chore: add examples and readme updates from hc

examples/README.md

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+These examples will be moved to another repo so that they can be tested/updated.
+`nargo` currently ascends the directory structure to the highest found Nargo.toml file.

examples/custom_modulus_example/Nargo.toml

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+[package]
+name = "custom_modulus_example"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.34.0"
+
+[dependencies]
+bignum = { path = "../.." }

examples/custom_modulus_example/README.md

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+# Custom (compile-time) modulus example
+
+This example assumes you have a modulus of 2048 bits and want to use it at compile-time. 
+
+## Generate parameters
+
+Generate parameters for BigNum library using the [paramgen tool](https://github.com/noir-lang/noir-bignum-paramgen). After cloning and building the repo, run for the desired modulus (replace value):
+```
+./target/release/paramgen instance 20456684803303048347530423697020751814399302502110667150078323718905269101172974800905356684386150937514420094865028635518785372677390301711491016948450207607846221304930728973797522378506999567139943652134234877561436074026268205649910255182621594085008284441316448901381890082286184776564686504132516001962221244304143672537300570792891349927421038112529453088606957312408143395552575078898208723518244571879698149587589463599543806422441264913231441141752207832659135584572831493077819920330893012366189267930979049769425009576366557305457228626690969723792844542486959558350660501921357172974372066702504546151411 RSA2048_example > out.txt
+```
+
+Result:
+```rust
+struct RSA2048_example_Params {}
+impl RuntimeBigNumParamsTrait<18> for RSA2048_example_Params {
+    pub fn modulus_bits() -> u64 {
+        2048
+    }
+}
+impl BigNumParamsTrait<18> for RSA2048_example_Params {
+    pub fn get_instance() -> BigNumInstance<18, Self> {
+        RSA2048_example_Instance
+    }
+    pub fn modulus_bits() -> u64 {
+        2048
+    }
+}
+global RSA2048_example_Instance: BigNumInstance<18, RSA2048_example_Params> = BigNumInstance {
+        modulus: [
+            0xca067270cbaa2f334deca1472a2bf3, 0x6610de1958b4206e3e34a14af22618, 0xe59e76000fcfb05956e5503a499841, 0x3d429c951fdec78afafacd2381782c, 0x617806aa0cf2d15b8c2a41bfd38ed9, 0xa4a58ff8737791b2d5571ea75c92d3, 0x49f85ae321bfbc161162bc1034f586, 0xa40bf68ca724dff0cf63605975cf48, 0xd5c5fe2ab9a1adf232351085a7d591, 0x1b0d1c9077b8f0794b2cb4af4d294f, 0x9401a841ae63b1f566fd6b69e62ccd, 0xd6958bad7bcf453c6cb489538831e0, 0x51bdf4c4cd4c92887a1d178d6bc3ce, 0xd153e9c43d7aa9b0d7a2c1af84ea38, 0x9deb9b61f4e25c1bd0d53e4e0e61e1, 0xbed33da20e52d7c916d486f235202b, 0x0c541718be73e9ce00f430c086b205, 0xa2
+        ],
+        double_modulus: [
+            0x01940ce4e197545e669bd9428e5457e6, 0x01cc21bc32b16840dc7c694295e44c30, 0x01cb3cec001f9f60b2adcaa074933081, 0x017a85392a3fbd8f15f5f59a4702f058, 0x01c2f00d5419e5a2b71854837fa71db1, 0x01494b1ff0e6ef2365aaae3d4eb925a5, 0x0193f0b5c6437f782c22c5782069eb0c, 0x014817ed194e49bfe19ec6c0b2eb9e8f, 0x01ab8bfc5573435be4646a210b4fab22, 0x01361a3920ef71e0f29659695e9a529e, 0x01280350835cc763eacdfad6d3cc5999, 0x01ad2b175af79e8a78d96912a71063c0, 0x01a37be9899a992510f43a2f1ad7879c, 0x01a2a7d3887af55361af45835f09d46f, 0x013bd736c3e9c4b837a1aa7c9c1cc3c2, 0x017da67b441ca5af922da90de46a4056, 0x0118a82e317ce7d39c01e861810d640a, 0x0143
+        ],
+        modulus_u60: U60Repr { limbs: ArrayX { segments: [[
+            0x034deca1472a2bf3, 0x0ca067270cbaa2f3, 0x0e3e34a14af22618, 0x06610de1958b4206, 0x0956e5503a499841, 0x0e59e76000fcfb05, 0x0afafacd2381782c, 0x03d429c951fdec78, 0x0b8c2a41bfd38ed9, 0x0617806aa0cf2d15, 0x02d5571ea75c92d3, 0x0a4a58ff8737791b, 0x061162bc1034f586, 0x049f85ae321bfbc1, 0xcf63605975cf48, 0x0a40bf68ca724dff, 0x0232351085a7d591, 0x0d5c5fe2ab9a1adf], [0x094b2cb4af4d294f, 0x01b0d1c9077b8f07, 0x0566fd6b69e62ccd, 0x09401a841ae63b1f, 0x0c6cb489538831e0, 0x0d6958bad7bcf453, 0x087a1d178d6bc3ce, 0x051bdf4c4cd4c928, 0xd7a2c1af84ea38, 0x0d153e9c43d7aa9b, 0x0bd0d53e4e0e61e1, 0x09deb9b61f4e25c1, 0x0916d486f235202b, 0x0bed33da20e52d7c, 0x0e00f430c086b205, 0xc541718be73e9c, 0xa2, 0x00]] } },
+        modulus_u60_x4: U60Repr { limbs: ArrayX { segments: [[
+            0x034deca1472a2bf3, 0x0ca067270cbaa2f3, 0x0e3e34a14af22618, 0x06610de1958b4206, 0x0956e5503a499841, 0x0e59e76000fcfb05, 0x0afafacd2381782c, 0x03d429c951fdec78, 0x0b8c2a41bfd38ed9, 0x0617806aa0cf2d15, 0x02d5571ea75c92d3, 0x0a4a58ff8737791b, 0x061162bc1034f586, 0x049f85ae321bfbc1, 0xcf63605975cf48, 0x0a40bf68ca724dff, 0x0232351085a7d591, 0x0d5c5fe2ab9a1adf], [0x094b2cb4af4d294f, 0x01b0d1c9077b8f07, 0x0566fd6b69e62ccd, 0x09401a841ae63b1f, 0x0c6cb489538831e0, 0x0d6958bad7bcf453, 0x087a1d178d6bc3ce, 0x051bdf4c4cd4c928, 0xd7a2c1af84ea38, 0x0d153e9c43d7aa9b, 0x0bd0d53e4e0e61e1, 0x09deb9b61f4e25c1, 0x0916d486f235202b, 0x0bed33da20e52d7c, 0x0e00f430c086b205, 0xc541718be73e9c, 0xa2, 0x00], [0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00], [0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00]] } },
+        redc_param: [
+            0x1fdcdafc0b4a1e7e76fea4d047f13c, 0xe9f3127f19f73583a5378c5e3fb4a4, 0x61d4721841efbeac85a785dd48b58e, 0x747ed55336a2218c1f011c5f730478, 0xf8e29e2ca49c4df7aa57cc4defb877, 0x67541ffb558ce4fea6409c61e5b96a, 0xc616b60523693e15226453d9c1a735, 0x8e609df55b7e417c40ec232ac85f2c, 0xec4a3a8702ea1a8836d5bf2d63dc4c, 0x73ee26edb819fac7c1a771ed2d2447, 0x84e660e9c009e0eadd88a0c2d957c5, 0xb9c718c32762b643b6c29606ea3766, 0x9077b6f87aa1ba000b853a4841ea29, 0x28e68f0218123e2f911bfe4f2afdb1, 0xc45f6a9c6e8f5c18e38415b847eba6, 0xfd74428019998d5f69bda9954cd6d4, 0x6c48d17d943d2d123a0cb7f1044254, 0x0194
+        ]
+};
+```
+
+## Add parameters to file
+
+Create a file `src/custom_params.nr` and add the generated code. 
+
+Slightly adjust parameters:
+- remove the `pub` keywords
+- add dependencies
+- change return type of `modulus_bits` to `u32`
+- add to `RuntimeBigNumParamsTrait` and `BigNumParamsTrait`:
+```rust 
+fn has_multiplicative_inverse() -> bool { false }
+``` 
+  > Note: if you are generating modulus parameters for a field, this is not needed since the multiplicative inverse exists for all elements.
+
+## Use the params in Noir program
+
+Import the parameters like this:
+```rust
+use custom_params::RSA2048_example_Params;
+```
+
+and define the type of the BigNums for this modulus:
+```rust
+type RSA2048 = BigNum<18, RSA2048_example_Params>;
+```
+
+Now arithmetic operations can be performed:
+```rust
+fn main(a: RSA2048, b: RSA2048, expected: RSA2048) {
+    let c = a + b; // modular addition (constrained)
+    assert(c == expected);
+}
+```
+
+Test this function
+```
+nargo test
+```

examples/custom_modulus_example/src/custom_params.nr

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+use dep::bignum::BigNum;
+use dep::bignum::BigNumParamsTrait;
+use dep::bignum::runtime_bignum::BigNumInstance;
+use dep::bignum::runtime_bignum::BigNumParamsTrait as RuntimeBigNumParamsTrait;
+use dep::bignum::utils::u60_representation::U60Repr;
+use dep::bignum::utils::arrayX::ArrayX;
+
+struct RSA2048_example_Params {}
+impl RuntimeBigNumParamsTrait<18> for RSA2048_example_Params {
+    fn modulus_bits() -> u32 {
+        2048
+    }
+    fn has_multiplicative_inverse() -> bool { false }
+}
+impl BigNumParamsTrait<18> for RSA2048_example_Params {
+    fn get_instance() -> BigNumInstance<18, Self> {
+        RSA2048_example_Instance
+    }
+    fn modulus_bits() -> u32 {
+        2048
+    }
+    fn has_multiplicative_inverse() -> bool { false }
+}
+global RSA2048_example_Instance: BigNumInstance<18, RSA2048_example_Params> = BigNumInstance {
+        modulus: [
+            0xca067270cbaa2f334deca1472a2bf3, 0x6610de1958b4206e3e34a14af22618, 0xe59e76000fcfb05956e5503a499841, 0x3d429c951fdec78afafacd2381782c, 0x617806aa0cf2d15b8c2a41bfd38ed9, 0xa4a58ff8737791b2d5571ea75c92d3, 0x49f85ae321bfbc161162bc1034f586, 0xa40bf68ca724dff0cf63605975cf48, 0xd5c5fe2ab9a1adf232351085a7d591, 0x1b0d1c9077b8f0794b2cb4af4d294f, 0x9401a841ae63b1f566fd6b69e62ccd, 0xd6958bad7bcf453c6cb489538831e0, 0x51bdf4c4cd4c92887a1d178d6bc3ce, 0xd153e9c43d7aa9b0d7a2c1af84ea38, 0x9deb9b61f4e25c1bd0d53e4e0e61e1, 0xbed33da20e52d7c916d486f235202b, 0x0c541718be73e9ce00f430c086b205, 0xa2
+        ],
+        double_modulus: [
+            0x01940ce4e197545e669bd9428e5457e6, 0x01cc21bc32b16840dc7c694295e44c30, 0x01cb3cec001f9f60b2adcaa074933081, 0x017a85392a3fbd8f15f5f59a4702f058, 0x01c2f00d5419e5a2b71854837fa71db1, 0x01494b1ff0e6ef2365aaae3d4eb925a5, 0x0193f0b5c6437f782c22c5782069eb0c, 0x014817ed194e49bfe19ec6c0b2eb9e8f, 0x01ab8bfc5573435be4646a210b4fab22, 0x01361a3920ef71e0f29659695e9a529e, 0x01280350835cc763eacdfad6d3cc5999, 0x01ad2b175af79e8a78d96912a71063c0, 0x01a37be9899a992510f43a2f1ad7879c, 0x01a2a7d3887af55361af45835f09d46f, 0x013bd736c3e9c4b837a1aa7c9c1cc3c2, 0x017da67b441ca5af922da90de46a4056, 0x0118a82e317ce7d39c01e861810d640a, 0x0143
+        ],
+        modulus_u60: U60Repr { limbs: ArrayX { segments: [[
+            0x034deca1472a2bf3, 0x0ca067270cbaa2f3, 0x0e3e34a14af22618, 0x06610de1958b4206, 0x0956e5503a499841, 0x0e59e76000fcfb05, 0x0afafacd2381782c, 0x03d429c951fdec78, 0x0b8c2a41bfd38ed9, 0x0617806aa0cf2d15, 0x02d5571ea75c92d3, 0x0a4a58ff8737791b, 0x061162bc1034f586, 0x049f85ae321bfbc1, 0xcf63605975cf48, 0x0a40bf68ca724dff, 0x0232351085a7d591, 0x0d5c5fe2ab9a1adf], [0x094b2cb4af4d294f, 0x01b0d1c9077b8f07, 0x0566fd6b69e62ccd, 0x09401a841ae63b1f, 0x0c6cb489538831e0, 0x0d6958bad7bcf453, 0x087a1d178d6bc3ce, 0x051bdf4c4cd4c928, 0xd7a2c1af84ea38, 0x0d153e9c43d7aa9b, 0x0bd0d53e4e0e61e1, 0x09deb9b61f4e25c1, 0x0916d486f235202b, 0x0bed33da20e52d7c, 0x0e00f430c086b205, 0xc541718be73e9c, 0xa2, 0x00]] } },
+        modulus_u60_x4: U60Repr { limbs: ArrayX { segments: [[
+            0x034deca1472a2bf3, 0x0ca067270cbaa2f3, 0x0e3e34a14af22618, 0x06610de1958b4206, 0x0956e5503a499841, 0x0e59e76000fcfb05, 0x0afafacd2381782c, 0x03d429c951fdec78, 0x0b8c2a41bfd38ed9, 0x0617806aa0cf2d15, 0x02d5571ea75c92d3, 0x0a4a58ff8737791b, 0x061162bc1034f586, 0x049f85ae321bfbc1, 0xcf63605975cf48, 0x0a40bf68ca724dff, 0x0232351085a7d591, 0x0d5c5fe2ab9a1adf], [0x094b2cb4af4d294f, 0x01b0d1c9077b8f07, 0x0566fd6b69e62ccd, 0x09401a841ae63b1f, 0x0c6cb489538831e0, 0x0d6958bad7bcf453, 0x087a1d178d6bc3ce, 0x051bdf4c4cd4c928, 0xd7a2c1af84ea38, 0x0d153e9c43d7aa9b, 0x0bd0d53e4e0e61e1, 0x09deb9b61f4e25c1, 0x0916d486f235202b, 0x0bed33da20e52d7c, 0x0e00f430c086b205, 0xc541718be73e9c, 0xa2, 0x00], [0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00], [0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00]] } },
+        redc_param: [
+            0x1fdcdafc0b4a1e7e76fea4d047f13c, 0xe9f3127f19f73583a5378c5e3fb4a4, 0x61d4721841efbeac85a785dd48b58e, 0x747ed55336a2218c1f011c5f730478, 0xf8e29e2ca49c4df7aa57cc4defb877, 0x67541ffb558ce4fea6409c61e5b96a, 0xc616b60523693e15226453d9c1a735, 0x8e609df55b7e417c40ec232ac85f2c, 0xec4a3a8702ea1a8836d5bf2d63dc4c, 0x73ee26edb819fac7c1a771ed2d2447, 0x84e660e9c009e0eadd88a0c2d957c5, 0xb9c718c32762b643b6c29606ea3766, 0x9077b6f87aa1ba000b853a4841ea29, 0x28e68f0218123e2f911bfe4f2afdb1, 0xc45f6a9c6e8f5c18e38415b847eba6, 0xfd74428019998d5f69bda9954cd6d4, 0x6c48d17d943d2d123a0cb7f1044254, 0x0194
+        ]
+};
+

examples/custom_modulus_example/src/main.nr

@@ -0,0 +1,25 @@
+mod custom_params;
+
+use custom_params::RSA2048_example_Params;
+use dep::bignum::BigNum;
+
+type RSA2048 = BigNum<18, RSA2048_example_Params>;
+
+fn main(a: RSA2048, b: RSA2048, expected: RSA2048) {
+    let c = a + b;
+    assert(c == expected);
+}
+
+#[test]
+
+fn test_main() {
+    let mut a: RSA2048 = BigNum::new();
+    a.limbs[0] = 10;
+    let mut b: RSA2048 = BigNum::new();
+    b.limbs[0] = 20;
+
+    let mut expected: RSA2048 = BigNum::new();
+    expected.limbs[0] = 30;
+
+    main(a, b, expected);
+}

examples/ed25519_example/Nargo.toml

@@ -0,0 +1,8 @@
+[package]
+name = "ed25519_example"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.34.0"
+
+[dependencies]
+bignum = { path = "../.." }

examples/ed25519_example/Prover.toml

@@ -0,0 +1,20 @@
+[a]
+limbs = ["1", "0", "0"]
+
+[b]
+limbs = ["2", "0", "0"]
+
+[c]
+limbs = ["3", "0", "0"]
+
+[d]
+limbs = ["4", "0", "0"]
+
+[e]
+limbs = ["5", "0", "0"]
+
+[expected]
+limbs = ["43", "0", "0"]
+
+[f]
+limbs = ["6", "0", "0"]

examples/ed25519_example/README.md

@@ -0,0 +1,31 @@
+# ed25519 Example
+
+## Testing
+Run test:
+```
+nargo test
+```
+
+## Execution
+The testvalues are also in the `Prover.toml`. Adjust them to prove with other values. 
+Execute the Noir program:
+```
+nargo execute example1
+```
+The witness is written to `./target/example1.gz`. 
+
+Prove valid execution (with default Barretenberg backend):
+```
+bb prove -b ./target/ed25519_example.json -w ./target/example1.gz -o ./target/proof
+```
+The generated proof will be in `./target/proof`. 
+
+Verify proof by computing the verification key:
+```
+bb write_vk -b ./target/ed25519_example.json -o ./target/vk
+```
+.. and then verifying the proof:
+```
+bb verify -k ./target/vk -p ./target/proof
+```
+If successful, you see nothing! Otherwise an error will show. 

examples/ed25519_example/src/main.nr

@@ -0,0 +1,57 @@
+use dep::bignum::fields::ed25519Fq::ED25519_Fq_Params;
+use dep::bignum::runtime_bignum::BigNumInstance;
+use dep::bignum::BigNum;
+
+// Prime field mod 2^255-19
+type Ed25519Fq = BigNum<3, ED25519_Fq_Params>;
+
+// Check that `a * b + (c + d) * e + f` equals `expected`
+// (this constrains `a * b + (c + d) * e + f - expected = 0`)
+fn main(
+    a: Ed25519Fq,
+    b: Ed25519Fq,
+    c: Ed25519Fq,
+    d: Ed25519Fq,
+    e: Ed25519Fq,
+    f: Ed25519Fq,
+    expected: Ed25519Fq
+) {
+    // Step 1: calculate g = a * b + (c + d) * e + f in unconstrained functions 
+    // First product term a * b
+    let t0 = a.__mul(b);
+    // Second product term (c + d) * e
+    let t1 = (c.__add(d)).__mul(e);
+    let g = t0.__add(t1).__add(f);
+
+    // Step 2: 
+    // product term 1 a * b. Rewrite to (a + 0) * b
+    // product term 2 (c + d) * e 
+    // linear terms f, g
+    BigNum::evaluate_quadratic_expression(
+        [[a, BigNum::new()], [c, d]],
+        [[false, false], [false, false]],
+        [[b], [e]],
+        [[false], [false]],
+        [f, g],
+        [false, true]
+    );
+
+    // Step 3: check res equals `expected`
+    assert(g == expected);
+}
+
+// Simple test
+#[test]
+fn test_main() {
+    // a=1, b=2, c=3, d=4, e=5, f=6
+    let a: Ed25519Fq = BigNum::from_array([1, 0, 0]);
+    let b: Ed25519Fq = BigNum::from_array([2, 0, 0]);
+    let c: Ed25519Fq = BigNum::from_array([3, 0, 0]);
+    let d: Ed25519Fq = BigNum::from_array([4, 0, 0]);
+    let e: Ed25519Fq = BigNum::from_array([5, 0, 0]);
+    let f: Ed25519Fq = BigNum::from_array([6, 0, 0]);
+    // 1*2 + (3+4) * 5 + 6 = 43
+    let expected: Ed25519Fq = BigNum::from_array([43, 0, 0]);
+
+    main(a, b, c, d, e, f, expected);
+}

examples/runtime_modulus_example/Nargo.toml

@@ -0,0 +1,8 @@
+[package]
+name = "runtime_modulus_example"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.34.0"
+
+[dependencies]
+bignum = { path = "../.." }

examples/runtime_modulus_example/Prover.toml

@@ -0,0 +1,12 @@
+modulus = ["0xca067270cbaa2f334deca1472a3e31", "0x6610de1958b4206e3e34a14af22618", "0xe59e76000fcfb05956e5503a499841", "0x3d429c951fdec78afafacd2381782c", "0x617806aa0cf2d15b8c2a41bfd38ed9", "0xa5bd5539ec839696d39cf8dfdd92d3", "0xe9bed0c391a2d4050d6d2367c6a943", "0x2d2150ff6bb93bf3a7d1163587d488", "0x852ed0ee51bf01f3"]
+redc_param = ["0xb6b68cb415434ea6273830091e7ba4", "0x9da115c204677051818a8c033cbc85", "0x8cb24ce7044f7c1fe2cd6d5425e397", "0x478c0c53482aacf0b50d3649c1b096", "0xc7b7c730e69ec6b458f1ea2faf1c28", "0x6b733155f8756e0d0cc2bce80185da", "0xf34103a45581aa68206e1aac5ecdca", "0x59c9f258f9e3905b9629a277b6752f", "0x01ec13452bee67c5dc"]
+
+[a]
+limbs = ["10", "0", "0", "0", "0", "0", "0", "0", "0"]
+
+[b]
+limbs = ["20", "0", "0", "0", "0", "0", "0", "0", "0"]
+
+[expected]
+limbs = ["30", "0", "0", "0", "0", "0", "0", "0", "0"]
+

examples/runtime_modulus_example/README.md

@@ -0,0 +1,64 @@
+# Runtime modulus example
+
+## Create program
+
+As explained in the general BigNum README, for a runtime modulus a `BigNumInstance` either has to be passed into the program or generated using the `modulus` and `redc_param`. In this example we'll do the latter.
+
+This is an example assuming RSA for a modulus of 1024 bits.
+
+At compile-time we know the number of bits of the modulus and can define the types:
+```rust
+struct RSA1024Params {}
+impl RuntimeBigNumParamsTrait<9> for RSA1024Params {
+    fn modulus_bits() -> u32 {
+        1024
+    }
+    fn has_multiplicative_inverse() -> bool { false }
+}
+
+type RSA1024 = BigNum<9, RSA1024Params>;
+type RSA1024Instance = BigNumInstance<9, RSA1024Params>;
+```
+
+The `main` function expects the `modulus` and `redc_param` in order to generate the `BigNumInstance` and consequently use it to perform arithmetic operations.
+
+```rust
+fn main(modulus: [Field; 9], redc_param: [Field; 9], a: RSA1024, b: RSA1024, expected: RSA1024) {
+    let BNInstance: RSA1024Instance = BigNumInstance::new(modulus, redc_param);
+
+    let c = BNInstance.add(a, b);
+    assert(BNInstance.eq(expected, c));
+}
+```
+
+## Test program
+
+The test defines `modulus` and `redc_param`, which have been obtained using [this tool](https://github.com/noir-lang/noir-bignum-paramgen). To run the test:
+```
+nargo test
+```
+
+## Execute, prove & verify program
+
+The testvalues are also in the `Prover.toml`. Adjust them to prove with other values. 
+Execute the Noir program:
+```
+nargo execute example2
+```
+The witness is written to `./target/example2.gz`. 
+
+Prove valid execution (with default Barretenberg backend):
+```
+bb prove -b ./target/custom_modulus_example.json -w ./target/example2.gz -o ./target/proof
+```
+The generated proof will be in `./target/proof`. 
+
+Verify proof by computing the verification key:
+```
+bb write_vk -b ./target/custom_modulus_example.json -o ./target/vk
+```
+.. and then verifying the proof:
+```
+bb verify -k ./target/vk -p ./target/proof
+```
+If successful, you see nothing! Otherwise an error will show.