list of resources added to readme

README.md

@@ -17,14 +17,28 @@
 ## Overview
 This repo is a reference on correctly testing and constraining circom circuits, with example workflows and reference patterns.
 
-todo: discuss tools
-
-WIP
-### Circomkit
-
-
-### Prerequisites
-todo
+### Tools and resources considered
+
+#### tools
+- [circomkit circom testing suite](https://github.com/erhant/circomkit)
+    - [examples](https://github.com/erhant/circomkit-examples)
+- [circomspect static analyzer and linter for circom](https://github.com/trailofbits/circomspect)
+    - [ToB blog: it pays to be circomspect](https://blog.trailofbits.com/2022/09/15/it-pays-to-be-circomspect/)
+    - [ToB blog: circomspect has more passes](https://blog.trailofbits.com/2023/03/21/circomspect-static-analyzer-circom-more-passes/)
+- [zksecurity circomscribe - demo playground, visualize constraints ](https://www.circomscribe.dev/)
+    - [blog post about circomscribe](https://www.zksecurity.xyz/blog/posts/circomscribe/)
+- [Picus QED uniqueness property underconstraint checker](https://github.com/Veridise/Picus)
+- [circom-mutator](https://github.com/aviggiano/circom-mutator) - test whether mutations of correct circuit actually fail by fuzzing
+
+#### more reading about underconstrained circuits
+- [circom constraining docs](https://docs.circom.io/circom-language/constraint-generation/)
+- [circom docs: --inspect option](https://docs.circom.io/circom-language/code-quality/inspect/)
+- [veridise blog: circom pairing](https://medium.com/veridise/circom-pairing-a-million-dollar-zk-bug-caught-early-c5624b278f25)
+- [dacian: exploiting under-constrained zk circuits](https://dacian.me/exploiting-under-constrained-zk-circuits)
+- [blockdev: tips for safe circom circuits](https://hackmd.io/@blockdev/Bk_-jRkXa)
+- [circom101 book by erhant, author of circomkit](https://github.com/erhant/circom101/tree/main)
+- [0xparc: circom workshop series](https://learn.0xparc.org/materials/circom/learning-group-1/intro-zkp)
+- [paper by veridise on underconstrained circuits](https://eprint.iacr.org/2023/512.pdf)
 
 ## License
 

circuits/multiplier.circom

@@ -4,7 +4,6 @@ template Multiplier() {
     signal input a;
     signal input b;
     signal output c;
-
     signal intermediary;
 
     // Constraint: a * b should equal c
@@ -15,4 +14,36 @@ template Multiplier() {
     assert(c == a * b);
 }
 
-// component main = Multiplier();
+template UnderconstrainedMultiplier1() {
+    signal input a;
+    signal input b;
+    signal output c;
+    signal intermediary;
+
+    intermediary <-- a * b;
+    c <== intermediary;
+    assert(c == a * b);
+}
+
+template UnderconstrainedMultiplier2() {
+    signal input a;
+    signal input b;
+    signal output c;
+    signal intermediary;
+
+    intermediary <== a * b;
+    c <-- intermediary;
+    assert(c == a * b);
+}
+
+template UnderconstrainedMultiplier3() {
+    signal input a;
+    signal input b;
+    signal output c;
+    signal intermediary;
+
+    intermediary <== a * b;
+    c <== intermediary;
+    // assert(c == a * b);
+}
+