diff --git a/src/plonk.rs b/src/plonk.rs
index 07368a1..639d149 100644
--- a/src/plonk.rs
+++ b/src/plonk.rs
@@ -9,7 +9,8 @@ use blake2b_simd::Params as Blake2bParams;
 
 use crate::arithmetic::{CurveAffine, FieldExt};
 use crate::poly::{
-    commitment::Params, Coeff, EvaluationDomain, ExtendedLagrangeCoeff, LagrangeCoeff, Polynomial,
+    commitment::Params, Coeff, EvaluationDomain, ExtendedLagrangeCoeff, LagrangeCoeff,
+    PinnedEvaluationDomain, Polynomial,
 };
 use crate::transcript::{ChallengeScalar, Transcript};
 
@@ -84,7 +85,7 @@ impl<C: CurveAffine> VerifyingKey<C> {
             .personal(C::BLAKE2B_PERSONALIZATION)
             .to_state();
 
-        let s = format!("{}", self);
+        let s = format!("{:?}", self.pinned());
 
         hasher.update(&(s.len() as u64).to_le_bytes());
         hasher.update(s.as_bytes());
@@ -94,30 +95,40 @@ impl<C: CurveAffine> VerifyingKey<C> {
 
         Ok(())
     }
-}
 
-impl<C: CurveAffine> std::fmt::Display for VerifyingKey<C> {
-    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> {
-        write!(
-            f,
-            "VerificationKey {{ \
-            base_modulus = {base_modulus}, \
-            scalar_modulus = {scalar_modulus}, \
-            domain = {domain}, \
-            cs = {cs}, \
-            fixed_commitments = {fixed_commitments:?}, \
-            permutations = {permutations:?}\
-        }}",
-            base_modulus = C::Base::MODULUS,
-            scalar_modulus = C::Scalar::MODULUS,
-            domain = self.domain,
-            cs = self.cs,
-            fixed_commitments = self.fixed_commitments,
-            permutations = self.permutations
-        )
+    /// Obtains a pinned representation of this verification key that contains
+    /// the minimal information necessary to reconstruct the verification key.
+    pub fn pinned(&self) -> PinnedVerificationKey<'_, C> {
+        PinnedVerificationKey {
+            domain: self.domain.pinned(),
+            fixed_commitments: &self.fixed_commitments,
+            permutations: &self.permutations,
+            cs: self.cs.pinned(),
+        }
     }
 }
 
+/// Minimal representation of a verification key that can be used to identify
+/// its active contents.
+pub struct PinnedVerificationKey<'a, C: CurveAffine> {
+    domain: PinnedEvaluationDomain<'a, C::Scalar>,
+    fixed_commitments: &'a Vec<C>,
+    permutations: &'a Vec<permutation::VerifyingKey<C>>,
+    cs: PinnedConstraintSystem<'a, C::Scalar>,
+}
+
+impl<'a, C: CurveAffine> std::fmt::Debug for PinnedVerificationKey<'a, C> {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> {
+        f.debug_struct("PinnedVerificationKey")
+            .field("base_modulus", &C::Base::MODULUS)
+            .field("scalar_modulus", &C::Scalar::MODULUS)
+            .field("domain", &self.domain)
+            .field("cs", &self.cs)
+            .field("fixed_commitments", self.fixed_commitments)
+            .field("permutations", self.permutations)
+            .finish()
+    }
+}
 /// This is a proving key which allows for the creation of proofs for a
 /// particular circuit.
 #[derive(Debug)]
@@ -564,7 +575,7 @@ fn test_proving() {
 
     // Initialize the proving key
     let vk = keygen_vk(&params, &empty_circuit).expect("keygen_vk should not fail");
-    assert_eq!(format!("{}", vk), "VerificationKey {base_modulus = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001, scalar_modulus = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001, domain = Domain {k = 5, extended_k = 7, omega = 0x0cc3380dc616f2e1daf29ad1560833ed3baea3393eceb7bc8fa36376929b78cc }, cs = ConstraintSystem {num_fixed_columns: 8, num_advice_columns: 5, num_instance_columns: 1, fixed_queries: [(Column { index: 6, column_type: Fixed }, Rotation(0)), (Column { index: 7, column_type: Fixed }, Rotation(0)), (Column { index: 0, column_type: Fixed }, Rotation(0)), (Column { index: 2, column_type: Fixed }, Rotation(0)), (Column { index: 3, column_type: Fixed }, Rotation(0)), (Column { index: 4, column_type: Fixed }, Rotation(0)), (Column { index: 1, column_type: Fixed }, Rotation(0)), (Column { index: 5, column_type: Fixed }, Rotation(0))], advice_queries: [(Column { index: 1, column_type: Advice }, Rotation(0)), (Column { index: 2, column_type: Advice }, Rotation(0)), (Column { index: 3, column_type: Advice }, Rotation(0)), (Column { index: 4, column_type: Advice }, Rotation(1)), (Column { index: 0, column_type: Advice }, Rotation(-1))], instance_queries: [(Column { index: 0, column_type: Instance }, Rotation(0))], permutations: [Argument { columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }, Column { index: 3, column_type: Advice }] }, Argument { columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }, Column { index: 3, column_type: Advice }] }], lookups: [Argument { input_columns: [Column { index: 1, column_type: Advice }], table_columns: [Column { index: 6, column_type: Fixed }] }, Argument { input_columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }], table_columns: [Column { index: 6, column_type: Fixed }, Column { index: 7, column_type: Fixed }] }], gates: [Sum(Sum(Sum(Sum(Product(Advice(0), Fixed(3)), Product(Advice(1), Fixed(4))), Product(Product(Advice(0), Advice(1)), Fixed(6))), Scaled(Product(Advice(2), Fixed(5)), 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000)), Product(Fixed(2), Product(Advice(3), Advice(4)))), Product(Fixed(7), Sum(Advice(0), Scaled(Instance(0), 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000))), ]}, fixed_commitments = [(0x3710f15f98bf0a7421343fdf390b9519506c67431a5c78678fbcc4db815c8547, 0x0a3c77f30ab2a2741b21cd45326c87dc7f7050b1a5e8c181a534b9c84a09cb02), (0x115235d6bd2467772db857d66e7f44837cd38bb6ac0c7b412c2997dd47cbbc3c, 0x0f339c20c40e10bed476699e2ddb88256092447582b250f329b0cbf6c3f66f17), (0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0), (0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0), (0x241d6d9c2060ce821d4b05ff2f9566c3947541f3d14a9aabcdb96c19158e8bc7, 0x39582cc6bdb1a4a88e89c050ad6db0ade34f45ec5791b07de6e694e9627ca66a), (0x1ee805e20232ba31eeae1fa345bd88ac7df81dc43ffb3967a218ea4defc9d17d, 0x2838c83c064d44e87e5c8d05a234ad24d2d4a502a370acb514b430f516c0f0bf), (0x37ead9904c760201ec4734ef398f0bdb5fe5a5e6e9db19c85e6b5483bdeb0a0b, 0x1dc08c38ed713b14f7a21a891a83b52160a3ffb0dccfbd70db7c7eb235dd193e), (0x2dc3d20553691216c988ecbb596c4bda329f27d50bd8a7c2fb0d84b423da3cb4, 0x025b40e800020458e15e3a57268562e6c08c6971d71262bd67c72437cfc60b4c)], permutations = [VerifyingKey { commitments: [(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06), (0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818), (0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc)] }, VerifyingKey { commitments: [(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06), (0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818), (0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc)] }]}");
+    assert_eq!(format!("{:?}", vk.pinned()), "PinnedVerificationKey { base_modulus: \"0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\", scalar_modulus: \"0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001\", domain: PinnedEvaluationDomain { k: 5, extended_k: 7, omega: 0x0cc3380dc616f2e1daf29ad1560833ed3baea3393eceb7bc8fa36376929b78cc }, cs: PinnedConstraintSystem { num_fixed_columns: 8, num_advice_columns: 5, num_instance_columns: 1, gates: [Sum(Sum(Sum(Sum(Product(Advice(0), Fixed(3)), Product(Advice(1), Fixed(4))), Product(Product(Advice(0), Advice(1)), Fixed(6))), Scaled(Product(Advice(2), Fixed(5)), 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000)), Product(Fixed(2), Product(Advice(3), Advice(4)))), Product(Fixed(7), Sum(Advice(0), Scaled(Instance(0), 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000)))], advice_queries: [(Column { index: 1, column_type: Advice }, Rotation(0)), (Column { index: 2, column_type: Advice }, Rotation(0)), (Column { index: 3, column_type: Advice }, Rotation(0)), (Column { index: 4, column_type: Advice }, Rotation(1)), (Column { index: 0, column_type: Advice }, Rotation(-1))], instance_queries: [(Column { index: 0, column_type: Instance }, Rotation(0))], fixed_queries: [(Column { index: 6, column_type: Fixed }, Rotation(0)), (Column { index: 7, column_type: Fixed }, Rotation(0)), (Column { index: 0, column_type: Fixed }, Rotation(0)), (Column { index: 2, column_type: Fixed }, Rotation(0)), (Column { index: 3, column_type: Fixed }, Rotation(0)), (Column { index: 4, column_type: Fixed }, Rotation(0)), (Column { index: 1, column_type: Fixed }, Rotation(0)), (Column { index: 5, column_type: Fixed }, Rotation(0))], permutations: [Argument { columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }, Column { index: 3, column_type: Advice }] }, Argument { columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }, Column { index: 3, column_type: Advice }] }], lookups: [Argument { input_columns: [Column { index: 1, column_type: Advice }], table_columns: [Column { index: 6, column_type: Fixed }] }, Argument { input_columns: [Column { index: 1, column_type: Advice }, Column { index: 2, column_type: Advice }], table_columns: [Column { index: 6, column_type: Fixed }, Column { index: 7, column_type: Fixed }] }] }, fixed_commitments: [(0x3710f15f98bf0a7421343fdf390b9519506c67431a5c78678fbcc4db815c8547, 0x0a3c77f30ab2a2741b21cd45326c87dc7f7050b1a5e8c181a534b9c84a09cb02), (0x115235d6bd2467772db857d66e7f44837cd38bb6ac0c7b412c2997dd47cbbc3c, 0x0f339c20c40e10bed476699e2ddb88256092447582b250f329b0cbf6c3f66f17), (0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0), (0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0), (0x241d6d9c2060ce821d4b05ff2f9566c3947541f3d14a9aabcdb96c19158e8bc7, 0x39582cc6bdb1a4a88e89c050ad6db0ade34f45ec5791b07de6e694e9627ca66a), (0x1ee805e20232ba31eeae1fa345bd88ac7df81dc43ffb3967a218ea4defc9d17d, 0x2838c83c064d44e87e5c8d05a234ad24d2d4a502a370acb514b430f516c0f0bf), (0x37ead9904c760201ec4734ef398f0bdb5fe5a5e6e9db19c85e6b5483bdeb0a0b, 0x1dc08c38ed713b14f7a21a891a83b52160a3ffb0dccfbd70db7c7eb235dd193e), (0x2dc3d20553691216c988ecbb596c4bda329f27d50bd8a7c2fb0d84b423da3cb4, 0x025b40e800020458e15e3a57268562e6c08c6971d71262bd67c72437cfc60b4c)], permutations: [VerifyingKey { commitments: [(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06), (0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818), (0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc)] }, VerifyingKey { commitments: [(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06), (0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818), (0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc)] }] }");
     let pk = keygen_pk(&params, vk, &empty_circuit).expect("keygen_pk should not fail");
 
     let mut pubinputs = pk.get_vk().get_domain().empty_lagrange();
diff --git a/src/plonk/circuit.rs b/src/plonk/circuit.rs
index 3360c05..eda0f1a 100644
--- a/src/plonk/circuit.rs
+++ b/src/plonk/circuit.rs
@@ -1,6 +1,6 @@
 use core::cmp::max;
 use core::ops::{Add, Mul};
-use ff::{Field, PrimeField};
+use ff::Field;
 use std::{
     convert::TryFrom,
     ops::{Neg, Sub},
@@ -380,33 +380,27 @@ pub struct ConstraintSystem<F> {
     pub(crate) lookups: Vec<lookup::Argument>,
 }
 
-impl<F: PrimeField> std::fmt::Display for ConstraintSystem<F> {
+/// Represents the minimal parameters that determine a `ConstraintSystem`.
+#[derive(Debug)]
+pub struct PinnedConstraintSystem<'a, F: Field> {
+    num_fixed_columns: &'a usize,
+    num_advice_columns: &'a usize,
+    num_instance_columns: &'a usize,
+    gates: PinnedGates<'a, F>,
+    advice_queries: &'a Vec<(Column<Advice>, Rotation)>,
+    instance_queries: &'a Vec<(Column<Instance>, Rotation)>,
+    fixed_queries: &'a Vec<(Column<Fixed>, Rotation)>,
+    permutations: &'a Vec<permutation::Argument>,
+    lookups: &'a Vec<lookup::Argument>,
+}
+
+struct PinnedGates<'a, F: Field>(&'a Vec<(&'static str, Expression<F>)>);
+
+impl<'a, F: Field> std::fmt::Debug for PinnedGates<'a, F> {
     fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> {
-        write!(
-            f,
-            "ConstraintSystem {{ \
-             num_fixed_columns: {num_fixed_columns}, \
-             num_advice_columns: {num_advice_columns}, \
-             num_instance_columns: {num_instance_columns}, \
-             fixed_queries: {fixed_queries:?}, \
-             advice_queries: {advice_queries:?}, \
-             instance_queries: {instance_queries:?}, \
-             permutations: {permutations:?}, \
-             lookups: {lookups:?}, \
-             gates: [",
-            num_fixed_columns = self.num_fixed_columns,
-            num_advice_columns = self.num_advice_columns,
-            num_instance_columns = self.num_instance_columns,
-            fixed_queries = &self.fixed_queries,
-            advice_queries = &self.advice_queries,
-            instance_queries = &self.instance_queries,
-            lookups = &self.lookups,
-            permutations = &self.permutations
-        )?;
-        for (_, expr) in self.gates.iter() {
-            write!(f, "{:?}, ", expr)?;
-        }
-        write!(f, "]}}")
+        f.debug_list()
+            .entries(self.0.iter().map(|&(_, ref expr)| expr))
+            .finish()
     }
 }
 
@@ -427,7 +421,24 @@ impl<F: Field> Default for ConstraintSystem<F> {
 }
 
 impl<F: Field> ConstraintSystem<F> {
-    /// Add a permutation argument for some columns
+    /// Obtain a pinned version of this constraint system; a structure with the
+    /// minimal parameters needed to determine the rest of the constraint
+    /// system.
+    pub fn pinned(&self) -> PinnedConstraintSystem<'_, F> {
+        PinnedConstraintSystem {
+            num_fixed_columns: &self.num_fixed_columns,
+            num_advice_columns: &self.num_advice_columns,
+            num_instance_columns: &self.num_instance_columns,
+            gates: PinnedGates(&self.gates),
+            fixed_queries: &self.fixed_queries,
+            advice_queries: &self.advice_queries,
+            instance_queries: &self.instance_queries,
+            permutations: &self.permutations,
+            lookups: &self.lookups,
+        }
+    }
+
+    /// Add a permutation argument for some advice columns
     pub fn permutation(&mut self, columns: &[Column<Any>]) -> usize {
         let index = self.permutations.len();
 
diff --git a/src/poly/domain.rs b/src/poly/domain.rs
index d4c8498..c6bd2ce 100644
--- a/src/poly/domain.rs
+++ b/src/poly/domain.rs
@@ -376,20 +376,23 @@ impl<G: Group> EvaluationDomain<G> {
     pub fn get_quotient_poly_degree(&self) -> usize {
         self.quotient_poly_degree as usize
     }
-}
 
-impl<G: Group> std::fmt::Display for EvaluationDomain<G> {
-    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> {
-        write!(
-            f,
-            "Domain {{ \
-            k = {k:?}, \
-            extended_k = {extended_k:?}, \
-            omega = {omega:?} \
-        }}",
-            k = self.k,
-            extended_k = self.extended_k,
-            omega = self.omega
-        )
+    /// Obtain a pinned version of this evaluation domain; a structure with the
+    /// minimal parameters needed to determine the rest of the evaluation
+    /// domain.
+    pub fn pinned(&self) -> PinnedEvaluationDomain<'_, G> {
+        PinnedEvaluationDomain {
+            k: &self.k,
+            extended_k: &self.extended_k,
+            omega: &self.omega,
+        }
     }
 }
+
+/// Represents the minimal parameters that determine an `EvaluationDomain`.
+#[derive(Debug)]
+pub struct PinnedEvaluationDomain<'a, G: Group> {
+    k: &'a u32,
+    extended_k: &'a u32,
+    omega: &'a G::Scalar,
+}