest, Adi Shamir, Leonard Adleman +- year 1977 +- one of the first public key cryptosystems +- based on the difficulty of factorization of the product of two big prime numbers + +--- +# Prime numbers +- We need an asymmetric key, in a way where we can decrypt a message encrypted with the asymetric key +- Without allowing to find the private key from the public key +- in RSA we resolve this with factorization of prime numbers +- using prime numbers for $p$ and $q$, it's difficult factorize $n$ to obtain $p$ and $q$, where $n=p*q$ + +--- + +Example: +If we know $n$ which we need to find the $p$ and $q$ values where $p*q=n$: +``` +n = 35 +``` +To obtain the possible factors, is needed to brute force trying different combinations, until we find: +``` +p = 5 +q = 7 +``` +In this case is easy as it's a simple example with small numbers. The idea is to do this with big prime numbers + +--- + +Another exmample with more bigger prime numbers: +``` +n = 272604817800326282194810623604278579733 +``` +From $n$, I don't have a 'direct' way to obtain $p$ and $q$. I need to try by brute force the different values until finding a correct combination. +``` +p = 17975460804519255043 +q = 15165386899666573831 +n = 17975460804519255043 * 15165386899666573831 = 272604817800326282194810623604278579733 +``` + +--- + +If we do this with non prime numbers: +``` +n = 32 +We can factorize 32 = 2 * 2 * 2 * 2 * 2 +combining that values in two values X * Y +for example (2*2*2) * (2*2) = 8*4 = 32 +we can also take 2 * (2*2*2*2) = 2 * 16 = 32 +... +``` + +--- + +One example with bigger non prime numbers: +``` +n = 272604817800326282227951471308464408608 +We can take: +p = 17975460804519255044 +q = 15165386899666573832 + +Or also: +p = 2 +q = 136302408900163141113975735654232204304 +... +``` + +In the real world: +- https://en.wikipedia.org/wiki/RSA_numbers +- https://en.wikipedia.org/wiki/RSA_Factoring_Challenge#The_prizes_and_records + +So, we are basing this in the fact that is not easy to factorize big numbers composed by big primes. + + +--- +# Keys generation +- PubK: $e$, $n$ +- PrivK: $d$, $n$ +- are choosen randomly 2 big prime numbers $p$ and $q$, that will be secrets +- $n = p * q$ +- $λ$ is the Carmichael function + - $λ(n) = (p − 1) * (q − 1)$ +- Choose a prime number $e$ that satisfies $1 < e < λ(n)$ and $gcd(e, λ(n))=1$ + - Usually in examples is used $e = 2^16 + 1 = 65537$ +- $d$ such as $e * d = 1 mod λ(n)$ + - $d = e^(-1) mod λ(n) = e modinv λ(n)$ + +--- + +### Example +- `p = 3` +- `q = 11` +- `e = 7` value choosen between 1 and λ(n)=20, where λ(n) is not divisible by this value +- `n = 3 * 11 = 33` +- `λ(n) = (3-1) * (11-1) = 2 * 10 = 20` +- `d` such as `7 * d = 1 mod 20` +- `d = 3` +- PubK: `e=7, n=33` +- PrivK: `d=3, n=33` + +--- + +### Naive code +```python +def egcd(a, b): + if a == 0: + return (b, 0, 1) + g, y, x = egcd(b%a,a) + return (g, x - (b//a) * y, y) + +def modinv(a, m): + g, x, y = egcd(a, m) + if g != 1: + raise Exception('No modular inverse') + return x%m +``` + +--- + +``` +def newKeys(): + p = number.getPrime(n_length) + q = number.getPrime(n_length) + + # pubK e, n + e = 65537 + n = p*q + pubK = PubK(e, n) + + # privK d, n + phi = (p-1) * (q-1) + d = modinv(e, phi) + privK = PrivK(d, n) + + return({'pubK': pubK, 'privK': privK}) +``` + +--- + +# Encryption +- Brenna wants to send the message `m` to Alice, so, will use the Public Key from Alice to encrypt `m` +- `m` powered at `e` of the public key from Alice +- evaluate at modulus of `n` + +### Example +- message to encrypt `m = 5` +- receiver public key: `e=7, n=33` +- `c = 5 ^ 7 mod 33 = 78125 mod 33 = 14` + +### Naive code +```python +def encrypt(pubK, m): + c = (m ** pubK.e) % pubK.n + return c +``` + +--- +# Decrypt +- from an encrypted value `c` +- `c` powered at `d` of the private key of the person to who the message was encrypted +- evaluate at modulus of `n` + +### Example +- receiver private key, PrivK: `d=3, n=33` +- `m = 14 ^ 3 mod 33 = 2744 mod 33 = 5` + +### Naive code +```python +def decrypt(privK, c): + m_d = (c ** privK.d) % privK.n + return m_d +``` + +--- +# What's going on when encrypting and decrypting? + +![encrdecr](https://cdn-images-1.medium.com/max/1600/1*4AXQNOrddQJud0fZJ3FNgg.png "encrdecr") + +--- + +``` +n = pq +e +phi = (p-1)(q-1) +d = e^-1 mod (phi) = e^-1 mod (p-1)(q-1) + +# encrypt +c = m^e mod n = m^e mod pq + +# decrypt +m' = c^d mod n = c ^(e^-1 mod (p-1)(q-1)) mod pq = + = (m^e)^(e^-1 mod (p-1)(q-1)) mod pq = + = m^(e * e^-1 mod (p-1)(q-1)) mod pq = + = m^(1 mod (p-1)(q-1)) mod pq = + [theorem in which we're not going into details] + a ^ (1 mod λ(N)) mod N = a mod N + [/theorem] + = m mod pq +``` + + +--- +# Signature +- encryption operation but using PrivK instead of PubK, and PubK instead of PrivK + +- having a message `m` +- power of `m` at `d` of the private key from the signer person +- evaluated at modulus `n` + +--- + +### Example +- private key of the person emitter of the signature: `d = 3, n = 33` +- message to be signed: `m=5` +- signature: `s = 5 ** 3 % 33 = 26` + +### Naive code +```python +def sign(privK, m): + s = (m ** privK.d) % privK.n + return s +``` + +--- + +# Verification of the signature +- having message `m` and the signature `s` +- elevate `m` at `e` of the public key from the signer +- evaluate at modulus of `n` + +--- + +### Example +- public key from the singer person `e=7, n=33` +- message `m=5` +- signature `s=26` +- verification `v = 26**7 % 33 = 5` +- check that we have recovered the message (that `m` is equivalent to `v`) `m = 5 = v = 5` + +### Naive code +```python +def verifySign(pubK, s, m): + v = (s ** pubK.e) % privK.n + return v==m +``` + +--- + +# Homomorphic Multiplication +- from two values $a$ and $b$ +- encrypted are $a_{encr}$ and $b_{encr}$ +- we can compute the multiplication of the two encrypted values, obtaining the result encrypted +- the encrypted result from the multiplication is calculated doing: $c_{encr} = a_{encr} * b_{encr} mod n$ +- we can decrypt $c_{encr}$ and we will obtain $c$, equivalent to $a * b$ + +- Why: +``` +((a^e mod n) * (b^e mod n)) mod n = += (a^e * b^e mod n) mod n = (a*b)^e mod n +``` + +--- + +### Example +- PubK: `e=7, n=33` +- PrivK: `d=3, n=33` +- `a = 5` +- `b = 8` +- `a_encr = 5^7 mod 33 = 78125 mod 33 = 14` +- `b_encr = 8^7 mod 33 = 2097152 mod 33 = 2` +- `c_encr = (14 * 2) mod 33 = 28 mod 33 = 28` +- `c = 28 ^ 3 mod 33 = 21952 mod 33 = 7` +- `c = 7 = a * b % n = 5 * 8 % 33 = 7`, on `5*8 mod 33 = 7` +- take a `n` enough big, if not the result will be cropped by the modulus + +--- + +### Naive code +```python +def homomorphic_mul(pubK, a, b): + c = (a*b) % pubK.n + return c +``` + +--- +# Small demo +[...] + +# And now... practical implementation +- full night long +- big ints are your friends diff --git a/src/shamirsecretsharing/shamirsecretsharing.md b/src/shamirsecretsharing/shamirsecretsharing.md new file mode 100644 index 0000000..0e96d3e --- /dev/null +++ b/src/shamirsecretsharing/shamirsecretsharing.md @@ -0,0 +1,176 @@ +# Shamir's Secret Sharing + + + +- https://arnaucube.com +- https://github.com/arnaucube +- https://twitter.com/arnaucube +


+
+ +
+ 2019-07-05 +
+ +--- + +# Intro +- I'm not an expert on the field, neither a mathematician. Just an engineer with interest for cryptography +- Short talk (15 min), with the objective to make a practical introduction to the Shamir's Secret Sharing algorithm +- Is not a talk about mathematical demostrations, is a talk with the objective to get the basic notions to be able to do a practical implementation of the algorithm +- After the talk, we will do a practical workshop to implement the concepts. We can offer support for Go, Rust, Python and Nodejs (you can choose any other language, but we will not be able to help) + +--- + +- Cryptographic algorithm +- Created by Adi Shamir, in 1979 + - also known by the $RSA$ cryptosystem + - explained in few months ago in a similar talk: https://github.com/arnaucube/slides/rsa + +--- +## What's this about? +- imagine having a password that you want to share with 5 persons, in a way that they need to join their parts to get the original password +- take the password, split it in 5 parts, and give one part to each one +- when they need to recover it, they just need to get together, put all the pieces and recover the password (the `secret`) +- this, has the problem that if a person looses its part, the secret will not be recovered anymore.. luckly we have a solution here: + +--- + +- Shamir's Secret Sharing: + - from a secret to be shared, we generate 5 parts, but we can specify a number of parts that are needed to recover the secret + - so for example, we generate 5 parts, where we will need only 3 of that 5 parts to recover the secret, and the order doesn't matter + - we have the ability to define the thresholds of $M$ parts to be created, and $N$ parts to be able the recover + +--- + +- 2 points are sufficient to define a line +- 3 points are sufficient to define a parabola +- 4 points are sufficient to define a cubic curve +- $K$ points are suficient to define a polynomial of degree $k-1$ + +We can create infinity of polynomials of degree 2, that goes through 2 points, but with 3 points, we can define a polynomial of degree 2 unique. + +![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/3_polynomials_of_degree_2_through_2_points.svg/220px-3_polynomials_of_degree_2_through_2_points.svg.png) + +--- + +## Naming + +- `s`: secret +- `m`: number of parts to be created +- `n`: number of minimum parts necessary to recover the secret +- `p`: random prime number, the Finite Field will be over that value + +--- + +## Secret generation +- we want that are necessary $n$ parts of $m$ to recover $s$ + - where $n
+ +![](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2c3a2ab16a8723c0446de6a30da839198fb04b) + +--- + +## Wikipedia example +*example over real numbers, in the practical world, we use the algorithm in the Finite Field over $p$ +(more details: https://en.wikipedia.org/wiki/Shamir's_Secret_Sharing#Problem)
+- $s=1234$ +- $m=6$ +- $n=3$ +- $f(x) = \alpha_0 + \alpha_1 x + \alpha_2 x^2$ + - $\alpha_0 = s = 1234$ + - $\alpha_1 = 166$ *(random)* + - $\alpha_2 = 94$ *(random)* +- $f(x) = 1234 + 166 x + 94 x^2$ + +--- + +- $f(x) = 1234 + 166 x + 94 x^2$ +- we calculate the points $P = (x, f(x))$ + - where $x$ is each one of the values between $1$ and $m$ + - $P_1=(1, f(1)) = (1, 1494)$ + - $P_2=(2, f(2)) = (2, 1942)$ + - $P_3=(3, f(3)) = (3, 2578)$ + - $P_4=(4, f(4)) = (4, 3402)$ + - $P_5=(5, f(5)) = (5, 4414)$ + - $P_6=(6, f(6)) = (6, 5614)$ +--- + +- to recover the secret, let's imagine that we take the packets 2, 4, 5 + - $(x_0, y_0) = (2, 1942)$ + - $(x_0, y_0) = (4, 3402)$ + - $(x_0, y_0) = (5, 4414)$ + +--- + +- let's calculate the Lagrange Interpolation + - ![](https://wikimedia.org/api/rest_v1/media/math/render/svg/388471f79b8d3bdb75851b99ed15e5849329cc84) + - ![](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c853bdf0daa2db92cd70a6ab21dfd858296cfdd) + - ![](https://wikimedia.org/api/rest_v1/media/math/render/svg/2013ee56aba68b07d8d4a2c6578e77ff8e8940ff) + - ![](https://wikimedia.org/api/rest_v1/media/math/render/svg/32fc145272d82d9ebf62b4e30a05eac2b7d2873a) +- obtaining $f(x) = \alpha_0 + \alpha_1 x + \alpha_2 x^2$, where $\alpha_0$ is the secret $s$ recovered + - where we eavluate the polynomial at $f(0)$, obtaining $\alpha_0 = s$ +- *we are not going into details now, but if you want in the practical workshop we can analyze the 'mathematical' part of all of this + +--- + +# And now... practical implementation +- full night long +- big ints are your friends +- $L(x) = \sum_{j=0}^{n} y_j l_j(x)$ +![](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2c3a2ab16a8723c0446de6a30da839198fb04b) + +# About + + + +- https://arnaucube.com +- https://github.com/arnaucube +- https://twitter.com/arnaucube +
+
+ +
+ 2019-07-05 +
diff --git a/src/zksnarks-from-scratch-a-technical-explanation/demo/go.mod b/src/zksnarks-from-scratch-a-technical-explanation/demo/go.mod new file mode 100644 index 0000000..f809562 --- /dev/null +++ b/src/zksnarks-from-scratch-a-technical-explanation/demo/go.mod @@ -0,0 +1,8 @@ +module demo + +go 1.12 + +require ( + github.com/arnaucube/go-snark v0.0.4 // indirect + github.com/stretchr/testify v1.4.0 // indirect +) diff --git a/src/zksnarks-from-scratch-a-technical-explanation/demo/go.sum b/src/zksnarks-from-scratch-a-technical-explanation/demo/go.sum new file mode 100644 index 0000000..4b78653 --- /dev/null +++ b/src/zksnarks-from-scratch-a-technical-explanation/demo/go.sum @@ -0,0 +1,15 @@ +github.com/arnaucube/go-snark v0.0.4 h1:JJbQx/wg0u1mzJk9Of/rqCkclPgXuvPrLWHfvgnoyEE= +github.com/arnaucube/go-snark v0.0.4/go.mod h1:m1VkAgz3F+Jdighf2n5eMLe670AR6fBhBGfVHwz2QRk= +github.com/davecgh/go-spew v1.1.0/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38= +github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c= +github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38= +github.com/pmezard/go-difflib v1.0.0 h1:4DBwDE0NGyQoBHbLQYPwSUPoCMWR5BEzIk/f1lZbAQM= +github.com/pmezard/go-difflib v1.0.0/go.mod h1:iKH77koFhYxTK1pcRnkKkqfTogsbg7gZNVY4sRDYZ/4= +github.com/stretchr/objx v0.1.0/go.mod h1:HFkY916IF+rwdDfMAkV7OtwuqBVzrE8GR6GFx+wExME= +github.com/stretchr/testify v1.2.2/go.mod h1:a8OnRcib4nhh0OaRAV+Yts87kKdq0PP7pXfy6kDkUVs= +github.com/stretchr/testify v1.4.0 h1:2E4SXV/wtOkTonXsotYi4li6zVWxYlZuYNCXe9XRJyk= +github.com/stretchr/testify v1.4.0/go.mod h1:j7eGeouHqKxXV5pUuKE4zz7dFj8WfuZ+81PSLYec5m4= +github.com/urfave/cli v1.20.0/go.mod h1:70zkFmudgCuE/ngEzBv17Jvp/497gISqfk5gWijbERA= +gopkg.in/check.v1 v0.0.0-20161208181325-20d25e280405/go.mod h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0= +gopkg.in/yaml.v2 v2.2.2 h1:ZCJp+EgiOT7lHqUV2J862kp8Qj64Jo6az82+3Td9dZw= +gopkg.in/yaml.v2 v2.2.2/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= diff --git a/src/zksnarks-from-scratch-a-technical-explanation/demo/main.go b/src/zksnarks-from-scratch-a-technical-explanation/demo/main.go new file mode 100644 index 0000000..76ad159 --- /dev/null +++ b/src/zksnarks-from-scratch-a-technical-explanation/demo/main.go @@ -0,0 +1,109 @@ +package main + +import ( + "encoding/json" + "fmt" + "math/big" + "strings" + "time" + + "github.com/arnaucube/go-snark" + "github.com/arnaucube/go-snark/circuitcompiler" +) + +func main() { + // circuit function + // y = x^5 + 2*x + 6 + code := ` + func exp5(private a): + b = a * a + c = a * b + d = a * c + e = a * d + return e + + func main(private s0, public s1): + s2 = exp5(s0) + s3 = s0 * 2 + s4 = s3 + s2 + s5 = s4 + 6 + equals(s1, s5) + out = 1 * 1 + ` + fmt.Print("\ncode of the circuit:") + fmt.Println(code) + + // parse the code + parser := circuitcompiler.NewParser(strings.NewReader(code)) + circuit, err := parser.Parse() + if err != nil { + panic(err) + } + fmt.Println("\ncircuit data:", circuit) + circuitJson, _ := json.Marshal(circuit) + fmt.Println("circuit:", string(circuitJson)) + + b8 := big.NewInt(int64(8)) + privateInputs := []*big.Int{b8} + b32790 := big.NewInt(int64(32790)) + publicSignals := []*big.Int{b32790} + + // wittness + w, err := circuit.CalculateWitness(privateInputs, publicSignals) + if err != nil { + panic(err) + } + + // code to R1CS + fmt.Println("\ngenerating R1CS from code") + a, b, c := circuit.GenerateR1CS() + fmt.Println("\nR1CS:") + fmt.Println("a:", a) + fmt.Println("b:", b) + fmt.Println("c:", c) + + // R1CS to QAP + // TODO zxQAP is not used and is an old impl, TODO remove + alphas, betas, gammas, _ := snark.Utils.PF.R1CSToQAP(a, b, c) + fmt.Println("qap") + fmt.Println(alphas) + fmt.Println(betas) + fmt.Println(gammas) + + _, _, _, px := snark.Utils.PF.CombinePolynomials(w, alphas, betas, gammas) + + // calculate trusted setup + setup, err := snark.GenerateTrustedSetup(len(w), *circuit, alphas, betas, gammas) + if err != nil { + panic(err) + } + fmt.Println("\nt:", setup.Toxic.T) + + // zx and setup.Pk.Z should be the same (currently not, the correct one is the calculation used inside GenerateTrustedSetup function), the calculation is repeated. TODO avoid repeating calculation + + proof, err := snark.GenerateProofs(*circuit, setup.Pk, w, px) + if err != nil { + panic(err) + } + + fmt.Println("\n proofs:") + fmt.Println(proof) + + // fmt.Println("public signals:", proof.PublicSignals) + fmt.Println("\nsignals:", circuit.Signals) + fmt.Println("witness:", w) + b32790Verif := big.NewInt(int64(32790)) + publicSignalsVerif := []*big.Int{b32790Verif} + before := time.Now() + if !snark.VerifyProof(setup.Vk, proof, publicSignalsVerif, true) { + fmt.Println("Verification not passed") + } + fmt.Println("verify proof time elapsed:", time.Since(before)) + + // check that with another public input the verification returns false + bOtherWrongPublic := big.NewInt(int64(34)) + wrongPublicSignalsVerif := []*big.Int{bOtherWrongPublic} + if snark.VerifyProof(setup.Vk, proof, wrongPublicSignalsVerif, true) { + fmt.Println("Verification should not have passed") + } +} diff --git a/src/zksnarks-from-scratch-a-technical-explanation/imgs/arnaucube.png b/src/zksnarks-from-scratch-a-technical-explanation/imgs/arnaucube.png new file mode 100644 index 0000000000000000000000000000000000000000..ebdcb2293470bce8d8dcdc380cd5114830b55f07 GIT binary patch literal 194441 zcmd3Ng;!Kx_x_z3y1N@B1?i3f2I)@ek`(EX96+Q?K#}fFC8R+?K%~2oZjf$%TfK3A`x!fJ@g-lDQbHi z0}okeC#RRMJrG|3K-TT0g~v;4I&XUqJ34tqRSj#dm#6?h2Pnd%b$sUb+x!-6Ug*+X zRcKFoj81w8VULwSe8`cgVXaWB)?YD=Jg0n{wwqhTge-&t4pb9l&s;UExOwsz$$S40 z@JaC@?QmtN=tlR>P{eLp5QWd?V4Gj(F>aCcD!9HSeRGbQXTR>bFw)8PS z`NM|~UY_$i`yq9R!GVGTgZaE2Hp5-1Bn8o+t%w4RGgViJWsVzgz`5*2BMEMPjo@40PyU zk5plFN}qfO?#SSNsAes|$}@)iQT?A>z?BfkeGnhKgNwmg8TuY!)(3!zlauoRPr7dl zbR`2^T9J-knFHAw67G2ilfVO`7k&~Yut3jH{p+>jUweSRH+iuIckiML^&gpLXqGKY z&@p6VuaG~iHBDtLl;XuKI={bT!xve*7`v5QPmdET{%a5cyr8hK&}0o7x_%3}65Y=u zzq$Mp;opm&IaOCqny}sU=YWB?$!r-|DP%YaM_2j3dxF=4F!hntPNebQaKWDq(kQhc zU8pOzo5HH}J2>QyA4OYO5Gw%vf?4X8!-sndE^ z6(g*F0{nG>pNDiGVXbm;bIqNqA9q5IB-}TDO#@f&w%3}L3N!22BvZdEpLRlU@1EHXWgrYZMOtZ%Xq9Mr z`oAA2M5dzEcnEm7yY_Q#{qsY}ob*7UCef=h^nGKNRM5lPYs6&w-X_-x>a5BCA|MV$ z7^zeB=*_U7&j4p*2FkJREQh^)`T_diDXs;#0Z#Brdjex{o9XyW^T*xy+4 zlH+8gLs+2yOW}bV{tWflbpK?Y7P3-1%V%%@aqxkvQ+D9tQ5blH!P*&Gp8RKlbCX!2 z4-#<1ALDfrf-?F|6_r~oF95g{vWo;ti3(U(BU^O-V)ESf_k@CT7^@$W(6 z(;w>U%H7$}mJXJguC*E52;qLP^;^P-|J|ZPx9(V*-*^9fUyFypB|}5(@i^e>z*R-OP4#;^xhFLgvh0Vg0fbqyR}{Vk4qE1b}S>RsE~-O*Zo&{-O_ zfy<#ed2;5XdcL3|v5+iw=$kR7Uc}Qw&*BP|28y=E8TtKJ-UK~#LdB-UsVey#g=5B+CCK0$D?PUGwOyl z+9MS)Wz}u=Dk^m#4I9lh8=nzMt`4|;mT111-h2~iomSKf_`LaS|KG=UL^Cju0p7tq ze0o35f4>C;vij|e%}G5T(1wo`@=k^s&>1j5K11lKSx2%b>EEsac!u(UAZ)SuYvriPA+UK?zMD zs=l9_YlmNW?j0^q-`hN&Qk>7Dy%F*B{dDebL_xOajuyOE-GV;+<_GDfbvsNsjMc>leW$^2*S9|@nZGIYT!XBBrf_3t=~Kc3Co9{ktV)c z)1pt+0zW$FP&?HYpZs)v*Yd*);4Q~g`T4u(`i}pM)C&0aX6sD-%-)0(L)yO6(qU1v zi{v@*cF%lZItoq1uA@btGb39TVA%bn*{K_iAF?YJAMi`n=Yr2PA?a}}I5Eg`Ip8wj z_n^@79n?$pKVB=x^bz0r7KXWnF!}xn5SNnf9o^4vKiH^DXjZlzD^=Qe6pTvZ+zt+V zT2Dv-#}2|axuq}M76X7)20E-f+_9*kLPw`)f{0An^FTsO7kmvbXiiQJMb?OVD&aHg!qROF~0vV`G-OO7jdZ112;2| z{I|@S4jSfPBOlj9+_@qwottzl{(^}(dmTnx2?75t@4ziT&_ck?*-r5WeRuQywsTuv zIZw{c(_6x~*wCj#>^veYY20i|%~xqotOm1r7fp`FuA^JjK<-n4Cm27T(&gUF%S^t0 z;d+%qt5#a7a`u}n>03w3CVz(ALDuT%w2x%@Vfx3{3DPW0TXZ~_&3(!DdXfG!xDI%X zubd+DL^|$%&w4xp8cov(wA_A9 zM;JatDuuuT?4Mf>oA=I6vo2f5XV{Xr{3L6Z-y9vSE#Hqik`(J1{_(~`A1PjB(3A+3 zWXq-~taNiO-UuqZndg0r&m5(d{zdpP)k3Sx+>uAA5?w%|s(chMO!W1m(q*dZLL$a9 zn$rjlQ;79SEm7Qg_Ic3^zFjIY@**0ZQ24l6SR%Fi{h~3d0j-8P_ci(>0-N-APJcYw3$<3c?~9N$LYefYkQQP zmY4&62e@RwYN_+NK%p{e5=MdOZ|NSY!swqyMvTBWM3!jD$~kiMsUa#T2@g`ATkm1VB{+7RPd-9&%)aQ9# z0gnG1RTiN5c%wZ8Qjfju@K`20mXCHpM*;#RRF^#gTxyMR zWfATZv4r!_er?p5HM7TVBrisyN9C0B(UXys9Kms0@~$YF5|?&f1F*}`6lv$WRxL(4 zV&MZ2-G(1wD>PTlk3()XmQ%&Iu0X4BLU?2+v~TR%xa@^ZLjkvd-YPsb5QT8({l&iV z934hnnyUU}Uc}8y&}Z6fO|#5r&dx8L{2C%!*$^lHzvpZ|rXahmWdAVI@v|D`M&H%? 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b/src/zksnarks-from-scratch-a-technical-explanation/zksnarks-from-scratch-a-technical-explanation.md new file mode 100644 index 0000000..f830610 --- /dev/null +++ b/src/zksnarks-from-scratch-a-technical-explanation/zksnarks-from-scratch-a-technical-explanation.md @@ -0,0 +1,877 @@ +# zkSNARKs from scratch, a technical explanation + +


+

+ +
+[iden3.io](https://iden3.io) +[github.com/iden3](https://github.com/iden3) +[twitter.com/identhree](https://twitter.com/identhree) + + +--- + +## Warning + +
+ +- I'm not a mathematician, this talk is not for mathematicians + +- In free time, have been studying zkSNARKS & implementing it in Go + +- Talk about a technical explaination from an engineer point of view +- The idea is to try to transmit the learnings from long night study hours during last winter +- Also at the end will briefly overview how we use zkSNARKs in iden3 +- This slides will be combined with + - parts of the code from https://github.com/arnaucube/go-snark + - whiteboard draws and writtings +- Don't use your own crypto. But it's fun to implement it (only for learning purposes) + +
+ +--- + +## Contents + +
+ +- Introduction + - zkSNARK overview + - zkSNARK flow + - Generating and verifying proofs +- Foundations + - Basics of modular arithmetic + - Groups + - Finite fields + - Elliptic curve operations +- Pairings + - Bilinear Pairings + - BLS signatures + +
+ +--- + +
+ +- zkSNARK (Pinocchio) + - Circuit compiler + - R1CS + - QAP + - Lagrange Interpolation + - Trusted Setup + - Proofs generation + - Proofs verification +- Groth16 +- How we use zkSNARKs in iden3 +- libraries +- Circuit languages +- utilities (Elliptic curve & Hash functions) inside the zkSNARK libraries + - BabyJubJub + - Mimc + - Poseidon +- References + +
+ +--- + +## Introduction +- zero knowledge concept +- examples +- some concept explanations + - https://en.wikipedia.org/wiki/Zero-knowledge_proof + - https://hackernoon.com/wtf-is-zero-knowledge-proof-be5b49735f27 + +--- + + +## zkSNARK overview + + +- protocol to prove the correctness of a computation +- useful for + - scalability + - privacy + - interoperability +- examples: + - Alice can prove to Brenna that knows $x$ such as $f(x) = y$ + - Brenna can prove to Alice that knows a certain input which $Hash$ results in a certain known value + - Carol can proof that is a member of an organization without revealing their identity + - etc + +--- + +### zkSNARK flow + +
+ +
+ +--- + + +### Generating and verifying proofs +Generating a proof: + + + + + +

+ +Verifying a proof: + + + + + + +--- + + +## Foundations +- Modular aritmetic +- Groups +- Finite fields +- Elliptic Curve Cryptography + +--- + +## Basics of modular arithmetic +- Modulus, `mod`, `%` +- Remainder after division of two numbers + +![clocks](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/220px-Clock_group.svg.png "clocks") + +```python +5 mod 12 = 5 +14 mod 12 = 2 +83 mod 10 = 3 +``` + +```python +5 + 3 mod 6 = 8 mod 6 = 2 +``` + + +--- + +## Groups +- a **set** with an **operation** + - **operation** must be *associative* +- neutral element ($identity$): adding the neutral element to any element gives the element +- inverse: $e$ + $e_{inverse}$ = $identity$ +- cyclic groups + - finite group with a generator element + - any element must be writable by a multiple of the generator element +- abelian group + - group with *commutative* operation + +--- + +## Finite fields +- algebraic structure like Groups, but with **two operations** +- extended fields concept (https://en.wikipedia.org/wiki/Field_extension) + +--- + +## Elliptic curve +- point addition + +$(x_1, y_1) + (x_2, y_2) = +(\dfrac{ +x_1 y_2 + x_2 y_1 +}{ +1 + d x_1 x_2 y_1 y_2 +}, +\dfrac{ +y_1 y_2 - x_1 x_2 +}{ +1-dx_1 x_2 y_1 y_2 +})$ +- G1 +- G2 + +*(whiteboard explanation)* + +--- + +## Pairings +- 3 typical types used for SNARKS: + - BN (Barreto Naehrig) - used in Ethereum + - BLS (Barreto Lynn Scott) - used in ZCash & Ethereum 2.0 + - MNT (Miyaji- Nakabayashi - Takano) - used in CodaProtocol +- $y^2 = x^3 + b$ with embedding degree 12 +- function that maps (pairs) two points from sets `S1` and `S2` into another set `S3` +- is a [bilinear](https://en.wikipedia.org/wiki/Bilinear_map) function +- $e(G_1, G_2) -> G_T$ +- the groups must be + - cyclic + - same prime order ($r$) + +--- + +- $F_q$, where $q=$`21888242871839275222246405745257275088696311157297823662689037894645226208583` +- $F_r$, where $r=$`21888242871839275222246405745257275088548364400416034343698204186575808495617` + +--- + +## Bilinear Pairings +$e(P_1 + P_2, Q_1) == e(P_1, Q_1) \cdot e(P_2, Q_1)$ + +$e(P_1, Q_1 + Q_2) == e(P_1, Q_1) \cdot e(P_1, Q_2)$ + +$e(aP, bQ) == e(P, Q)^{ab} == e(bP, aQ)$ + + + +$e(g_1, g_2)^6 == e(g_1, 6 \cdot g_2)$ + +$e(g_1, g_2)^6 == e(6 \cdot g_1, g_2)$ + +$e(g_1, g_2)^6 == e(3 \cdot g_1, 2 g_2)$ + +$e(g_1, g_2)^6 == e(2 \cdot g_1, 3 g_2)$ + + +--- + +### BLS signatures +*(small overview, is offtopic here, but is interesting)* +- key generation + - random private key $x$ in $[0, r-1]$ + - public key $g^x$ +- signature + - $h=Hash(m)$ (over G2) + - signature $\sigma=h^x$ +- verification + - check that: $e(g, \sigma) == e(g^x, Hash(m))$ + $e(g, h^x) == e(g^x, h)$ + + +--- + +- aggregate signatures + - $s = s0 + s1 + s2 ...$ +- verify aggregated signatures + +
+ +$e(G,S) == e(P, H(m))$ + +$e(G, s0+s1+s2...) == e(p0, H(m)) \cdot e(p1, H(m)) \cdot e(p2, H(m)) ...$ + +
+ + +More info: https://crypto.stanford.edu/~dabo/pubs/papers/BLSmultisig.html + +--- + +## Circuit compiler +- not a software compiler -> a constraint prover + - what this means +- constraint concept + - `value0` == `value1` `` `value2` +- want to proof that a certain computation has been done correctly +- graphic of circuit with gates (whiteboard) + +- about high level programing languages for zkSNARKS, by *Harry Roberts*: https://www.youtube.com/watch?v=nKrBJo3E3FY + +--- + +Circuit code example: +$f(x) = x^5 + 2\cdot x + 6$ + +``` +func exp5(private a): + b = a * a + c = a * b + d = a * c + e = a * d + return e + +func main(private s0, public s1): + s2 = exp5(s0) + s3 = s0 * 2 + s4 = s3 + s2 + s5 = s4 + 6 + equals(s1, s5) + out = 1 * 1 +``` + +--- + +## Inputs and Witness +For a certain circuit, with the inputs that we calculate the Witness for the circuit signals +- private inputs: `[8]` + - in this case the private input is the 'secret' $x$ value that computed into the equation gives the expected $f(x)$ +- public inputs: `[32790]` + - in this case the public input is the result of the equation +- signals: `[one s1 s0 b0 c0 d0 s2 s3 s4 s5 out]` +- witness: `[1 32790 8 64 512 4096 32768 16 32784 32790 1]` + +--- + +## R1CS +- Rank 1 Constraint System +- way to write down the constraints by 3 linear combinations +- 1 constraint per operation +- $(A, B, C) = A.s \cdot B.s - C.s = 0$ +- from flat code constraints we can generate the R1CS + +--- + +## R1CS + +
+ +$(a_{11}s_1 + a_{12}s_2 + ... + a_{1n}s_n) \cdot (b_{11}s_1 + b_{12}s_2 + ... + b_{1n}s_n) - (c_{11}s_1 + c_{12}s_2 + ... + c_{1n}s_n) = 0$ + +$(a_{21}s_1 + a_{22}s_2 + ... + a_{2n}s_n) \cdot (b_{21}s_1 + b_{22}s_2 + ... + b_{2n}s_n) - (c_{21}s_1 + c_{22}s_2 + ... + c_{2n}s_n) = 0$ +$(a_{31}s_1 + a_{32}s_2 + ... + a_{3n}s_n) \cdot (b_{31}s_1 + b_{32}s_2 + ... + b_{3n}s_n) - (c_{31}s_1 + c_{32}s_2 + ... + c_{3n}s_n) = 0$ +[...] +$(a_{m1}s_1 + a_{m2}s_2 + ... + a_{mn}s_n) \cdot (b_{m1}s_1 + b_{m2}s_2 + ... + b_{mn}s_n) - (c_{m1}s_1 + c_{m2}s_2 + ... + c_{mn}s_n) = 0$ + +*where $s$ are the signals of the circuit, and we need to find $a, b, c$ that satisfies the equations +
+ +--- + +R1CS constraint example: +- signals: `[one s1 s0 b0 c0 d0 s2 s3 s4 s5 out]` +- witness: `[1 32790 8 64 512 4096 32768 16 32784 32790 1]` +- First constraint flat code: `b0 == s0 * s0` +- R1CS first constraint: +$A_1 = [00100000000]$ +$B_1 = [00100000000]$ +$C_1 = [00010000000]$ + +--- + +R1CS example: + +| $A$| $B$ | $C$: | +|-|-|-| +| $[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 0 0 0 0 1 1 0 0 0]$
$[6 0 0 0 0 0 0 0 1 0 0]$
$[0 0 0 0 0 0 0 0 0 1 0]$
$[0 1 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$ | $[0 0 1 0 0 0 0 0 0 0 0]$
$[0 0 0 1 0 0 0 0 0 0 0]$
$[0 0 0 0 1 0 0 0 0 0 0]$
$[0 0 0 0 0 1 0 0 0 0 0]$
$[2 0 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$
$[1 0 0 0 0 0 0 0 0 0 0]$ | $[0 0 0 1 0 0 0 0 0 0 0]$
$[0 0 0 0 1 0 0 0 0 0 0]$
$[0 0 0 0 0 1 0 0 0 0 0]$
$[0 0 0 0 0 0 1 0 0 0 0]$
$[0 0 0 0 0 0 0 1 0 0 0]$
$[0 0 0 0 0 0 0 0 1 0 0]$
$[0 0 0 0 0 0 0 0 0 1 0]$
$[0 1 0 0 0 0 0 0 0 0 0]$
$[0 0 0 0 0 0 0 0 0 1 0]$
$[0 0 0 0 0 0 0 0 0 0 1]$ | + + +--- + +## QAP +- Quadratic Arithmetic Programs +- 3 polynomials, linear combinations of R1CS +- very good article about QAP by Vitalik Buterin https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649 + +--- + +![qap](imgs/qap-screenshot.png) + +--- + +### Lagrange Interpolation +(Polynomial Interpolation) +- for a group of points, we can find the smallest degree polynomial that goees through all that points + - this polynomial is unique for each group of points + +![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Lagrange_polynomial.svg/440px-Lagrange_polynomial.svg.png) + +--- + +$L(x) = \sum_{j=0}^{n} y_j l_j(x)$ + +

+ +![](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2c3a2ab16a8723c0446de6a30da839198fb04b) + + +--- + +#### Shamir's Secret Sharing +*(small overview, is offtopic here, but is interesting)* + +- from a secret to be shared, we generate 5 parts, but we can specify a number of parts that are needed to recover the secret +- so for example, we generate 5 parts, where we will need only 3 of that 5 parts to recover the secret, and the order doesn't matter +- we have the ability to define the thresholds of $M$ parts to be created, and $N$ parts to be able the recover + +--- + +##### Shamir's Secret Sharing - Secret generation +- we want that are necessary $n$ parts of $m$ to recover $s$ + - where $n + +$(\alpha_1(x)s_1 + \alpha_2(x)s_2 + ... + \alpha_n(x)s_n) \cdot (\beta_1(x)s_1 + \beta_2(x)s_2 + ... + \beta_n(x)s_n) - (\gamma_1(x)s_1 + \gamma_2(x)s_2 + ... + \gamma_n(x)s_n) = P(x)$ + +|----------------------- $A(x)$ -----------------------|------------------------ $B(x)$ -----------------------|------------------------ $C(x)$ ------------------------| + + + +
+ +- $P(x) = A(x)B(x)-C(x)$ +- $P(x) = Z(x) h(x)$ +- $Z(x)$: divisor polynomial + - $Z(x) = (x - x_1)(x-x_2)...(x-x_m) => ...=> (x_1, 0), (x_2, 0), ..., (x_m, 0)$ + - optimizations with FFT +- $h(x) = P(x) / Z(x)$ + +
+ +--- + +*The following explanation is for the [Pinocchio protocol](https://eprint.iacr.org/2013/279.pdf), all the examples will be for this protocol. The [Groth16](https://eprint.iacr.org/2016/260.pdf) is explained also in the end of this slides.* + +--- + +## Trusted Setup +- concept +- $\tau$ (Tau) +- "Toxic waste" +- Proving Key +- Verification Key + +--- + +$g_1 t^0, g_1 t^1, g_1 t^2, g_1 t^3, g_1 t^4, ...$ +$g_2 t^0, g_2 t^1, g_2 t^2, g_2 t^3, g_2 t^4, ...$ + +--- + +Proving Key: +$pk = (C, pk_A, pk_A', pk_B, pk_B', pk_C, pk_C', pk_H)$ where: +- $pk_A = \{ A_i(\tau) \rho_A P_1 \}^{m+3}_{i=0}$ +- $pk_A' = \{ A_i(\tau) \alpha_A \rho_A P_1 \}^{m+3}_{i=n+1}$ +- $pk_B = \{ B_i(\tau) \rho_B P_2 \}^{m+3}_{i=0}$ +- $pk_B' = \{ B_i(\tau) \alpha_B \rho_B P_1 \}^{m+3}_{i=0}$ +- $pk_C = \{ C_i(\tau) \rho_C P_1 \}^{m+3}_{i=0} = \{C_i(\tau) \rho_A \rho_B P_1\}^{m+3}_{i=0}$ +- $pk_C' = \{ C_i(\tau) \alpha_C \rho_C P_1 \}^{m+3}_{i=0} = \{ C_i(\tau) \alpha_C \rho_A \rho_B P_1 \}^{m+3}_{i=0}$ +- $pk_K = \{ \beta (A_i(\tau) \rho_A + B_i(\tau) \rho_B C_i(\tau) \rho_A \rho_B) P_1 \} ^{m+3}_{i=0}$ +- $pk_H = \{ \tau^i P_1 \}^d_{i=0}$ + +where: +- $d$: degree of polynomial $Z(x)$ +- $m$: number of circuit signals + +--- + +Verification Key: +$vk = (vk_A, vk_B, vk_C, vk_\gamma, vk^1_{\beta\gamma}, vk^2_{\beta\gamma}, vk_Z, vk_{IC})$ +- $vk_A = \alpha_A P_2$, $vk_B = \alpha_B P_1$, $vk_C = \alpha_C P_2$ +- $vk_{\beta\gamma} = \gamma P_2$, $vk^1_{\beta\gamma} = \beta\gamma P_1$, $vk^2_{\beta\gamma} = \beta\gamma P_2$ +- $vk_Z = Z(\tau) \rho_A \rho_B P_2$, $vk_{IC} = (A_i(\tau) \rho_A P_1)^n_{i=0}$ + +--- + +```go +type Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH) + G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H + A [][3]*big.Int + B [][3][2]*big.Int + C [][3]*big.Int + Kp [][3]*big.Int + Ap [][3]*big.Int + Bp [][3]*big.Int + Cp [][3]*big.Int + Z []*big.Int +} + +type Vk struct { + Vka [3][2]*big.Int + Vkb [3]*big.Int + Vkc [3][2]*big.Int + IC [][3]*big.Int + G1Kbg [3]*big.Int // g1 * Kbeta * Kgamma + G2Kbg [3][2]*big.Int // g2 * Kbeta * Kgamma + G2Kg [3][2]*big.Int // g2 * Kgamma + Vkz [3][2]*big.Int +} +``` + +--- + +```go +// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed +type Setup struct { + Toxic struct { + T *big.Int // trusted setup secret + Ka *big.Int + Kb *big.Int + Kc *big.Int + Kbeta *big.Int + Kgamma *big.Int + RhoA *big.Int + RhoB *big.Int + RhoC *big.Int + } + Pk Pk + Vk Vk +} +``` + +--- + +## Proofs generation +- $A, B, C, Z$ (from the QAP) +- random $\delta_1, \delta_2, \delta_3$ +- $H(z)= \dfrac{A(z)B(z)-C(z)}{Z(z)}$ + - $A(z) = A_0(z) + \sum_{i=1}^m s_i A_i(x) + \delta_1 Z(z)$ + - $B(z) = B_0(z) + \sum_{i=1}^m s_i B_i(x) + \delta_2 Z(z)$ + - $C(z) = C_0(z) + \sum_{i=1}^m s_i B_i(x) + \delta_2 Z(z)$ +(where $m$ is the number of public inputs) + +--- + +- $\pi_A = $ +- $\pi_A' = $ +- $\pi_B = $ + - example: + ```go + for i := 0; i < circuit.NVars; i++ { + proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(pk.B[i], w[i])) + proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(pk.Bp[i], w[i])) + } + ``` +($c=1+witness+\delta_1+\delta_2+\delta_3$ +- $\pi_B' = $ +- $\pi_C = $ +- $\pi_C' = $ +- $\pi_K = $ +- $\pi_H = $ +- proof: $\pi = (\pi_A, \pi_A', \pi_B, \pi_B', \pi_C, \pi_C', \pi_K, \pi_H$ + +--- + +## Proofs verification + + + +- $vk_{kx} = vk_{IC,0} + \sum_{i=1}^n x_i vk_{IC,i}$ + +Verification: +- $e(\pi_A, vk_a) == e(\pi_{A'}, g_2)$ +- $e(vk_b, \pi_B) == e(\pi_{B'}, g_2)$ +- $e(\pi_C, vk_c) == e(\pi_{C'}, g_2)$ +- $e(vk_{kx}+\pi_A, \pi_B) == e(\pi_H, vk_{kz}) \cdot e(\pi_C, g_2)$ +- $e(vk_{kx} + \pi_A + \pi_C, V_{\beta\gamma}^2) \cdot e(vk_{\beta\gamma}^1, \pi_B) == e(\pi_k, vk_{\gamma}^1)$ + +--- + + +
+Example (whiteboard): +

+ +$\dfrac{ + e(\pi_A, \pi_B) +}{ + e(\pi_C, g_2) +} += e(g_1 h(t), g_2 z(t)) +$ +
+$\dfrac{ + e(A_1 + A_2 + ... + A_n, B_1 + B_2 + ... + B_n) +}{ + e(C_1 + C_2 + ... + C_n, g_2) +} += e(g_1 h(t), g_2 z(t)) +$ +
+$\dfrac{ + e(g_1 \alpha_1(t) s_1 + g_1 \alpha_2(t) s_2 + ... + g_1 \alpha_n(t) s_n, g_2 \beta_1(t)s_1 + g_2 \beta_2(t) s_2 + ... + g_2 \beta_n(t) s_n) +}{ + e(g_1 \gamma_1(t) s_1 + g_1 \gamma_2(t) s_2 + ... + g_1 \gamma_n(t) s_n, g_2) +} += e(g_1 h(t), g_2 z(t)) +$ +
+$ +e(g_1 \alpha_1(t) s_1 + g_1 \alpha_2(t) s_2 + ... + g_1 \alpha_n(t) s_n, g_2 \beta_1(t)s_1 + g_2 \beta_2(t) s_2 + ... + g_2 \beta_n(t) s_n)$ +$= e(g_1 h(t), g_2 z(t)) \cdot e(g_1 \gamma_1(t) s_1 + g_1 \gamma_2(t) s_2 + ... + g_1 \gamma_n(t) s_n, g_2) +$ + +
+ + +--- + +## Groth16 + + + + +### Trusted Setup +$\tau = \alpha, \beta, \gamma, \delta, x$ + +$\sigma_1 =$ +- $\alpha, \beta, \delta, \{ x^i\}_{i=0}^{n-1}$ + +- $\{ +\dfrac{ +\beta u_i(x) + \alpha v_i(x) + w_i(x) +}{ +\gamma +} +\}_{i=0}^l$ + +- $\{ +\dfrac{ +\beta u_i(x) + \alpha v_i(x) + w_i(x) +}{ +\delta +} +\}_{i=l+1}^m$ + +- $\{ +\dfrac{x^i t(x)}{\delta} +\}_{i=0}^{n-2}$ + +$\sigma_2 = (\beta, \gamma, \delta, \{ x^i \}_{i=0}^{n-1})$ + +*(where $u_i(x), v_i(x), w_i(x)$ are the $QAP$)* + +--- + +```go +type Pk struct { // Proving Key + BACDelta [][3]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / δ } from l+1 to m + Z []*big.Int + G1 struct { + Alpha [3]*big.Int + Beta [3]*big.Int + Delta [3]*big.Int + At [][3]*big.Int // {a(τ)} from 0 to m + BACGamma [][3]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m + } + G2 struct { + Beta [3][2]*big.Int + Gamma [3][2]*big.Int + Delta [3][2]*big.Int + BACGamma [][3][2]*big.Int // {( βui(x)+αvi(x)+wi(x) ) / γ } from 0 to m + } + PowersTauDelta [][3]*big.Int // powers of τ encrypted in G1 curve, divided by δ +} +``` + +--- + +```go +type Vk struct { + IC [][3]*big.Int + G1 struct { + Alpha [3]*big.Int + } + G2 struct { + Beta [3][2]*big.Int + Gamma [3][2]*big.Int + Delta [3][2]*big.Int + } +} +``` + + +--- + +```go +// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed +type Setup struct { + Toxic struct { + T *big.Int // trusted setup secret + Kalpha *big.Int + Kbeta *big.Int + Kgamma *big.Int + Kdelta *big.Int + } + Pk Pk + Vk Vk +} +``` + +--- + +# +## Proofs Generation +$\pi_A=\alpha + \sum_{i=0}^m \alpha_i u_i(x) + r \delta$ +$\pi_B=\beta + \sum_{i=0}^m \alpha_i v_i(x) + s \delta$ + +
+ +$\pi_C = \dfrac{ +\sum_{i=l+1}^m a_i(\beta u_i(x) + \alpha v_i(x) + w_i(x)) + h(x)t(x) +}{ +\delta +} + \pi_As + \pi_Br -rs\delta$ + +
+ +$\pi=\pi_A^1, \pi_B^1, \pi_C^2$ + +--- + +### Proof Verification + +
+ +$[\pi_A]_1 \cdot [\pi_B]_2 = [\alpha]_1 \cdot [\beta]_2 + +\sum_{i=0}^l a_i [ +\dfrac{ +\beta u_i(x) + \alpha v_i(x) + w_i(x) +}{ +\gamma +} +]_1 +\cdot [\gamma]_2 + [\pi_C]_1 \cdot [\delta]_2 +$ + +
+ +$e(\pi_A, \pi_B) = e(\alpha, \beta) \cdot e(pub, \gamma) \cdot e(\pi_C, \delta)$ + + +--- + +## How we use zkSNARKs in iden3 +- proving a credentials without revealing it's content +- proving that an identity has a claim issued by another identity, without revealing all the data +- proving any property of an identity +- $ITF$ (Identity Transition Function), a way to prove with a zkSNARK that an identity has been updated following the defined protocol + - identities can not cheat when issuing claims +- etc + +## Other ideas for free time side project +- Zendermint (Tendermint + zkSNARKs) + +--- + + + + +## zkSNARK libraries +- [bellman](https://github.com/zkcrypto/bellman) (rust) +- [libsnark](https://github.com/scipr-lab/libsnark) (c++) +- [snarkjs](https://github.com/iden3/snarkjs) (javascript) +- [websnark](https://github.com/iden3/websnark) (wasm) +- [go-snark](https://github.com/arnaucube/go-snark) (golang) [do not use in production] + +## Circuit languages +| language | snark library with which plugs in | +|-----|-----| +| [Zokrates](https://github.com/Zokrates/ZoKrates) | libsnark, bellman | +| [Snarky](https://github.com/o1-labs/snarky) | libsnark | +| [circom](https://github.com/iden3/circom) | snarkjs, websnark, bellman | +| [go-snark-circuit](https://github.com/arnaucube/go-snark) | go-snark | + +--- + +## Utilities (Elliptic curve & Hash functions) inside the zkSNARK +- we work over $F_r$, where $r=$`21888242871839275222246405745257275088548364400416034343698204186575808495617` + +- BabyJubJub +- Mimc +- Poseidon + +--- + +##### *Utilities (Elliptic curve & Hash functions) inside the zkSNARK* + +### BabyJubJub +- explaination: https://medium.com/zokrates/efficient-ecc-in-zksnarks-using-zokrates-bd9ae37b8186 +- implementations: + - go: https://github.com/iden3/go-iden3-crypto + - javascript & circom: https://github.com/iden3/circomlib + - rust: https://github.com/arnaucube/babyjubjub-rs + - c++: https://github.com/barryWhiteHat/baby_jubjub_ecc + +--- + +##### *Utilities (Elliptic curve & Hash functions) inside the zkSNARK* + +### Mimc7 +- explaination: https://eprint.iacr.org/2016/492.pdf +- implementations in: + - go: https://github.com/iden3/go-iden3-crypto + - javascript & circom: https://github.com/iden3/circomlib + - rust: https://github.com/arnaucube/mimc-rs + +--- + +##### *Utilities (Elliptic curve & Hash functions) inside the zkSNARK* + +### Poseidon +- explaination: https://eprint.iacr.org/2019/458.pdf +- implementations in: + - go: https://github.com/iden3/go-iden3-crypto + - javascript & circom: https://github.com/iden3/circomlib + +--- + +# References +- `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf +- `Pinocchio: Nearly practical verifiable computation`, Bryan Parno, Craig Gentry, Jon Howell, Mariana Raykova https://eprint.iacr.org/2013/279.pdf +- `On the Size of Pairing-based Non-interactive Arguments`, Jens Groth https://eprint.iacr.org/2016/260.pdf +- (also all the links through the slides) + +--- + +
+Thank you very much +
+ +
+ + +
+
+ +[arnaucube.com](https://arnaucube.com) +[github.com/arnaucube](https://github.com/arnaucube) +[twitter.com/arnaucube](https://twitter.com/arnaucube) + + + 2019-08-20 +
+ +
+[iden3.io](https://iden3.io) +[github.com/iden3](https://github.com/iden3) +[twitter.com/identhree](https://twitter.com/identhree) diff --git a/zksnarks-from-scratch-a-technical-explanation.pdf b/zksnarks-from-scratch-a-technical-explanation.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8a11969dc6f2af13e9d6bfecdce77da58b8d3f04 GIT binary patch literal 1194298 zcmce-c{o(>A2;46HMX|Vj+E3KrAT&CsZ1r(IB3j?EE!?!OEZK-R3t-4q$G2gV@bv~ zvK1plMp?2ine4j}en+40bN&8%{(P>hKU~w9Ip@C5`+ncA?Upe(f8p54C46&$(DzxH-8<9XWT$#mT|W-Qg%k-oVAl%Kf%A`k|4%orOI5T`318 zRrEjpU2Ny%c;3RzTI$GoHH8x=RZplWD4kSLJb6k1IB`N2{fAz6)yeAreuw!ZC#kvDR;bo=*-o`Z$0HAWt^vvRv5rE=l~M*fnuo$VdB z|NcJbbl1tnh+=UY{qA{d54+pe*Dsz$zkSuh)m};w{q^e@`LlLzt_IdF=bRiUPL9@& z=r=L)7wqo3p+l6va2FlSdF$IwR_HyhSUcLHH&IYlKB;(8OH0bt4V@kbjMvNIUvu$9 zE?u6wk~c;hTdXTqbRC&^ta(wHLIX{qHGw&hWVh5f7{-xCO|H#nt~kpms@P3IF-0s$N3L ziRS-3vqT_%%ieHrnI)%={|&*tQ;By^WBS;NCqobj`%-MVqFnODkM@7=CurP&^xj)3 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