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Agreement parameter alpha+ #2

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206 changes: 60 additions & 146 deletions scripts/FRI_soundness_list_decoding.sage
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Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
from math import log2


def commit_soundness_error_list(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0):
def commit_soundness_error_list(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m):
# Returns the log2 soundness error of the FRI commit phase

# Parameters:
Expand All @@ -11,20 +11,19 @@ def commit_soundness_error_list(field_bits, lde_domain_bits, trace_domain_bits,
# fold_arities: A list of the log2 of the folding factors used in the FRI protocol.
# num_polys: The degree of the batching plus 1 e.g. for linear batching it is equal to 2.
# m: The proximity paramater to the Johnson bound.
# quotient_degree: The maximal degree of the denominators appearing in the evaluation quotients.

if m < 3:
print("Proximity parameter m cannot be less than 3")
return

trace_len = 2**trace_domain_bits
domain = 2**lde_domain_bits
rho_plus = (trace_len + quotient_degree)/domain
rho = trace_len / domain

return log2(((num_polys - 0.5) * (domain**2) * (m + 0.5)**7 * rho_plus**(-3/2) / 3) + (2*m + 1) * (domain + 1) * rho_plus**(-1/2) * sum([2**a for a in fold_arities])) - field_bits
return log2(((num_polys - 0.5) * (domain**2) * (m + 0.5)**7 * rho**(-3/2) / 3) + (2*m + 1) * (domain + 1) * rho**(-1/2) * sum([2**a for a in fold_arities])) - field_bits


def min_achievable_soundness_error(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree=0):
def min_achievable_soundness_error(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys):
# Returns the minimal achievable log2 soundness error of the FRI commit phase

# Parameters:
Expand All @@ -33,10 +32,9 @@ def min_achievable_soundness_error(field_bits, lde_domain_bits, trace_domain_bit
# trace_domain_bits: The log2 of the trace domain.
# fold_arities: A list of the log2 of the folding factors used in the FRI protocol.
# num_polys: The degree of the batching plus 1 e.g. for linear batching it is equal to 2.
# quotient_degree: The maximal degree of the denominators appearing in the evaluation quotients.

m = 3
return commit_soundness_error_list(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree) + 1
return commit_soundness_error_list(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m) + 1


def query_soundness_error_list(lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=0):
Expand Down Expand Up @@ -82,7 +80,7 @@ def num_samples_list(security_bits, lde_domain_bits, trace_domain_bits, m, quoti
return num_queries


def proximity_parameter(security_bits, field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m_max=10**7, quotient_degree=0):
def proximity_parameter(security_bits, field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m_max=10**7):
# Returns the largest proximity parameter that is sufficient to reach a certain security number of bits.

# Parameters:
Expand All @@ -93,22 +91,70 @@ def proximity_parameter(security_bits, field_bits, lde_domain_bits, trace_domain
# fold_arities: A list of the log2 of the folding factors used in the FRI protocol.
# num_polys: The degree of the batching plus 1 e.g. for linear batching it is equal to 2.
# m_max: An upper bound on the size of the proximity parameter.
# quotient_degree: The maximal degree of the denominators appearing in the evaluation quotients.

m = 3
current_security_bits = - commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree)
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m)

if current_security_bits < security_bits:
print("Target security bits is not reachable from the current parameters")
return -1
while current_security_bits > security_bits and m < m_max:
m += 1
current_security_bits = - commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree)
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m)
return m-1



# Helper function for making examples
def example(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree, target_security):
print("Field number of bits", field_bits)
print("LDE domain size 2^", lde_domain_bits)
print("Trace domain size 2^", trace_domain_bits)
print("Fold arities ", fold_arities)
print("Number of polynomials in the batch ", num_polys)
print("Correction factor of rho+ ", quotient_degree)

min_error = min_achievable_soundness_error(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys)
print("Smallest achievable soundness error 2^", min_error)
if target_security > -min_error:
m = 3
print("Target security is unachievable, the maximum achievable is", -min_error)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits,
trace_domain_bits, m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
else:
m = proximity_parameter(target_security + 1, field_bits, lde_domain_bits,
trace_domain_bits, fold_arities, num_polys, m_max=10**7)
print("Target security is achievable with proximity parameter m", m)

num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")


print("==============================================================")
print("==========================Example 1===========================")
print("==============================================================")
Expand All @@ -118,53 +164,9 @@ lde_domain_bits = 15+3
trace_domain_bits = 15
fold_arities = [5, 4, 1]
num_polys = 2
m_max = 10**7
quotient_degree = 0

print("Field number of bits", field_bits)
print("LDE domain size 2^", lde_domain_bits)
print("Trace domain size 2^", trace_domain_bits)
print("Fold arities ", fold_arities)
print("Number of polynomials in the batch ", num_polys)
print("Correction factor of rho+ ", quotient_degree)

min_error = min_achievable_soundness_error(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree)
print("Smallest achievable soundness error 2^", min_error)
if target_security > -min_error:
m = 3
print("Target security is unachievable, the maximum achievable is", -min_error)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits,
trace_domain_bits, m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
else:
m = proximity_parameter(target_security + 1, field_bits, lde_domain_bits,
trace_domain_bits, fold_arities, num_polys, m_max=10**7, quotient_degree=0)
print("Target security is achievable with proximity parameter m", m)

num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
example(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree, target_security)


print("==============================================================")
Expand All @@ -178,53 +180,9 @@ lde_domain_bits = 15+3
trace_domain_bits = 15
fold_arities = [5, 4, 1]
num_polys = 2
m_max = 10**7
quotient_degree = 0

print("Field number of bits", field_bits)
print("LDE domain size 2^", lde_domain_bits)
print("Trace domain size 2^", trace_domain_bits)
print("Fold arities ", fold_arities)
print("Number of polynomials in the batch ", num_polys)
print("Correction factor of rho+ ", quotient_degree)

min_error = min_achievable_soundness_error(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree)
print("Smallest achievable soundness error 2^", min_error)
if target_security > -min_error:
m = 3
print("Target security is unachievable, the maximum achievable is", -min_error)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits,
trace_domain_bits, m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
else:
m = proximity_parameter(target_security + 1, field_bits, lde_domain_bits,
trace_domain_bits, fold_arities, num_polys, m_max=10**7, quotient_degree=0)
print("Target security is achievable with proximity parameter m", m)

num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
example(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree, target_security)

print("==============================================================")
print("==========================Example 3===========================")
Expand All @@ -237,50 +195,6 @@ lde_domain_bits = 20+5
trace_domain_bits = 20
fold_arities = [5, 4, 1]
num_polys = 2
m_max = 10**7
quotient_degree = 0

print("Field number of bits", field_bits)
print("LDE domain size 2^", lde_domain_bits)
print("Trace domain size 2^", trace_domain_bits)
print("Fold arities ", fold_arities)
print("Number of polynomials in the batch ", num_polys)
print("Correction factor of rho+ ", quotient_degree)

min_error = min_achievable_soundness_error(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree)
print("Smallest achievable soundness error 2^", min_error)
if target_security > -min_error:
m = 3
print("Target security is unachievable, the maximum achievable is", -min_error)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits,
trace_domain_bits, m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(-min_error + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
else:
m = proximity_parameter(target_security + 1, field_bits, lde_domain_bits,
trace_domain_bits, fold_arities, num_polys, m_max=10**7, quotient_degree=0)
print("Target security is achievable with proximity parameter m", m)

num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=0, max_num_queries=200)
print("The number of queries required to achieve this with 0-bits of grinding", num_queries)
num_queries = num_samples_list(target_security + 1, lde_domain_bits, trace_domain_bits,
m, quotient_degree, grinding_bits=16, max_num_queries=200)
print("The number of queries required to achieve this with 16-bits of grinding", num_queries)
commit_error = commit_soundness_error_list(
field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, m, quotient_degree=0)
print("Commit phase soundness", -commit_error)
query_error = query_soundness_error_list(
lde_domain_bits, trace_domain_bits, num_queries, m, quotient_degree=0, grinding_bits=16)
print("Query phase soundness", -query_error)
print("The total soundness of FRI is",int(min(-query_error, -commit_error)) - 1, "bits")
example(field_bits, lde_domain_bits, trace_domain_bits, fold_arities, num_polys, quotient_degree, target_security)